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p-Laplacian Poisson problem with Robin +boundary conditions +Alba Lia Masiello*, Gloria Paoli +Abstract +Let Ω ⊂ Rn be an open, bounded and Lipschitz set. We consider the Poisson problem for +the p−Laplace operator associated to Ω with Robin boundary conditions. In this setting, we +study the equality case in the Talenti-type comparison stated in [6]. We prove that the equality +is achieved only if Ω is a ball and both the function u and the right hand side f of the Poisson +equation are radial. +Keywords: Robin boundary conditions, p-Laplace operator, rigidity result, Talenti compari- +son. +MSC 2020: 35J92, 35J25, 46E30. +E-mail address, A.L. Masiello (corresponding author): albalia.masiello@unina.it +Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli studi di Napoli +Federico II, Via Cintia, Complesso Universitario Monte S. Angelo, 80126 Napoli, Italy. +E-mail address, G. Paoli: gloria.paoli@fau.de +Department of Data Science (DDS) Chair in Dynamics, Control and Numerics (Alexander von +Humboldt-Professorship), Cauerstr. 11, 91058 Erlangen, Germany. +1 +Introduction +Symmetrization techniques in the context of qualitative properties of solutions to second-order +elliptic boundary value problems are introduced by Talenti in [25]. In this seminal paper, the +author considers an open, bounded and Lipschitz set Ω ⊂ Rn, the ball Ω♯ with the same measure +as Ω and the solutions u and v to the following problems +® +−∆uD = f +in Ω, +uD = 0 +on ∂Ω, +® +−∆vD = f ♯ +in Ω♯, +vD = 0 +on ∂Ω♯, +(1) +where f ∈ L2(Ω) is a positive function and f ♯ is its Schwarz rearrangement of f (see Definition +2.3). In this setting, Talenti proves the following point-wise estimate: +u♯ +D(x) ≤ vD(x), +for all x ∈ Ω♯. +(2) +For sake of completeness, we observe that this result is proved more generally for a uniformly +elliptic linear operator in divergence form. +1 + +1 +INTRODUCTION +2 +A version of this result for nonlinear operators in divergence form is contained in [26], which +includes as a special instance the case of the p-Laplace operator. Moreover, these results are later +extended, for instance, to the anisotropic elliptic operators in [2], to the parabolic case in [4], and +to higher order operators in [9, 28]. +Once a comparison result holds, it is natural to ask whether the equality cases can be character- +ized and, so, if a rigidity result is in force. In [3], the rigidity result linked to problem (1) is proved. +Indeed, the authors prove that if equality holds in (2), then Ω is a ball, u is radially symmetric +and decreasing and f = f ♯. Rigidity results for a generic linear, elliptic second order operator can +be found in [17] and [18]. To the best of our knowledge, rigidity results for nonlinear operators +with Dirichlet boundary conditions are not present in the literature. In this paper, we obtain, as a +corollary of our results, the rigidity for the p−Laplace operator with Dirichlet boundary conditions +in any dimension (see Corollary 3.3). +For a long time, it was believed that comparison results could not be proved by means of +spherical rearrangement argument when dealing with Robin boundary conditions, until the recent +paper [5]. The authors consider the following problems + + + + + +−∆u = f +in Ω, +∂u +∂ν + βu = 0 +on ∂Ω, + + + + + +−∆v = f ♯ +in Ω♯, +∂v +∂ν + βv = 0 +on ∂Ω♯, +and they prove a comparison result involving Lorentz norms of u and v whenever f is a non negative +function in L2(Ω) and β is a positive parameter. In particular, in the case f ≡ 1, they prove +∥u∥Lp(Ω) ≤ ∥v∥Lp(Ω♯), +p = 1, 2, +and, if n = 2, the pointwise comparison holds: +u♯(x) ≤ v(x), +for all x ∈ Ω♯. +(3) +In [21], it is proved that (3) is rigid, i.e. the equality case is possible if and only if Ω is a ball +and u is a radial and decreasing function. +Generalizations of the results contained in [5] can be found for the anisotropic case in [24], for +mixed boundary conditions in [1], in the case of the Hermite operator in [13]. +In the present paper, we focus our study on the rigidity for p−Laplace operator. In this case, +the comparison results are obtained in [6] and the setting is the following. +Let Ω be a bounded, open and Lipschitz set in Rn, with n ≥ 2. Let p ∈ (1, +∞) and let +f ∈ Lp′(Ω) be a positive function, where p′ = p/(p − 1). The Poisson problem for the p−Laplace +operator with Robin boundary conditions is + + + + + +−∆pu := −div(|∇u|p−2∇u) = f +in Ω +|∇u|p−2 ∂u +∂ν + β|u|p−2u = 0 +on ∂Ω, +(4) +where ν is the unit exterior normal to ∂Ω and β > 0. A function u ∈ W 1,p(Ω) is a weak solution +to (4) if +ˆ +Ω +|∇u|p−2∇u∇ϕ dx + β +ˆ +∂Ω +|u|p−2uϕ dHn−1(x) = +ˆ +Ω +fϕ dx, +∀ϕ ∈ W 1,p(Ω). +(5) + +1 +INTRODUCTION +3 +The symmetrized problem associated to (4) is the following + + + + + +−∆pv = f ♯ +in Ω♯ +|∇v|p−2 ∂v +∂ν + β|v|p−2v = 0 +on ∂Ω♯. +(6) +In [6] the authors establish a comparison result between suitable Lorentz norms (see Definition +2.4) of the solutions u and v to problems (4) and (6) respectively. In particular, they prove +∥u∥Lpk,p(Ω) ≤ ∥v∥Lpk,p(Ω♯), +∀ 0 < k ≤ +n(p − 1) +(n − 2)p + n, +(7) +and in the case f ≡ 1, they prove +u♯(x) ≤ v(x), +1 ≤ p ≤ +n +n − 1 +(8) +and +∥u∥Lpk,p(Ω) ≤ ∥v∥Lpk,p(Ω♯), +∀ 0 < k ≤ +n(p − 1) +(n − 2)p + n, +∀p > 1. +(9) +In the present paper, we want to characterize the equality case in (7) and (9), answering to +the open problem contained in [21]. For simplicity, we state the main Theorem only in the case +f ∈ Lp′(Ω) positive, since in the case f ≡ 1 the proof is analogous, as we observe in Remark 4.1. +Theorem 1.1. Let Ω ⊂ Rn be a bounded, open and Lipschitz set and let Ω♯ be the ball centered at +the origin with the same measure as Ω. Let u be the solution to (4) and let v be a solution to (6). +If +∥u∥Lpk,p(Ω) = ∥v∥Lpk,p(Ω♯), +for some k ∈ +ò +0, +n(p − 1) +(n − 2)p + n +ò +(10) +then, there exists x0 ∈ Rn such that +Ω = Ω♯ + x0, +u(· + x0) = v(·), +f(· + x0) = f ♯(·). +The idea of the proof of Theorem 1.1 is the following. First of all, we prove that hypothesis +(10) implies that the superlevel sets of u are balls. The main difficulty is to prove that these balls +are concentric. +Differently from the case of the Laplace operator with Dirichlet boundary conditions studied in +[4, 16], we can’t apply directly the steepest descent method introduced in [8], because it strongly +relays on the continuity of both the solution and of its gradient. In the case of the p−Laplace +equation, the continuity of the solution up to the boundary depends on the regularity of the given +datum f. To overcome this regularity issue we show that u is a solution to a suitable Dirichlet +problem and it satisfies the Pólya-Szegő inequality with equality sign. Then, we can conclude that +u is radially symmetric and decreasing, using the classical result contained in [10]. We make use +of Lemma 3.2, where the rigidity of the Poisson problem for the p-Laplace operator with Dirichlet +boundary condition is proved under the assumption f ∈ Lp′(Ω) and positive. Up to our knowledge, +Corollary 3.3 seems to be new in the literature. +The paper is organized as follows. In Section 2 we recall some definitions about rearrangement +of functions and we state some lemmas that we will need in the proof of the main theorem. Section +3 is dedicated to the proof of the main result and we conclude with a list of open problems. + +2 +NOTATION AND PRELIMINARIES +4 +2 +Notation and Preliminaries +Throughout this article we will denote by |Ω| the Lebesgue measure of an open and bounded +Lipschitz set of Rn, with n ≥ 2, and by P(Ω) the perimeter of Ω. Since we are assuming that +∂Ω is Lipschitz, we have that P(Ω) = Hn−1(∂Ω), where Hn−1 denotes the (n − 1)−dimensional +Hausdorff measure. +We recall the classical isoperimetric inequality and and we refer the reader, for example, to +[22, 11, 12, 27] and to the original paper by De Giorgi [15]. +Theorem 2.1 (Isoperimetric Inequality). Let E ⊂ Rn be a set of finite perimeter. Then, +nω +1 +nn |E| +n−1 +n +≤ P(E), +(11) +where ωn is the measure of the unit ball in Rn. Equality occurs if and only if E is (equivalent to) +a Ball. +For the following theorem, we refer to [7]. +Theorem 2.2 (Coarea formula). Let Ω ⊂ Rn be an open set with Lipschitz boundary. Let f ∈ +W 1,1 +loc (Ω) and let u : Ω → R be a measurable function. Then, +ˆ +Ω +u(x)|∇f(x)|dx = +ˆ +R +dt +ˆ +Ω∩f−1(t) +u(y) dHn−1(y). +(12) +Let us recall some basic notions about rearrangements. For a general overview, see, for instance, +[19]. +Definition 2.1. Let u : Ω → R be a measurable function, the distribution function of u is the +function µ : [0, +∞[ → [0, +∞[ defined as the measure of the superlevel sets of u, i.e. +µ(t) = |{ x ∈ Ω : |u(x)| > t }|. +Definition 2.2. Let u : Ω → R be a measurable function, the decreasing rearrangement of u is +the distribution function of µ. We will denote it by u∗(·). +Remark 2.1. Let us notice that the function µ(·) is decreasing and right continuous and the +function u∗(·) is its generalized inverse. +Definition 2.3. The Schwartz rearrangement of u is the function u♯ whose superlevel sets are +balls with the same measure as the superlevel sets of u. +We have the following relation between u♯ and u∗: +u♯(x) = u∗(ωn|x|n), +where ωn is the measure of the unit ball in Rn, and one can easily check that the functions u, u∗ +e u♯ are equi-distributed, i.e. they have the same distribution function, and it holds +∥u∥Lp(Ω) = ∥u∗∥Lp(0,|Ω|) = ∥u♯∥Lp(Ω♯), +for all p ≥ 1. + +2 +NOTATION AND PRELIMINARIES +5 +We also recall the Hardy-Littlewood inequality, an important propriety of the decreasing rear- +rangement, +ˆ +Ω +|h(x)g(x)| dx ≤ +ˆ |Ω| +0 +h∗(s)g∗(s) ds. +So, by choosing h(·) = χ{|u|>t}, one has +ˆ +|u|>t +|g(x)| dx ≤ +ˆ µ(t) +0 +g∗(s) ds. +We now introduce the Lorentz spaces (see [28] for more details on this topic). +Definition 2.4. Let 0 < p < +∞ and 0 < q ≤ +∞. The Lorentz space Lp,q(Ω) is the space of +those functions such that the quantity: +∥u∥Lp,q = + + + + + + + +p +1 +q +ň ∞ +0 +tqµ(t) +q +p dt +t +ã 1 +q +0 < q < ∞ +sup +t>0 +(tpµ(t)) +q = ∞ +is finite. +Let us observe that for p = q the Lorentz space coincides with the Lp space, as a consequence +of the Cavalieri’s Principle +ˆ +Ω +|u|p = p +ˆ +∞ +0 +tp−1µ(t) dt. +The solutions u to problem (4) and v to problem (6) are both p-superharmonic and, as a conse- +quence of the strong maximum principle and the lower semicontinuity (see [29, 20]), they achieve +their minima on the boundary. If we set +um = min +Ω u, +vm = min +Ω♯ v +the positiveness of β and the Robin boundary conditions leads to um ≥ 0 and vm ≥ 0. Hence, u +and v are strictly positive in the interior of Ω. Moreover, we can observe that +um = min +Ω u ≤ min +Ω♯ v = vm, +(13) +indeed, +vp−1 +m P(Ω♯) = +ˆ +∂Ω♯ v(x)p−1 dHn−1(x) = 1 +β +ˆ +Ω♯ f ♯ dx = 1 +β +ˆ +Ω +f dx += +ˆ +∂Ω +u(x)p−1 dHn−1(x) +≥ up−1 +m P(Ω) ≥ ump−1P(Ω♯). +Moreover, it holds +µ(t) ≤ φ(t) = |Ω|, +∀t ≤ vm. +(14) + +2 +NOTATION AND PRELIMINARIES +6 +Now, for t ≥ 0, we introduce the following notations: +Ut = {x ∈ Ω : u(x) > t} +∂Uint +t += ∂Ut ∩ Ω, +∂Uext +t += ∂Ut ∩ ∂Ω, +µ(t) = |Ut| +and +Vt = +¶ +x ∈ Ω♯ : v(x) > t +© +, +∂V int +t += ∂Vt ∩ Ω, +∂V ext +t += ∂Vt ∩ ∂Ω, +φ(t) = |Vt|. +Because of the invariance of the p−Laplacian and of the Schwarz rearrangement of f by rotation, +the solution v to (6) is radial, so the set Vt are balls. +Now, we recall some technical Lemmas, proved in [6], that we need in what follows. We recall +the proof of Lemma 2.3 for reader’s convenience, while we omit the proof of Lemma 2.4 and Lemma +2.5. +Lemma 2.3. Let u be the solution to (4) and let v be the solution to (6). Then, for almost every +t > 0, we have +γnµ(t)(1− 1 +n) +p +p−1 ≤ +Lj µ(t) +0 +f ∗(s) ds +å +1 +p−1 Ç +−µ′(t) + +1 +β +1 +p−1 +ˆ +∂Uext +t +1 +u dHn−1(x) +å +(15) +and +γnφ(t)(1− 1 +n) +p +p−1 = +Lj φ(t) +0 +f ∗(s) ds +å +1 +p−1 Ç +−φ′(t) + +1 +β +1 +p−1 +ˆ +∂V ext +t +1 +v dHn−1(x) +å +. +(16) +where γn = +Ä +nω1/n +n +ä +p +p−1. +Proof. Let t > 0 and h > 0. In the weak formulation (5), we choose the following test function +ϕ(x) = + + + + + +0 +if u < t +u − t +if t < u < t + h +h +if u > t + h, +(17) +obtaining +ˆ +Ut\Ut+h +|∇u|p dx + βh +ˆ +∂Uext +t+h +up−1 dHn−1(x) + β +ˆ +∂Uext +t +\∂Uext +t+h +up−1(u − t) dHn−1(x) += +ˆ +Ut\Ut+h +f(u − t) dx + h +ˆ +Ut+h +f dx. +(18) +Dividing (18) by h, using coarea formula and letting h go to 0, we have that for a.e. t > 0 +ˆ +∂Ut +g(x) dHn−1 = +ˆ +Ut +f dx, +where +® +|∇u|p−1 +if x ∈ ∂Uint +t +, +βup−1 +if x ∈ ∂Uext +t +. +(19) + +2 +NOTATION AND PRELIMINARIES +7 +Using the isoperimetric inequality, for a.e. t ∈ [0, +∞) we have +nω +1 +nn µ(t) +n−1 +n +≤ P(Ut) = +ˆ +∂Ut +dHn−1 +(20) +≤ +ň +∂Ut +g dHn−1(x) +ã 1 +p Lj +∂Ut +1 +g +1 +p−1 +dHn−1(x) +å1− 1 +p +(21) += +ň +∂Ut +g dHn−1(x) +ã 1 +p Lj +∂Uint +t +1 +|∇u| dHn−1(x) + +1 +β +1 +p−1 +ˆ +∂Uext +t +1 +u dHn−1(x) +å1− 1 +p +(22) +≤ +Lj µ(t) +0 +f ∗(s) ds +å 1 +p Ç +−µ′(t) + +1 +β +1 +p−1 +ˆ +∂Uext +t +1 +u dHn−1(x) +å1− 1 +p +, +(23) +and, so, (15) follows. Finally, we notice that, if v is the solution to (6), then all the inequalities +above are equalities, and, consequently, we have (16). +Lemma 2.4. For all τ ≥ vm, we have +ˆ τ +0 +tp−1 +Lj +∂Uext +t +1 +u(x) dHn−1(x) +å +dt ≤ 1 +pβ +ˆ |Ω| +0 +f ∗(s) ds. +(24) +Moreover, +ˆ τ +0 +tp−1 +ň +∂Vt∩∂Ω♯ +1 +v(x) dHn−1(x) +ã +dt = 1 +pβ +ˆ |Ω| +0 +f ∗(s) ds, +(25) +Lemma 2.5 (Gronwall). Let ξ(τ) be a continuously differentiable function, let q > 1 and let C be +a non negative constant C such that the following differential inequality holds +τξ′(τ) ≤ (q − 1)ξ(τ) + C +∀τ ≥ τ0 > 0. +Then, we have +ξ(τ) ≤ +Å +ξ(τ0) + +C +q − 1 +ã Å τ +τ0 +ãq−1 +− +C +q − 1 +∀τ ≥ τ0, +(26) +and +ξ′(τ) ≤ +Å(q − 1)ξ(τ0) + C +τ0 +ã Å τ +τ0 +ãq−2 +∀τ ≥ τ0. +(27) +The following Lemma is contained in [4]. +Lemma 2.6. Let f, g ∈ L2(Ω) be two positive functions. If +ˆ +Ω +fg dx = +ˆ +Ω♯ f ♯g♯ dx, +(28) +then, for every τ ≥ 0 there exists t ≥ 0 such that we have, up to zero measure set, +{g > τ} = {f > t}. +(29) + +3 +PROOF OF THEOREM ?? +8 +We conclude this preliminary session, recalling the classical results contained in [10] (see The- +orem 1.1 and Lemma 2.3). In particular, the result contained in (iii) of Lemma (2.7) gives the +rigidity of the Pólya-Szegő inequality (see [23]): +ˆ +Rn |∇u♯|p dx ≤ +ˆ +Rn |∇u|p dx, +∀u ∈ W 1,p(Rn). +(30) +Theorem 2.7. Let w ∈ W 1,p(Rn), let σ(t) be its distribution function and let +wM := +® +∥w∥∞ +if w ∈ L∞(Ω) ++∞ +otherwise. +Then, the following are true: +i. For almost all t ∈ (0, wM), +∞ > −σ′(t) ≥ +ˆ +w−1(t) +1 +|∇w|dHn−1 +(31) +ii. σ is absolutely continuous if and only if +��� +¶ +|∇w♯| = 0 +© +∩ +¶ +0 < w♯ < wM +©��� = 0. +(32) +iii. If +ˆ +Rn |∇w|p = +ˆ +Rn |∇w♯|p, +(33) +and (32) holds, then there exist a translate of w♯ which is almost everywhere equal to w. +Remark 2.2. We observe that in [14], it is proved that the condition +|{ |∇w| = 0 } ∩ { 0 < w < wM }| = 0 +(34) +implies (32). So, by (iii) in Lemma 2.7, if we have (33) and (34), there exists a translated of w♯ +which is almost everywhere equal to w. +Remark 2.3. We observe that the Pólya -Szegő inequality (30) and the relative rigidity result +(iii) contained in Lemma 2.7 hold also if we assume w ∈ W 1,p +0 (Ω). Indeed, it is easily proved that +for every w ∈ W 1,p +0 (Ω) one has w♯ ∈ W 1,p +0 (Ω♯). +3 +Proof of Theorem 1.1 +In order to prove the main Theorem 1.1, we divide the proof into the following steps. First of all, +we prove that, under the assumptions of Theorem 1.1, equality holds in (15) and this is the content +of Proposition 3.1. Then, in Proposition 3.4, we prove that equality in (15) implies the fact that Ω +is a ball and u and f are radial functions. In order to prove this last step, we need the key Lemma +3.2. + +3 +PROOF OF THEOREM ?? +9 +Proposition 3.1. Let u be the solution to (4) and let v be the solution to (6). If there exists k +k ∈ +ò +0, +n(p − 1) +(n − 2)p + n +ò +such that +∥u∥Lpk,p(Ω) = ∥v∥Lpk,p(Ω♯), +then equality holds in (15) for almost every t. +Proof. Since we are assuming that ∥u∥Lpk,p(Ω) = ∥v∥Lpk,p(Ω♯), we have that +ˆ +∞ +0 +tp−1µ(t) +1 +k dt = +ˆ +∞ +0 +tp−1φ(t) +1 +k dt. +(35) +Let us multiply (15) by tp−1µ(t)α, where α = 1 +k − +Å +1 − 1 +n +ã +p +p − 1, and let us integrate from 0 to ++∞: +γn +ˆ +∞ +0 +tp−1µ +1 +k (t) dt +≤ +ˆ +∞ +0 +Lj µ(t) +0 +f ∗(s) ds +å +1 +p−1 Ç +−µ′(t) + +1 +β +1 +p−1 +ˆ +∂Uext +t +1 +u dHn−1 +å +tp−1µ(t)α dt +≤ +ˆ +∞ +0 +tp−1µ(t)α +Lj µ(t) +0 +f ∗(s) ds +å +1 +p−1 +(−µ′(t)) dt + |Ω|α +pβ +p +p−1 +Lj |Ω| +0 +f ∗(s) ds +å +p +p−1 +, +(36) +where in the last inequality we have used µ(t) ≤ |Ω| and (24) in Lemma 2.4. As far as v is +concerned, it holds +γn +ˆ +∞ +0 +tp−1φ +1 +k (t) dt += +ˆ +∞ +0 +Lj φ(t) +0 +f ∗(s) ds +å +1 +p−1 Ç +−φ′(t) + +1 +β +1 +p−1 +ˆ +∂V ext +t +1 +u dHn−1 +å +tp−1φ(t)α dt += +ˆ +∞ +0 +tp−1φ(t)α +Lj φ(t) +0 +f ∗(s) ds +å +1 +p−1 +(−φ′(t)) dt + |Ω|α +pβ +p +p−1 +Lj |Ω| +0 +f ∗(s) ds +å +p +p−1 +. +(37) +We observe that the left-hand-side of (36) and the left-hand-side of (37) are equal from (35). So, +it follows +ˆ +∞ +0 +tp−1φ(t)α +Lj φ(t) +0 +f ∗(s) ds +å +1 +p−1 +(−φ′(t)) dt ≤ +ˆ +∞ +0 +tp−1µ(t)α +Lj µ(t) +0 +f ∗(s) ds +å +1 +p−1 +(−µ′(t)) dt. +(38) +Setting F(l) = +ˆ l +0 +ωδ +�ˆ ω +0 +f ∗(s) ds +� +1 +p−1 +dω, and integrating (38) by parts, we get +ˆ ∞ +0 +tp−2F(φ(t)) dt ≤ +ˆ ∞ +0 +tp−2F(µ(t)) dt, +being µ(t) = φ(t) = 0 for t > vM. In [6] (see the proof of Theorem 1.1), it is proved that +ˆ ∞ +0 +tp−2F(µ(t)) dt ≤ +ˆ ∞ +0 +tp−2F(φ(t)) dt. +(39) + +3 +PROOF OF THEOREM ?? +10 +and we recall here the proof for the reader’s convenience. In order to do that, we multiply (15) +by tp−1F(µ(t))µ(t)− (n−1)p +n(p−1) and we integrate between 0 and τ > vm. First, we observe that, by +the hypothesis k ≤ +n(p − 1) +(n − 2)p + n, it follows that the function h(l) = F(l)l− (n−1)p +n(p−1) is non decreasing. +Hence, we obtain +ˆ τ +0 +γntp−1F(µ(t)) dt ≤ +ˆ τ +0 +� +−µ′(t) +� +tp−1µ(t)− (n−1)p +n(p−1) F(µ(t)) +Lj µ(t) +0 +f ∗(s) ds +å +1 +p−1 +dt ++ F(|Ω|)|Ω|− p(n−1) +n(p−1) +pβ +p +p−1 +Lj |Ω| +0 +f ∗(s) ds +å +p +p−1 +. +If we integrate by parts both sides of the last expression and we set +C = F(|Ω|)|Ω|− p(n−1) +n(p−1) +pβ +p +p−1 +Lj |Ω| +0 +f ∗(s) ds +å +p +p−1 +, +we obtain +τ +ˆ τ +0 +γntp−2F(µ(t)) dt + τHµ(τ) ≤ +ˆ τ +0 +ˆ t +0 +rp−2F(µ(r)) drdt + +ˆ τ +0 +Hµ(t) dt + C, +(40) +where +Hµ(τ) = − +ˆ +∞ +τ +tp−2µ(t)− p(n−1) +n(p−1) F(µ(t)) +ň µ(t) +0 +f ∗(s) ds +ã +1 +p−1 +dµ(t). +Setting now +ξ(τ) = +ˆ τ +0 +ˆ t +0 +γnrp−2F(µ(r)) dr + +ˆ t +0 +Hµ(t) dt, +inequality (40) becomes +τξ′(τ) ≤ ξ(τ) + C. +So, Lemma 2.5, with τ0 = vm and q=2, gives +ˆ τ +0 +γntp−2F(µ(t)) dt + Hµ(τ) ≤ +܈ vm +0 +tp−2F(µ(t) dt + Hµ(vm) + C +vm +ê +. +Of course, the inequality holds as equality if we replace µ(t) with φ(t), so we get: +ˆ τ +0 +γntp−2F(µ(t)) dt + Hµ(τ) ≤ +ˆ τ +0 +γnF(φ(t)) dt + Hφ(τ), +keeping in mind that µ(t) ≤ φ(t) = |Ω| for t ≤ vm. Now, letting τ → ∞, one has +ˆ ∞ +0 +tp−2F(µ(t))dt ≤ +ˆ ∞ +0 +tp−2F(φ(t))dt, +since Hµ(τ), Hφ(τ) → 0. + +3 +PROOF OF THEOREM ?? +11 +So, we get equality in (36) and, consequently, in (15) for almost every t, indeed +γn +ˆ +∞ +0 +tp−1µ +1 +k (t) dt +≤ +ˆ +∞ +0 +Lj µ(t) +0 +f ∗(s) ds +å +1 +p−1 Ç +−µ′(t) + +1 +β +1 +p−1 +ˆ +∂Uext +t +1 +u dHn−1 +å +tp−1µ(t)α dt +≤ +ˆ +∞ +0 +tp−2F(µ(t)) dt + |Ω|α +pβ +p +p−1 +Lj |Ω| +0 +f ∗(s) ds +å +p +p−1 += +ˆ +∞ +0 +tp−2F(φ(t)) dt + |Ω|α +pβ +p +p−1 +Lj |Ω| +0 +f ∗(s) ds +å +p +p−1 += +ˆ +∞ +0 +Lj φ(t) +0 +f ∗(s) ds +å +1 +p−1 Ç +−φ′(t) + +1 +β +1 +p−1 +ˆ +∂V ext +t +1 +v dHn−1 +å +tp−1φ(t)α dt += γn +ˆ +∞ +0 +tp−1φ +1 +k (t) dt = γn +ˆ +∞ +0 +tp−1µ +1 +k (t) dt. +In the following Lemma we prove that a solution to a Dirichlet problem, such that its distribu- +tion function satisfies the differential equation (42), is necessarily defined on a ball and it has to +be radial and decreasing. +Lemma 3.2. Let Ω ⊂ Rn be an open, bounded and Lipschitz set. Let f ∈ Lp′(Ω) be a positive +function, let w be a weak solution to +® +−∆pw = f +in Ω +w = 0 +on ∂Ω, +(41) +and let σ be the distribution function of w. If σ satisfies the following condition +γnσ(t)(1− 1 +n) +p +p−1 = +Lj σ(t) +0 +f ∗(s) ds +å +1 +p−1 � +−σ′(t) +� +, +for a.e. t ∈ [0, wM] +(42) +then, there exists x0 such that +Ω = Ω♯ + x0, +w(· + x0) = w♯(·), +f(· + x0) = f ♯(·). +Proof. First of all, we recall that w is a weak solution to (41) if and only if +ˆ +Ω +|∇w|p−2∇w∇ϕ dx = +ˆ +Ω +fϕ dx, +∀ϕ ∈ W 1,p +0 (Ω). +(43) +Arguing as in the proof of (15) in Lemma 2.3, choosing the same test function ϕ, defined in (17), +ϕ(x) = + + + + + +0 +if w < t +w − t +if t < w < t + h +h +if w > t + h, + +3 +PROOF OF THEOREM ?? +12 +one obtains +ˆ +∂Wt +|∇w|p−1 dHn−1 = +ˆ +Wt +f(x) dx ≤ +ˆ σ(t) +0 +f ⋆(s) ds, +(44) +where Wt = {x ∈ Ω : w(x) > t}. +If we apply the isoperimetric inequality to the superlevel set Wt, the Hölder inequality and the +Hardy-Littlewood inequality, we get, for almost every t, +nω +1 +nn σ(t) +n−1 +n +≤ P(Wt) = +ˆ +∂Wt +dHn−1 +(45) +≤ +ň +∂Wt +|∇w|p−1 dHn−1(x) +ã 1 +p ň +∂Wt +1 +|∇w| dHn−1(x) +ã1− 1 +p +(46) +≤ +Lj σ(t) +0 +f ∗(s) ds +å 1 +p � +−σ′(t) +�1− 1 +p . +(47) +So, hypothesis (42) ensures us that equality holds in the isoperimetric inequality (45), in the Hölder +inequality (46) and in the Hardy-Littlewood inequality (47). +We now divide the proof into three steps. +Step 1. Let us prove that the superlevel set { w > t } is a ball for all t ∈ [0, wM). Equality in (45) +implies that, for almost every t, Wt is a ball. On the other hand, for all t ∈ [0, wM), there exists a +sequence { tk } such that +1. tk → t; +2. tk > tk+1; +3. {w > tk} is a ball for all k. +Since { w > t } = ∪k { w > tk } can be written as an increasing union of balls, {w > t} is a ball for +all t and, in particular, Ω = {w > 0} is a ball too and we obtain that Ω = x0 + Ω♯. +From now on, we can assume without loss of generality that x0 = 0. +Step 2. Let us prove that the superlevel sets are concentric balls. +Equality in (46) implies also equality in Hölder inequality, i.e. +ˆ +∂Wt +dHn−1 = +ň +∂Wt +|∇w|p−1 dHn−1(x) +ã 1 +p ň +∂Wt +1 +|∇w| dHn−1(x) +ã1− 1 +p +. +This means that, for almost every t, |∇w| is constant Hn−1−almost everywhere on ∂Wt , and we +denote by Ct the (Hn−1−a.e.) constant value of |∇w| on ∂Wt. We claim that Ct ̸= 0 for almost +every t. Indeed, (44) and the positivity of f ensure us that +P(Wt)Cp−1 +t += +ˆ +∂Wt +|∇w|p−1 dHn−1 = +ˆ +Wt +f(x) dx > 0. +Integrating (42), we obtain w♯(x) = z(x), for all x ∈ Ω♯, where z is the solution to +® +−∆pz = f ♯ +in Ω♯ +z = 0 +on ∂Ω♯, +(48) + +3 +PROOF OF THEOREM ?? +13 +and it has the following explicit form: +z(x) = +ˆ |Ω| +ωn|x|n +1 +γn +ň s +0 +f ⋆(r) dr +ã1/(p−1) +1 +s(1−1/n)(p/(p−1)) ds, +so it easily follows that +��� +¶ +|∇w♯| = 0 +© +∩ +¶ +0 < w♯ < wM +©��� = 0. +(49) +Using (ii) in Lemma 2.7, we have that (49) implies the absolutely continuity of σ. +Now, we denote by C♯ +t the (Hn−1−a.e.) constant value of +��∇w♯�� on ∂W ♯ +t . We recall that it +holds +−σ′(t) = +ˆ +∂W ♯ +t +1 +|∇w♯| = P(∂W ♯ +t ) +C♯ +t +. +and, by the absolutely continuity of σ, we have +−σ′(t) = +ˆ +∂Wt +1 +|∇w| = P(∂Wt) +Ct +. +Since w and w♯ are equi-distributed, we have, +P(∂Wt) +Ct += P(∂W ♯ +t ) +C♯ +t +Moreover, since P(∂Wt) = P(∂W ♯ +t ), we have that Ct = C♯ +t. So, by the coarea formula, we get +ˆ +Ω +|∇w|p dx = +ˆ +∞ +0 +ˆ +∂Wt +|∇w|p−1 dHn−1 = +ˆ +∞ +0 +Cp−1 +t +P(Wt) dt dHn−1 += +ˆ +∞ +0 +Ä +C♯ +t +äp−1 P(Wt) dt dHn−1 = +ˆ +∞ +0 +ˆ +∂W ♯ +t +|∇w♯|p−1 dHn−1 = +ˆ +Ω♯|∇w♯|p dx. +By (iii) in Lemma 2.7, we conclude that u = u♯. +Step 3. Let us prove that f is radial and decreasing. +Equality in (47) reads, for almost every t, +ˆ +Wt +f(x) dx = +ˆ σ(t) +0 +f ∗(s) ds. +Moreover, for all τ ∈ [0, wM), there exists a sequence { τk } such that +1. τk → τ; +2. τk > τk+1; +3. +ˆ +Wτk +f(x) dx = +ˆ σ(τk) +0 +f ∗(s) ds, + +3 +PROOF OF THEOREM ?? +14 +and, by the continuity of σ(·), we have +ˆ σ(τ) +0 +f ∗(s) ds = lim +k +ˆ σ(τk) +0 +f ∗(s) = lim +k +ˆ +Wτk +f(x) dx = +ˆ +Wτ +f(x) dx. +By Lemma 2.6, we have that for all τ, there exists ατ such that +{w > τ} = {f > ατ}. +Consequently, we have that also f is radial and decreasing, so f = f ♯. +As a direct consequence of Lemma 3.2, we obtain the rigidity for the p−Laplace operator with +Dirichlet boundary conditions. +Corollary 3.3. Let Ω ⊂ Rn be an open, bounded and Lipschitz set. Let f ∈ Lp′(Ω) be a positive +function and let w and z be weak solutions respectively to +® +−∆pw = f +in Ω +w = 0 +on ∂Ω, +® +−∆pz = f ♯ +in Ω♯ +z = 0 +on ∂Ω♯. +(50) +If w♯(x) = z(x), for all x ∈ Ω♯, then there exists x0 ∈ Rn such that +Ω = Ω♯ + x0, +w(· + x0) = z(·), +f(· + x0) = f ♯(·). +Proof. From the proof of Lemma 3.2, it follows that the distribution function of w, denoted by σ, +satisfies +nω +1 +nn σ(t) +n−1 +n +≤ +Lj σ(t) +0 +f ∗(s) ds +å 1 +p �−σ′(t)�1− 1 +p . +(51) +Now, we integrate (51) from 0 to t, obtaining +u∗(t) = +ˆ |Ω| +σ(t) +1 +γn +ň s +0 +f ⋆(r) dr +ã1/(p−1) +1 +s(1−1/n)(p/(p−1)) ds = z∗(t). +So, if w♯ = z, we have w∗ = z∗, and consequently we obtain equality in (51) for almost every +t ∈ [0, wM]. We can conclude by applying Lemma 3.2. +Now, using Lemma 3.2, we are in position to conclude the proof of the main Theorem. +Proposition 3.4. Let Ω ⊂ Rn be an open, bounded and Lipschitz set and let Ω♯ be the ball with the +same measure as Ω. Let u be the solution to (4) and let µ be its distribution function. If equality +holds in (15), then there exists x0 ∈ Rn such that +Ω = Ω♯ + x0, +u(· + x0) = v(·), +f(· + x0) = f ♯(·). + +3 +PROOF OF THEOREM ?? +15 +Proof. Firstly, we claim that the superlevel sets { u > t } are balls for every t ∈ [0, uM). Equality +in (15) implies the equality in (20), i.e. +nω +1 +nn µ(t) +n−1 +n += P(Ut), +for a. e. t ∈ [0, uM] +that means that almost every superlevel set is a ball. Arguing as in Step 1 of Lemma 3.2, we +can conclude that every superlevel set is a ball, so, Ω = {u > um} is a ball and we obtain that +Ω = x0 + Ω♯. +Let us observe that for every t, s ∈ [um, uM] with t < s, as both Ut and Us are balls, we have +that ∂Ut ∩ ∂Us contains at most one point. In particular, the function w = u − um is a weak +solution to the Dirichlet problem (41) in Ω. +We claim that σ(t) = |{ w > t }| satisfies (42). +Since { w > t } = { u > t + um }, we have +σ(t) = µ(t + um) for all t ∈ [0, uM − um]. Moreover, we have +ˆ +∂Ut +1 +u dHn−1 = 0, +∀t > um +So, using the fact that we have equality in (15) by hypothesis, we get +γnσ(t)(1− 1 +n) +p +p−1 = γnµ(t + um)(1− 1 +n) +p +p−1 += +Lj µ(t+um) +0 +f ∗(s) ds +å +1 +p−1 Ç +−µ′(t + um) + +1 +β +1 +p−1 +ˆ +∂Uext +t+um +1 +u dHn−1(x) +å += +Lj σ(t) +0 +f ∗(s) ds +å +1 +p−1 �−σ′(t)� , +for all t ∈ (0, uM − um). So, we can conclude by Lemma 3.2. +We conclude now with the proof of the main Theorem. +Proof of Theorem 1.1. From Proposition 3.1, we have that the hypothesis of Theorem 1.1 +∥u∥Lpk,p(Ω) = ∥v∥Lpk,p(Ω♯), +for some k ∈ +ò +0, +n(p − 1) +(n − 2)p + n +ò +implies the following equality for almost every t ∈ (0, uM) +γnµ(t)(1− 1 +n) +p +p−1 = +Lj µ(t) +0 +f ∗(s) ds +å +1 +p−1 Ç +−µ′(t) + +1 +β +1 +p−1 +ˆ +∂Uext +t +1 +u dHn−1(x) +å +, +where µ(t) is the distribution function of u. +Now, we are in position to apply Proposition 3.4, and, so, there exists x0 ∈ Rn such that +Ω = Ω♯ + x0, +u(· + x0) = v(·), +f(· + x0) = f ♯(·). + +4 +REMARKS AND OPEN PROBLEMS +16 +4 +Remarks and open problems +Remark 4.1. In [6] the authors also prove that in the case f ≡ 1, it holds +∥u∥Lpk,p(Ω) ≤ ∥v∥Lpk,p(Ω♯), +if 0 < k ≤ +n(p − 1) +n(p − 1) − p. +(52) +We stress that the proof of Theorem 1.1 can be adapted to case f ≡ 1, regardless of the fact that +now the admissible k varies in a wider range. +Open problem 4.2. Below we present a list of open problems and work in progress. +• Generalize the rigidity results in the anisotropic setting, starting from the comparison proved +in [24]. +• Generalize the rigidity results to other problems, such as the ones investigated in [1], [13]. +Acknowledgements +The authors Alba Lia Masiello and Gloria Paoli are supported by GNAMPA of INdAM. The +author Gloria Paoli is supported by the Alexander von Humboldt Foundation with an Alexander +von Humboldt research fellowship. +References +[1] A. Alvino, F. Chiacchio, C. Nitsch, and C. Trombetti. Sharp estimates for solutions to elliptic +problems with mixed boundary conditions. J. Math. Pures Appl., 152:251—261, 2021. +[2] A. Alvino, V. Ferone, G. Trombetti, and P.-L. Lions. Convex symmetrization and applications. +Ann. Inst. H. Poincaré C Anal. Non Linéaire, 14(2):275–293, 1997. +[3] A. Alvino, P.-L. Lions, and G. Trombetti. A remark on comparison results via symmetrization. +Proc. Roy. Soc. Edinburgh Sect. A, 102(1-2):37–48, 1986. +[4] A. Alvino, P.-L. Lions, and G. Trombetti. +Comparison results for elliptic and parabolic +equations via Schwarz symmetrization. Ann. Inst. H. Poincaré Anal. Non Linéaire, 7(2):37– +65, 1990. +[5] A. Alvino, C. Nitsch, and C. Trombetti. A Talenti comparison result for solutions to elliptic +problems with Robin boundary conditions. to appear on Comm. Pure Appl. Math. +[6] V. Amato, A. Gentile, and A. L. Masiello. +Comparison results for solutions to p-Laplace +equations with Robin boundary conditions. +Ann. Mat. Pura Appl. (4), 201(3):1189–1212, +2022. +[7] L. Ambrosio, N. Fusco, and D. Pallara. Functions of bounded variation and free discontinuity +problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, +New York, 2000. + +REFERENCES +17 +[8] G. Aronsson and G. Talenti. Estimating the integral of a function in terms of a distribution +function of its gradient. Boll. Un. Mat. Ital. B (5), 18(3):885–894, 1981. +[9] M. S. Ashbaugh and R. D. Benguria. On Rayleigh’s conjecture for the clamped plate and +its generalization to three dimensions. +In Differential equations and mathematical physics +(Birmingham, AL, 1994), pages 17–27. Int. Press, Boston, MA, 1995. +[10] J. E. Brothers and W. P. Ziemer. Minimal rearrangements of Sobolev functions. J. Reine +Angew. Math., 384:153–179, 1988. +[11] Y. D. Burago and V. A. Zalgaller. +Geometric inequalities, volume 285 of Grundlehren +der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. +Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinski˘ı, Springer Series +in Soviet Mathematics. +[12] I. Chavel. Isoperimetric inequalities, volume 145 of Cambridge Tracts in Mathematics. Cam- +bridge University Press, Cambridge, 2001. Differential geometric and analytic perspectives. +[13] F. Chiacchio, N. Gavitone, C. Nitsch, and C. Trombetti. Sharp estimates for the gaussian +torsional rigidity with Robin boundary conditions. Potential Analysis, pages 1–10, 2022. +[14] A. Cianchi and N. Fusco. Steiner symmetric extremals in Pólya-Szegö type inequalities. Adv. +Math., 203(2):673–728, 2006. +[15] E. De Giorgi. Sulla proprietà isoperimetrica dell’ipersfera, nella classe degli insiemi aventi +frontiera orientata di misura finita. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. +Sez. Ia (8), 5:33–44, 1958. +[16] A. Ferone and R. Volpicelli. Minimal rearrangements of Sobolev functions: a new proof. Ann. +Inst. H. Poincaré C Anal. Non Linéaire, 20(2):333–339, 2003. +[17] V. Ferone and M. R. Posteraro. A remark on a comparison theorem. Comm. Partial Differ- +ential Equations, 16(8-9):1255–1262, 1991. +[18] S. Kesavan. On a comparison theorem via symmetrisation. Proc. Roy. Soc. Edinburgh Sect. +A, 119(1-2):159–167, 1991. +[19] S. Kesavan. Symmetrization & applications, volume 3 of Series in Analysis. World Scientific +Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. +[20] P. Lindqvist. On the definition and properties of p-superharmonic functions. J. Reine Angew. +Math., 365:67–79, 1986. +[21] A. L. Masiello and G. Paoli. A rigidity result for the robin torsion problem. arXiv preprint +arXiv:2209.06706, 2022. +[22] R. Osserman. The isoperimetric inequality. Bull. Amer. Math. Soc., 84(6):1182–1238, 1978. +[23] G. Pólya and G. Szegö. Isoperimetric Inequalities in Mathematical Physics. Annals of Math- +ematics Studies, No. 27. Princeton University Press, Princeton, N. J., 1951. + +REFERENCES +18 +[24] R. Sannipoli. Comparison results for solutions to the anisotropic Laplacian with Robin bound- +ary conditions. Nonlinear Anal., 214:Paper No. 112615, 21, 2022. +[25] G. Talenti. Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), +3(4):697–718, 1976. +[26] G. Talenti. Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces. Ann. +Mat. Pura Appl. (4), 120:160–184, 1979. +[27] G. Talenti. The standard isoperimetric theorem. In Handbook of convex geometry, Vol. A, B, +pages 73–123. North-Holland, Amsterdam, 1993. +[28] G. Talenti. Inequalities in rearrangement invariant function spaces. In Nonlinear analysis, +function spaces and applications, Vol. 5 (Prague, 1994), pages 177–230. Prometheus, Prague, +1994. +[29] J. L. Vázquez. A strong maximum principle for some quasilinear elliptic equations. Appl. +Math. Optim., 12(3):191–202, 1984. + diff --git a/09E2T4oBgHgl3EQfigc1/content/tmp_files/load_file.txt b/09E2T4oBgHgl3EQfigc1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..72b33674bff3674a4c5a5e33a512a296f035f358 --- /dev/null +++ b/09E2T4oBgHgl3EQfigc1/content/tmp_files/load_file.txt @@ -0,0 +1,594 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf,len=593 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='03958v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='AP] 10 Jan 2023 Rigidity results for the p-Laplacian Poisson problem with Robin boundary conditions Alba Lia Masiello*, Gloria Paoli Abstract Let Ω ⊂ Rn be an open, bounded and Lipschitz set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' We consider the Poisson problem for the p−Laplace operator associated to Ω with Robin boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In this setting, we study the equality case in the Talenti-type comparison stated in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' We prove that the equality is achieved only if Ω is a ball and both the function u and the right hand side f of the Poisson equation are radial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Keywords: Robin boundary conditions, p-Laplace operator, rigidity result, Talenti compari- son.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' MSC 2020: 35J92, 35J25, 46E30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' E-mail address, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Masiello (corresponding author): albalia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='masiello@unina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='it Dipartimento di Matematica e Applicazioni “R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Caccioppoli”, Università degli studi di Napoli Federico II, Via Cintia, Complesso Universitario Monte S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Angelo, 80126 Napoli, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' E-mail address, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Paoli: gloria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='paoli@fau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='de Department of Data Science (DDS) Chair in Dynamics, Control and Numerics (Alexander von Humboldt-Professorship), Cauerstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 11, 91058 Erlangen, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 1 Introduction Symmetrization techniques in the context of qualitative properties of solutions to second-order elliptic boundary value problems are introduced by Talenti in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In this seminal paper, the author considers an open, bounded and Lipschitz set Ω ⊂ Rn, the ball Ω♯ with the same measure as Ω and the solutions u and v to the following problems ® −∆uD = f in Ω, uD = 0 on ∂Ω, ® −∆vD = f ♯ in Ω♯, vD = 0 on ∂Ω♯, (1) where f ∈ L2(Ω) is a positive function and f ♯ is its Schwarz rearrangement of f (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In this setting, Talenti proves the following point-wise estimate: u♯ D(x) ≤ vD(x), for all x ∈ Ω♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (2) For sake of completeness, we observe that this result is proved more generally for a uniformly elliptic linear operator in divergence form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 1 1 INTRODUCTION 2 A version of this result for nonlinear operators in divergence form is contained in [26], which includes as a special instance the case of the p-Laplace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Moreover, these results are later extended, for instance, to the anisotropic elliptic operators in [2], to the parabolic case in [4], and to higher order operators in [9, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Once a comparison result holds, it is natural to ask whether the equality cases can be character- ized and, so, if a rigidity result is in force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In [3], the rigidity result linked to problem (1) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Indeed, the authors prove that if equality holds in (2), then Ω is a ball, u is radially symmetric and decreasing and f = f ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Rigidity results for a generic linear, elliptic second order operator can be found in [17] and [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' To the best of our knowledge, rigidity results for nonlinear operators with Dirichlet boundary conditions are not present in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In this paper, we obtain, as a corollary of our results, the rigidity for the p−Laplace operator with Dirichlet boundary conditions in any dimension (see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' For a long time, it was believed that comparison results could not be proved by means of spherical rearrangement argument when dealing with Robin boundary conditions, until the recent paper [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' The authors consider the following problems \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 −∆u = f in Ω, ∂u ∂ν + βu = 0 on ∂Ω, \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 −∆v = f ♯ in Ω♯, ∂v ∂ν + βv = 0 on ∂Ω♯, and they prove a comparison result involving Lorentz norms of u and v whenever f is a non negative function in L2(Ω) and β is a positive parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In particular, in the case f ≡ 1, they prove ∥u∥Lp(Ω) ≤ ∥v∥Lp(Ω♯), p = 1, 2, and, if n = 2, the pointwise comparison holds: u♯(x) ≤ v(x), for all x ∈ Ω♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (3) In [21], it is proved that (3) is rigid, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' the equality case is possible if and only if Ω is a ball and u is a radial and decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Generalizations of the results contained in [5] can be found for the anisotropic case in [24], for mixed boundary conditions in [1], in the case of the Hermite operator in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In the present paper, we focus our study on the rigidity for p−Laplace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In this case, the comparison results are obtained in [6] and the setting is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let Ω be a bounded, open and Lipschitz set in Rn, with n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let p ∈ (1, +∞) and let f ∈ Lp′(Ω) be a positive function, where p′ = p/(p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' The Poisson problem for the p−Laplace operator with Robin boundary conditions is \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 −∆pu := −div(|∇u|p−2∇u) = f in Ω |∇u|p−2 ∂u ∂ν + β|u|p−2u = 0 on ∂Ω, (4) where ν is the unit exterior normal to ∂Ω and β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' A function u ∈ W 1,p(Ω) is a weak solution to (4) if ˆ Ω |∇u|p−2∇u∇ϕ dx + β ˆ ∂Ω |u|p−2uϕ dHn−1(x) = ˆ Ω fϕ dx, ∀ϕ ∈ W 1,p(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (5) 1 INTRODUCTION 3 The symmetrized problem associated to (4) is the following \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 −∆pv = f ♯ in Ω♯ |∇v|p−2 ∂v ∂ν + β|v|p−2v = 0 on ∂Ω♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (6) In [6] the authors establish a comparison result between suitable Lorentz norms (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='4) of the solutions u and v to problems (4) and (6) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In particular, they prove ∥u∥Lpk,p(Ω) ≤ ∥v∥Lpk,p(Ω♯), ∀ 0 < k ≤ n(p − 1) (n − 2)p + n, (7) and in the case f ≡ 1, they prove u♯(x) ≤ v(x), 1 ≤ p ≤ n n − 1 (8) and ∥u∥Lpk,p(Ω) ≤ ∥v∥Lpk,p(Ω♯), ∀ 0 < k ≤ n(p − 1) (n − 2)p + n, ∀p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (9) In the present paper, we want to characterize the equality case in (7) and (9), answering to the open problem contained in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' For simplicity, we state the main Theorem only in the case f ∈ Lp′(Ω) positive, since in the case f ≡ 1 the proof is analogous, as we observe in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let Ω ⊂ Rn be a bounded, open and Lipschitz set and let Ω♯ be the ball centered at the origin with the same measure as Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let u be the solution to (4) and let v be a solution to (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' If ∥u∥Lpk,p(Ω) = ∥v∥Lpk,p(Ω♯), for some k ∈ ò 0, n(p − 1) (n − 2)p + n ò (10) then, there exists x0 ∈ Rn such that Ω = Ω♯ + x0, u(· + x0) = v(·), f(· + x0) = f ♯(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' The idea of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='1 is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' First of all, we prove that hypothesis (10) implies that the superlevel sets of u are balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' The main difficulty is to prove that these balls are concentric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Differently from the case of the Laplace operator with Dirichlet boundary conditions studied in [4, 16], we can’t apply directly the steepest descent method introduced in [8], because it strongly relays on the continuity of both the solution and of its gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In the case of the p−Laplace equation, the continuity of the solution up to the boundary depends on the regularity of the given datum f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' To overcome this regularity issue we show that u is a solution to a suitable Dirichlet problem and it satisfies the Pólya-Szegő inequality with equality sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Then, we can conclude that u is radially symmetric and decreasing, using the classical result contained in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' We make use of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='2, where the rigidity of the Poisson problem for the p-Laplace operator with Dirichlet boundary condition is proved under the assumption f ∈ Lp′(Ω) and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Up to our knowledge, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='3 seems to be new in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In Section 2 we recall some definitions about rearrangement of functions and we state some lemmas that we will need in the proof of the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Section 3 is dedicated to the proof of the main result and we conclude with a list of open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 2 NOTATION AND PRELIMINARIES 4 2 Notation and Preliminaries Throughout this article we will denote by |Ω| the Lebesgue measure of an open and bounded Lipschitz set of Rn, with n ≥ 2, and by P(Ω) the perimeter of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Since we are assuming that ∂Ω is Lipschitz, we have that P(Ω) = Hn−1(∂Ω), where Hn−1 denotes the (n − 1)−dimensional Hausdorff measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' We recall the classical isoperimetric inequality and and we refer the reader, for example, to [22, 11, 12, 27] and to the original paper by De Giorgi [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='1 (Isoperimetric Inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let E ⊂ Rn be a set of finite perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Then, nω 1 nn |E| n−1 n ≤ P(E), (11) where ωn is the measure of the unit ball in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Equality occurs if and only if E is (equivalent to) a Ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' For the following theorem, we refer to [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='2 (Coarea formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let Ω ⊂ Rn be an open set with Lipschitz boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let f ∈ W 1,1 loc (Ω) and let u : Ω → R be a measurable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Then, ˆ Ω u(x)|∇f(x)|dx = ˆ R dt ˆ Ω∩f−1(t) u(y) dHn−1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (12) Let us recall some basic notions about rearrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' For a general overview, see, for instance, [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let u : Ω → R be a measurable function, the distribution function of u is the function µ : [0, +∞[ → [0, +∞[ defined as the measure of the superlevel sets of u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' µ(t) = |{ x ∈ Ω : |u(x)| > t }|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let u : Ω → R be a measurable function, the decreasing rearrangement of u is the distribution function of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' We will denote it by u∗(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let us notice that the function µ(·) is decreasing and right continuous and the function u∗(·) is its generalized inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' The Schwartz rearrangement of u is the function u♯ whose superlevel sets are balls with the same measure as the superlevel sets of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' We have the following relation between u♯ and u∗: u♯(x) = u∗(ωn|x|n), where ωn is the measure of the unit ball in Rn, and one can easily check that the functions u, u∗ e u♯ are equi-distributed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' they have the same distribution function, and it holds ∥u∥Lp(Ω) = ∥u∗∥Lp(0,|Ω|) = ∥u♯∥Lp(Ω♯), for all p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 2 NOTATION AND PRELIMINARIES 5 We also recall the Hardy-Littlewood inequality, an important propriety of the decreasing rear- rangement, ˆ Ω |h(x)g(x)| dx ≤ ˆ |Ω| 0 h∗(s)g∗(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' So, by choosing h(·) = χ{|u|>t}, one has ˆ |u|>t |g(x)| dx ≤ ˆ µ(t) 0 g∗(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' We now introduce the Lorentz spaces (see [28] for more details on this topic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let 0 < p < +∞ and 0 < q ≤ +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' The Lorentz space Lp,q(Ω) is the space of those functions such that the quantity: ∥u∥Lp,q = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 p 1 q ň ∞ 0 tqµ(t) q p dt t ã 1 q 0 < q < ∞ sup t>0 (tpµ(t)) q = ∞ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let us observe that for p = q the Lorentz space coincides with the Lp space, as a consequence of the Cavalieri’s Principle ˆ Ω |u|p = p ˆ +∞ 0 tp−1µ(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' The solutions u to problem (4) and v to problem (6) are both p-superharmonic and, as a conse- quence of the strong maximum principle and the lower semicontinuity (see [29, 20]), they achieve their minima on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' If we set um = min Ω u, vm = min Ω♯ v the positiveness of β and the Robin boundary conditions leads to um ≥ 0 and vm ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Hence, u and v are strictly positive in the interior of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Moreover, we can observe that um = min Ω u ≤ min Ω♯ v = vm, (13) indeed, vp−1 m P(Ω♯) = ˆ ∂Ω♯ v(x)p−1 dHn−1(x) = 1 β ˆ Ω♯ f ♯ dx = 1 β ˆ Ω f dx = ˆ ∂Ω u(x)p−1 dHn−1(x) ≥ up−1 m P(Ω) ≥ ump−1P(Ω♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Moreover, it holds µ(t) ≤ φ(t) = |Ω|, ∀t ≤ vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (14) 2 NOTATION AND PRELIMINARIES 6 Now, for t ≥ 0, we introduce the following notations: Ut = {x ∈ Ω : u(x) > t} ∂Uint t = ∂Ut ∩ Ω, ∂Uext t = ∂Ut ∩ ∂Ω, µ(t) = |Ut| and Vt = ¶ x ∈ Ω♯ : v(x) > t © , ∂V int t = ∂Vt ∩ Ω, ∂V ext t = ∂Vt ∩ ∂Ω, φ(t) = |Vt|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Because of the invariance of the p−Laplacian and of the Schwarz rearrangement of f by rotation, the solution v to (6) is radial, so the set Vt are balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Now, we recall some technical Lemmas, proved in [6], that we need in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' We recall the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='3 for reader’s convenience, while we omit the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='4 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let u be the solution to (4) and let v be the solution to (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Then, for almost every t > 0, we have γnµ(t)(1− 1 n) p p−1 ≤ Lj µ(t) 0 f ∗(s) ds å 1 p−1 Ç −µ′(t) + 1 β 1 p−1 ˆ ∂Uext t 1 u dHn−1(x) å (15) and γnφ(t)(1− 1 n) p p−1 = Lj φ(t) 0 f ∗(s) ds å 1 p−1 Ç −φ′(t) + 1 β 1 p−1 ˆ ∂V ext t 1 v dHn−1(x) å .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (16) where γn = Ä nω1/n n ä p p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let t > 0 and h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In the weak formulation (5), we choose the following test function ϕ(x) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 0 if u < t u − t if t < u < t + h h if u > t + h, (17) obtaining ˆ Ut\\Ut+h |∇u|p dx + βh ˆ ∂Uext t+h up−1 dHn−1(x) + β ˆ ∂Uext t \\∂Uext t+h up−1(u − t) dHn−1(x) = ˆ Ut\\Ut+h f(u − t) dx + h ˆ Ut+h f dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (18) Dividing (18) by h, using coarea formula and letting h go to 0, we have that for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' t > 0 ˆ ∂Ut g(x) dHn−1 = ˆ Ut f dx, where ® |∇u|p−1 if x ∈ ∂Uint t , βup−1 if x ∈ ∂Uext t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (19) 2 NOTATION AND PRELIMINARIES 7 Using the isoperimetric inequality, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' t ∈ [0, +∞) we have nω 1 nn µ(t) n−1 n ≤ P(Ut) = ˆ ∂Ut dHn−1 (20) ≤ ň ∂Ut g dHn−1(x) ã 1 p Lj ∂Ut 1 g 1 p−1 dHn−1(x) å1− 1 p (21) = ň ∂Ut g dHn−1(x) ã 1 p Lj ∂Uint t 1 |∇u| dHn−1(x) + 1 β 1 p−1 ˆ ∂Uext t 1 u dHn−1(x) å1− 1 p (22) ≤ Lj µ(t) 0 f ∗(s) ds å 1 p Ç −µ′(t) + 1 β 1 p−1 ˆ ∂Uext t 1 u dHn−1(x) å1− 1 p , (23) and, so, (15) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Finally, we notice that, if v is the solution to (6), then all the inequalities above are equalities, and, consequently, we have (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' For all τ ≥ vm, we have ˆ τ 0 tp−1 Lj ∂Uext t 1 u(x) dHn−1(x) å dt ≤ 1 pβ ˆ |Ω| 0 f ∗(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (24) Moreover, ˆ τ 0 tp−1 ň ∂Vt∩∂Ω♯ 1 v(x) dHn−1(x) ã dt = 1 pβ ˆ |Ω| 0 f ∗(s) ds, (25) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='5 (Gronwall).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let ξ(τ) be a continuously differentiable function, let q > 1 and let C be a non negative constant C such that the following differential inequality holds τξ′(τ) ≤ (q − 1)ξ(τ) + C ∀τ ≥ τ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Then, we have ξ(τ) ≤ Å ξ(τ0) + C q − 1 ã Å τ τ0 ãq−1 − C q − 1 ∀τ ≥ τ0, (26) and ξ′(τ) ≤ Å(q − 1)ξ(τ0) + C τ0 ã Å τ τ0 ãq−2 ∀τ ≥ τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (27) The following Lemma is contained in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let f, g ∈ L2(Ω) be two positive functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' If ˆ Ω fg dx = ˆ Ω♯ f ♯g♯ dx, (28) then, for every τ ≥ 0 there exists t ≥ 0 such that we have, up to zero measure set, {g > τ} = {f > t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (29) 3 PROOF OF THEOREM ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 8 We conclude this preliminary session, recalling the classical results contained in [10] (see The- orem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In particular, the result contained in (iii) of Lemma (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='7) gives the rigidity of the Pólya-Szegő inequality (see [23]): ˆ Rn |∇u♯|p dx ≤ ˆ Rn |∇u|p dx, ∀u ∈ W 1,p(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (30) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let w ∈ W 1,p(Rn), let σ(t) be its distribution function and let wM := ® ∥w∥∞ if w ∈ L∞(Ω) +∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Then, the following are true: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' For almost all t ∈ (0, wM), ∞ > −σ′(t) ≥ ˆ w−1(t) 1 |∇w|dHn−1 (31) ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' σ is absolutely continuous if and only if ��� ¶ |∇w♯| = 0 © ∩ ¶ 0 < w♯ < wM ©��� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (32) iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' If ˆ Rn |∇w|p = ˆ Rn |∇w♯|p, (33) and (32) holds, then there exist a translate of w♯ which is almost everywhere equal to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' We observe that in [14], it is proved that the condition |{ |∇w| = 0 } ∩ { 0 < w < wM }| = 0 (34) implies (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' So, by (iii) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='7, if we have (33) and (34), there exists a translated of w♯ which is almost everywhere equal to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' We observe that the Pólya -Szegő inequality (30) and the relative rigidity result (iii) contained in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='7 hold also if we assume w ∈ W 1,p 0 (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Indeed, it is easily proved that for every w ∈ W 1,p 0 (Ω) one has w♯ ∈ W 1,p 0 (Ω♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 3 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='1 In order to prove the main Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='1, we divide the proof into the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' First of all, we prove that, under the assumptions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='1, equality holds in (15) and this is the content of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Then, in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='4, we prove that equality in (15) implies the fact that Ω is a ball and u and f are radial functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In order to prove this last step, we need the key Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 3 PROOF OF THEOREM ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 9 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let u be the solution to (4) and let v be the solution to (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' If there exists k k ∈ ò 0, n(p − 1) (n − 2)p + n ò such that ∥u∥Lpk,p(Ω) = ∥v∥Lpk,p(Ω♯), then equality holds in (15) for almost every t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Since we are assuming that ∥u∥Lpk,p(Ω) = ∥v∥Lpk,p(Ω♯), we have that ˆ +∞ 0 tp−1µ(t) 1 k dt = ˆ +∞ 0 tp−1φ(t) 1 k dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (35) Let us multiply (15) by tp−1µ(t)α, where α = 1 k − Å 1 − 1 n ã p p − 1, and let us integrate from 0 to +∞: γn ˆ +∞ 0 tp−1µ 1 k (t) dt ≤ ˆ +∞ 0 Lj µ(t) 0 f ∗(s) ds å 1 p−1 Ç −µ′(t) + 1 β 1 p−1 ˆ ∂Uext t 1 u dHn−1 å tp−1µ(t)α dt ≤ ˆ +∞ 0 tp−1µ(t)α Lj µ(t) 0 f ∗(s) ds å 1 p−1 (−µ′(t)) dt + |Ω|α pβ p p−1 Lj |Ω| 0 f ∗(s) ds å p p−1 , (36) where in the last inequality we have used µ(t) ≤ |Ω| and (24) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' As far as v is concerned, it holds γn ˆ +∞ 0 tp−1φ 1 k (t) dt = ˆ +∞ 0 Lj φ(t) 0 f ∗(s) ds å 1 p−1 Ç −φ′(t) + 1 β 1 p−1 ˆ ∂V ext t 1 u dHn−1 å tp−1φ(t)α dt = ˆ +∞ 0 tp−1φ(t)α Lj φ(t) 0 f ∗(s) ds å 1 p−1 (−φ′(t)) dt + |Ω|α pβ p p−1 Lj |Ω| 0 f ∗(s) ds å p p−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (37) We observe that the left-hand-side of (36) and the left-hand-side of (37) are equal from (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' So, it follows ˆ +∞ 0 tp−1φ(t)α Lj φ(t) 0 f ∗(s) ds å 1 p−1 (−φ′(t)) dt ≤ ˆ +∞ 0 tp−1µ(t)α Lj µ(t) 0 f ∗(s) ds å 1 p−1 (−µ′(t)) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (38) Setting F(l) = ˆ l 0 ωδ �ˆ ω 0 f ∗(s) ds � 1 p−1 dω, and integrating (38) by parts, we get ˆ ∞ 0 tp−2F(φ(t)) dt ≤ ˆ ∞ 0 tp−2F(µ(t)) dt, being µ(t) = φ(t) = 0 for t > vM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In [6] (see the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='1), it is proved that ˆ ∞ 0 tp−2F(µ(t)) dt ≤ ˆ ∞ 0 tp−2F(φ(t)) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (39) 3 PROOF OF THEOREM ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 10 and we recall here the proof for the reader’s convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In order to do that, we multiply (15) by tp−1F(µ(t))µ(t)− (n−1)p n(p−1) and we integrate between 0 and τ > vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' First, we observe that, by the hypothesis k ≤ n(p − 1) (n − 2)p + n, it follows that the function h(l) = F(l)l− (n−1)p n(p−1) is non decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Hence, we obtain ˆ τ 0 γntp−1F(µ(t)) dt ≤ ˆ τ 0 � −µ′(t) � tp−1µ(t)− (n−1)p n(p−1) F(µ(t)) Lj µ(t) 0 f ∗(s) ds å 1 p−1 dt + F(|Ω|)|Ω|− p(n−1) n(p−1) pβ p p−1 Lj |Ω| 0 f ∗(s) ds å p p−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' If we integrate by parts both sides of the last expression and we set C = F(|Ω|)|Ω|− p(n−1) n(p−1) pβ p p−1 Lj |Ω| 0 f ∗(s) ds å p p−1 , we obtain τ ˆ τ 0 γntp−2F(µ(t)) dt + τHµ(τ) ≤ ˆ τ 0 ˆ t 0 rp−2F(µ(r)) drdt + ˆ τ 0 Hµ(t) dt + C, (40) where Hµ(τ) = − ˆ +∞ τ tp−2µ(t)− p(n−1) n(p−1) F(µ(t)) ň µ(t) 0 f ∗(s) ds ã 1 p−1 dµ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Setting now ξ(τ) = ˆ τ 0 ˆ t 0 γnrp−2F(µ(r)) dr + ˆ t 0 Hµ(t) dt, inequality (40) becomes τξ′(τ) ≤ ξ(τ) + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' So, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='5, with τ0 = vm and q=2, gives ˆ τ 0 γntp−2F(µ(t)) dt + Hµ(τ) ≤ ܈ vm 0 tp−2F(µ(t) dt + Hµ(vm) + C vm ê .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Of course, the inequality holds as equality if we replace µ(t) with φ(t), so we get: ˆ τ 0 γntp−2F(µ(t)) dt + Hµ(τ) ≤ ˆ τ 0 γnF(φ(t)) dt + Hφ(τ), keeping in mind that µ(t) ≤ φ(t) = |Ω| for t ≤ vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Now, letting τ → ∞, one has ˆ ∞ 0 tp−2F(µ(t))dt ≤ ˆ ∞ 0 tp−2F(φ(t))dt, since Hµ(τ), Hφ(τ) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 3 PROOF OF THEOREM ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 11 So,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' we get equality in (36) and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' consequently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' in (15) for almost every t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' indeed γn ˆ +∞ 0 tp−1µ 1 k (t) dt ≤ ˆ +∞ 0 Lj µ(t) 0 f ∗(s) ds å 1 p−1 Ç −µ′(t) + 1 β 1 p−1 ˆ ∂Uext t 1 u dHn−1 å tp−1µ(t)α dt ≤ ˆ +∞ 0 tp−2F(µ(t)) dt + |Ω|α pβ p p−1 Lj |Ω| 0 f ∗(s) ds å p p−1 = ˆ +∞ 0 tp−2F(φ(t)) dt + |Ω|α pβ p p−1 Lj |Ω| 0 f ∗(s) ds å p p−1 = ˆ +∞ 0 Lj φ(t) 0 f ∗(s) ds å 1 p−1 Ç −φ′(t) + 1 β 1 p−1 ˆ ∂V ext t 1 v dHn−1 å tp−1φ(t)α dt = γn ˆ +∞ 0 tp−1φ 1 k (t) dt = γn ˆ +∞ 0 tp−1µ 1 k (t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In the following Lemma we prove that a solution to a Dirichlet problem, such that its distribu- tion function satisfies the differential equation (42), is necessarily defined on a ball and it has to be radial and decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let Ω ⊂ Rn be an open, bounded and Lipschitz set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let f ∈ Lp′(Ω) be a positive function, let w be a weak solution to ® −∆pw = f in Ω w = 0 on ∂Ω, (41) and let σ be the distribution function of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' If σ satisfies the following condition γnσ(t)(1− 1 n) p p−1 = Lj σ(t) 0 f ∗(s) ds å 1 p−1 � −σ′(t) � , for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' t ∈ [0, wM] (42) then, there exists x0 such that Ω = Ω♯ + x0, w(· + x0) = w♯(·), f(· + x0) = f ♯(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' First of all, we recall that w is a weak solution to (41) if and only if ˆ Ω |∇w|p−2∇w∇ϕ dx = ˆ Ω fϕ dx, ∀ϕ ∈ W 1,p 0 (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (43) Arguing as in the proof of (15) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='3, choosing the same test function ϕ, defined in (17), ϕ(x) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 0 if w < t w − t if t < w < t + h h if w > t + h, 3 PROOF OF THEOREM ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 12 one obtains ˆ ∂Wt |∇w|p−1 dHn−1 = ˆ Wt f(x) dx ≤ ˆ σ(t) 0 f ⋆(s) ds, (44) where Wt = {x ∈ Ω : w(x) > t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' If we apply the isoperimetric inequality to the superlevel set Wt, the Hölder inequality and the Hardy-Littlewood inequality, we get, for almost every t, nω 1 nn σ(t) n−1 n ≤ P(Wt) = ˆ ∂Wt dHn−1 (45) ≤ ň ∂Wt |∇w|p−1 dHn−1(x) ã 1 p ň ∂Wt 1 |∇w| dHn−1(x) ã1− 1 p (46) ≤ Lj σ(t) 0 f ∗(s) ds å 1 p � −σ′(t) �1− 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (47) So, hypothesis (42) ensures us that equality holds in the isoperimetric inequality (45), in the Hölder inequality (46) and in the Hardy-Littlewood inequality (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' We now divide the proof into three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let us prove that the superlevel set { w > t } is a ball for all t ∈ [0, wM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Equality in (45) implies that, for almost every t, Wt is a ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' On the other hand, for all t ∈ [0, wM), there exists a sequence { tk } such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' tk → t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' tk > tk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' {w > tk} is a ball for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Since { w > t } = ∪k { w > tk } can be written as an increasing union of balls, {w > t} is a ball for all t and, in particular, Ω = {w > 0} is a ball too and we obtain that Ω = x0 + Ω♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' From now on, we can assume without loss of generality that x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let us prove that the superlevel sets are concentric balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Equality in (46) implies also equality in Hölder inequality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' ˆ ∂Wt dHn−1 = ň ∂Wt |∇w|p−1 dHn−1(x) ã 1 p ň ∂Wt 1 |∇w| dHn−1(x) ã1− 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' This means that, for almost every t, |∇w| is constant Hn−1−almost everywhere on ∂Wt , and we denote by Ct the (Hn−1−a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=') constant value of |∇w| on ∂Wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' We claim that Ct ̸= 0 for almost every t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Indeed, (44) and the positivity of f ensure us that P(Wt)Cp−1 t = ˆ ∂Wt |∇w|p−1 dHn−1 = ˆ Wt f(x) dx > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Integrating (42), we obtain w♯(x) = z(x), for all x ∈ Ω♯, where z is the solution to ® −∆pz = f ♯ in Ω♯ z = 0 on ∂Ω♯, (48) 3 PROOF OF THEOREM ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 13 and it has the following explicit form: z(x) = ˆ |Ω| ωn|x|n 1 γn ň s 0 f ⋆(r) dr ã1/(p−1) 1 s(1−1/n)(p/(p−1)) ds, so it easily follows that ��� ¶ |∇w♯| = 0 © ∩ ¶ 0 < w♯ < wM ©��� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (49) Using (ii) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='7, we have that (49) implies the absolutely continuity of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Now, we denote by C♯ t the (Hn−1−a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=') constant value of ��∇w♯�� on ∂W ♯ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' We recall that it holds −σ′(t) = ˆ ∂W ♯ t 1 |∇w♯| = P(∂W ♯ t ) C♯ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' and, by the absolutely continuity of σ, we have −σ′(t) = ˆ ∂Wt 1 |∇w| = P(∂Wt) Ct .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Since w and w♯ are equi-distributed, we have, P(∂Wt) Ct = P(∂W ♯ t ) C♯ t Moreover, since P(∂Wt) = P(∂W ♯ t ), we have that Ct = C♯ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' So, by the coarea formula, we get ˆ Ω |∇w|p dx = ˆ +∞ 0 ˆ ∂Wt |∇w|p−1 dHn−1 = ˆ +∞ 0 Cp−1 t P(Wt) dt dHn−1 = ˆ +∞ 0 Ä C♯ t äp−1 P(Wt) dt dHn−1 = ˆ +∞ 0 ˆ ∂W ♯ t |∇w♯|p−1 dHn−1 = ˆ Ω♯|∇w♯|p dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' By (iii) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='7, we conclude that u = u♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let us prove that f is radial and decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Equality in (47) reads, for almost every t, ˆ Wt f(x) dx = ˆ σ(t) 0 f ∗(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Moreover, for all τ ∈ [0, wM), there exists a sequence { τk } such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' τk → τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' τk > τk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' ˆ Wτk f(x) dx = ˆ σ(τk) 0 f ∗(s) ds, 3 PROOF OF THEOREM ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 14 and, by the continuity of σ(·), we have ˆ σ(τ) 0 f ∗(s) ds = lim k ˆ σ(τk) 0 f ∗(s) = lim k ˆ Wτk f(x) dx = ˆ Wτ f(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='6, we have that for all τ, there exists ατ such that {w > τ} = {f > ατ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Consequently, we have that also f is radial and decreasing, so f = f ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' As a direct consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='2, we obtain the rigidity for the p−Laplace operator with Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let Ω ⊂ Rn be an open, bounded and Lipschitz set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let f ∈ Lp′(Ω) be a positive function and let w and z be weak solutions respectively to ® −∆pw = f in Ω w = 0 on ∂Ω, ® −∆pz = f ♯ in Ω♯ z = 0 on ∂Ω♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (50) If w♯(x) = z(x), for all x ∈ Ω♯, then there exists x0 ∈ Rn such that Ω = Ω♯ + x0, w(· + x0) = z(·), f(· + x0) = f ♯(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' From the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='2, it follows that the distribution function of w, denoted by σ, satisfies nω 1 nn σ(t) n−1 n ≤ Lj σ(t) 0 f ∗(s) ds å 1 p �−σ′(t)�1− 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (51) Now, we integrate (51) from 0 to t, obtaining u∗(t) = ˆ |Ω| σ(t) 1 γn ň s 0 f ⋆(r) dr ã1/(p−1) 1 s(1−1/n)(p/(p−1)) ds = z∗(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' So, if w♯ = z, we have w∗ = z∗, and consequently we obtain equality in (51) for almost every t ∈ [0, wM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' We can conclude by applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Now, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='2, we are in position to conclude the proof of the main Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let Ω ⊂ Rn be an open, bounded and Lipschitz set and let Ω♯ be the ball with the same measure as Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let u be the solution to (4) and let µ be its distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' If equality holds in (15), then there exists x0 ∈ Rn such that Ω = Ω♯ + x0, u(· + x0) = v(·), f(· + x0) = f ♯(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 3 PROOF OF THEOREM ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Firstly, we claim that the superlevel sets { u > t } are balls for every t ∈ [0, uM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Equality in (15) implies the equality in (20), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' nω 1 nn µ(t) n−1 n = P(Ut), for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' t ∈ [0, uM] that means that almost every superlevel set is a ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Arguing as in Step 1 of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='2, we can conclude that every superlevel set is a ball, so, Ω = {u > um} is a ball and we obtain that Ω = x0 + Ω♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Let us observe that for every t, s ∈ [um, uM] with t < s, as both Ut and Us are balls, we have that ∂Ut ∩ ∂Us contains at most one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In particular, the function w = u − um is a weak solution to the Dirichlet problem (41) in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' We claim that σ(t) = |{ w > t }| satisfies (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Since { w > t } = { u > t + um }, we have σ(t) = µ(t + um) for all t ∈ [0, uM − um].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Moreover, we have ˆ ∂Ut 1 u dHn−1 = 0, ∀t > um So, using the fact that we have equality in (15) by hypothesis, we get γnσ(t)(1− 1 n) p p−1 = γnµ(t + um)(1− 1 n) p p−1 = Lj µ(t+um) 0 f ∗(s) ds å 1 p−1 Ç −µ′(t + um) + 1 β 1 p−1 ˆ ∂Uext t+um 1 u dHn−1(x) å = Lj σ(t) 0 f ∗(s) ds å 1 p−1 �−σ′(t)� , for all t ∈ (0, uM − um).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' So, we can conclude by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' We conclude now with the proof of the main Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='1, we have that the hypothesis of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='1 ∥u∥Lpk,p(Ω) = ∥v∥Lpk,p(Ω♯), for some k ∈ ò 0, n(p − 1) (n − 2)p + n ò implies the following equality for almost every t ∈ (0, uM) γnµ(t)(1− 1 n) p p−1 = Lj µ(t) 0 f ∗(s) ds å 1 p−1 Ç −µ′(t) + 1 β 1 p−1 ˆ ∂Uext t 1 u dHn−1(x) å , where µ(t) is the distribution function of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Now, we are in position to apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='4, and, so, there exists x0 ∈ Rn such that Ω = Ω♯ + x0, u(· + x0) = v(·), f(· + x0) = f ♯(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' 4 REMARKS AND OPEN PROBLEMS 16 4 Remarks and open problems Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' In [6] the authors also prove that in the case f ≡ 1, it holds ∥u∥Lpk,p(Ω) ≤ ∥v∥Lpk,p(Ω♯), if 0 < k ≤ n(p − 1) n(p − 1) − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' (52) We stress that the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='1 can be adapted to case f ≡ 1, regardless of the fact that now the admissible k varies in a wider range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Open problem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Below we present a list of open problems and work in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Generalize the rigidity results in the anisotropic setting, starting from the comparison proved in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Generalize the rigidity results to other problems, such as the ones investigated in [1], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Acknowledgements The authors Alba Lia Masiello and Gloria Paoli are supported by GNAMPA of INdAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' The author Gloria Paoli is supported by the Alexander von Humboldt Foundation with an Alexander von Humboldt research fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Alvino, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Chiacchio, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Nitsch, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Trombetti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Sharp estimates for solutions to elliptic problems with mixed boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Pures Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=', 152:251—261, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Alvino, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Ferone, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Trombetti, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Lions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Convex symmetrization and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Poincaré C Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Non Linéaire, 14(2):275–293, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Alvino, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Lions, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Trombetti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' A remark on comparison results via symmetrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Edinburgh Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Nitsch, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Trombetti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' to appear on Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfigc1/content/2301.03958v1.pdf'} +page_content=' Math.' metadata={'source': 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Private Graph Neural +Networks with Random Walk Sampling +Morgane Ayle +Technical University of Munich +morgane.ayle@tum.de +Jan Schurchardt +Technical University of Munich +j.schuchardt@tum.de +Lukas Gosch +Technical University of Munich +l.gosch@tum.de +Daniel Zügner +Technical University of Munich +zuegnerd@in.tum.de +Stephan Günnemann +Technical University of Munich +s.guennemann@tum.de +Abstract +Deep learning models are known to put the privacy of their training data at risk, +which poses challenges for their safe and ethical release to the public. Differentially +private stochastic gradient descent is the de facto standard for training neural +networks without leaking sensitive information about the training data. However, +applying it to models for graph-structured data poses a novel challenge: unlike +with i.i.d. data, sensitive information about a node in a graph cannot only leak +through its gradients, but also through the gradients of all nodes within a larger +neighborhood. In practice, this limits privacy-preserving deep learning on graphs +to very shallow graph neural networks. We propose to solve this issue by training +graph neural networks on disjoint subgraphs of a given training graph. We develop +three random-walk-based methods for generating such disjoint subgraphs and +perform a careful analysis of the data-generating distributions to provide strong +privacy guarantees. Through extensive experiments, we show that our method +greatly outperforms the state-of-the-art baseline on three large graphs, and matches +or outperforms it on four smaller ones. +1 +Introduction +The introduction of Graph Neural Networks (GNNs) has enabled the training of Deep Learning (DL) +models on graph-structured data and for various tasks such as node classification, link prediction or +graph classification. However, similar to DL models trained on image [1] or text data [2, 3], GNNs +leak information about their training data [4–6], such as the features of a node, or which nodes are +connected by an edge. +In this paper, we analyze the privacy of GNNs under the lens of Differential Privacy (DP) +[7]. In particular, we ensure the privacy of all nodes’ features in a graph. While DP-SGD [8] is the de +facto standard for training DL models with DP, its transfer to GNNs is not straightforward given the +non-i.i.d. nature of the data. Indeed, since an L-layer GNN typically uses the L-hop neighborhood +of a node during the forward pass, the gradient of a node does not depend on that node alone, but +on all nodes in its neighborhood. While some works [9, 10] have attempted to apply DP to GNNs, +most of them focus on edge-level DP. Methods that can be applied to feature-level DP suffer from +2022 Trustworthy and Socially Responsible Machine Learning (TSRML 2022) co-located with NeurIPS 2022. +arXiv:2301.00738v1 [cs.LG] 2 Jan 2023 + +loose privacy guarantees [9], or rely on custom GNN architectures [10]. We propose an adaptation +of DP-SGD to train GNNs with feature-level DP while attenuating the aforementioned problem +and preserving a high model utility. We experimentally demonstrate that our method can offer +significantly stronger privacy guarantees than prior work, particularly on large graphs. +2 +Background +2.1 +Differential privacy +(ϵ, δ)-DP +Differential Privacy (DP) [7] is a notion of privacy that allows data analysts to extract +useful statistics from a dataset, without leaking too much information about the samples in it. More +formally, given two neighboring datasets D and D′ – denoted D ∼ D′ – that differ by one sample +(either by deleting, adding or modifying a sample), a randomized algorithm M with co-domain Y +is (ϵ, δ)-DP if for all O ⊆ Y , and for all D ∼ D′, Pr[M(D) ∈ O] ≤ exp(ϵ)Pr[M(D′) ∈ O] + δ. +The parameters ϵ and δ are the privacy budget parameters: the smaller their values, the better the +privacy guarantees. +(α, γ)-RDP +An alternative definition of DP is Rényi Differential Privacy (RDP) [11]. A randomized +algorithm M is said to be γ-RDP of order α – or (α, γ)-RDP – if for any D ∼ D′ it holds that +Dα(M(D), M(D′)) ≤ γ, where Dα = +1 +α−1 log Ex∼Q +� +P (x) +Q(x) +�α +is the Rényi divergence of order α +which measures the similarity of the distributions P and Q. Note that if M is (α, γ)-RDP, then it +is also (ϵ, δ)-DP for any 0 < δ < 1 where ϵ = fRDP→DP(α, γ, δ) = γ + log( α−1 +α ) − log δ+log α +α−1 +[12]. +We rely on (α, γ)-RDP during our analysis, but report our results in terms of (ϵ, δ)-DP following +prior work. +The Gaussian mechanism +Given an algorithm A with real-valued output space A : ND → Rd, +the Gaussian mechanism privatizes the algorithm by adding Gaussian noise to the outputs of A, +i.e. M = Gσ (A (D)) = A(D) + N(0, σ2). Given that the ℓ2 sensitivity of A is ∆2A(D) = +maxD∼D′ ∥A(D) − A(D′)∥2, the mechanism satisfies (α, γ(α))-RDP, with γ(α) = α(∆2A)2 +2σ2 +. +Intuitively, this indicates that the larger the sensitivity of the function, the more noise needs to be +added to obtain a small privacy budget, and therefore the worse the final performance will be. A +small sensitivity is therefore desirable. +Amplification by sub-sampling +A useful property of DP (and RDP) is that, given a mechanism S +that samples a sub-set of the dataset D, applying a private mechanism to S(D) leads to better privacy +guarantees than applying it to the entire dataset D. Intuitively, this is due to the fact that subsampling +introduces a non-zero chance of an added or modified sample to not be processed by the randomized +algorithm. Typically, S is assumed to be a Poisson or uniform sampling over the dataset. Poisson +sampling is typically used when the neighboring datasets differ in size, while uniform sampling is +used otherwise. In this paper, we rely on uniform sampling. +2.2 +Differential privacy in deep learning +Differentially Private Stochastic Gradient Descent (DP-SGD) [13, 14, 8] is the foundation of many +works [9, 2, 15] that apply DP to deep learning. It privatizes the weights of a model with respect to +the input dataset at every iteration of training, and then accumulates the privacy budget being spent +over all iterations. One private training iteration consists of batching a set of samples, computing the +gradient on each sample independently, clipping the norm of each gradient vector to a maximum norm +C, calculating the entire gradient by adding calibrated Gaussian noise, and finally performing an +update step. The clipping step is used to bound the sensitivity of the gradients to changes in the input. +Then, assuming that two neighboring datasets D and D′ differ in the features of one sample, the +sensitivity of the total gradient on a batch of i.i.d. samples is bounded by 2C. Through batching (i.e. +sub-sampling the dataset using a sampling mechanism S), amplification by sub-sampling theorems +[16, 17] can be exploited to get better privacy guarantees at every iteration. Finally, assuming each +iteration t is (α, γt)-RDP, the overall training is then (α, �T +t=0 γt)-RDP [11] where T is the total +number of iterations. +2 + +2.3 +Graph neural networks +Definition +In the following, we define a graph as G = {X, A}, where X ∈ RN×d is the feature +matrix in which each row corresponds to one node’s feature vector, and A ∈ {0, 1}N×N is the +adjacency matrix in which Aij is 1 if there exists an edge between nodes i and j and 0 otherwise. +Note that we only consider undirected graphs, therefore A = AT . Graph Neural Networks (GNNs) +are a class of models that learn a mapping f : G → Z ∈ RN×d′, where Z is an updated feature +matrix of G that can be used for various downstream tasks. Each layer of a GNN typically consists of +two steps: 1) in the aggregation step, information about the neighborhood of every node is gathered; +2) in the update step, the feature vector of every node is updated based on its current feature vector +and the aggregated neighborhood information. +The receptive field +The receptive field of a node in a GNN is defined as the region in the input +graph that influences the GNN’s predictions for that specific node. For a GNN with L layers, the +receptive field of a node v is the L-hop neighborhood of v. Thus, for a graph with maximum node +degree K, the largest possible receptive field size of any node v is RF(v) = �L +l=0 Kl = KL+1−1 +K−1 +, +i.e. the receptive field grows exponentially with the number of layers of the GNN. +2.4 +Differential privacy in graph neural networks +Given that graphs contain two types of attributes – node features and edges – multiple levels of DP +[18, 9, 10] can be considered: edge-level DP, where the edges between nodes are private; feature-level +DP, where the features of nodes are private; and node-level DP, where both the features and edges of +nodes are private. In this work, we focus on feature-level DP using DP-SGD. Contrary to traditional +i.i.d. datasets, samples in a graph (i.e. nodes) are not independent: changing the features of one +node affects the gradients of all nodes within the receptive field of the modified node. In fact, the +sensitivity of the total gradient on a graph is bounded by 2 KL+1−1 +K−1 +C (see Appendix A), which +grows exponentially with the number of layers L. Given that the Gaussian mechanism adds noise +proportional to the sensitivity of the total gradient, this can lead to large amounts of noise being +added during training, which in turn leads to poor final model utility. +3 +Related work +In [19], a node-level differentially private GNN is trained by perturbing features and edges locally +before sending them to a global server. This setup is called local DP, and differs from our notion of +DP where a central learner is trusted with the real data. The authors in [15] propose to split the graph +into disjoint sub-graphs using uniform node sampling, then treat each sub-graph as an independent +sample. Note that, contrary to our method which considers privacy at the individual node feature +level, their approach treats the entire graph as a datapoint to privatize, rather than providing privacy +for the individual nodes in the graph. The method in [10] privatizes GNNs at both the node-level +and edge-level. However, their approach only applies to the GNN architecture they propose and +not to arbitrary GNNs, unlike our proposed method. Furthermore, it does not resolve the issue of +exponentially growing sensitivity in transductive learning scenarios. For a survey on DP on graph +data, refer to [20]. Finally, the authors of [9] propose to reduce the sensitivity of a GNN’s gradients +by bounding the maximum degree K of the graph. However, this does not resolve the exponential +growth with the number of layers. Therefore, they still obtain loose privacy guarantees (ϵ = 20). +Since this method is the closest to our setup, we compare our approach to theirs in our experiments. +4 +Methodology +4.1 +Approach +We propose to adapt DP-SGD to the graph domain to ensure that the weights of a GNN are private +with respect to the nodes’ features, while overcoming the problem of requiring exponentially more +noise with a growing network depth. In the following, we define two graphs G and G′ as neighbors if +they share the same structure A and number of nodes N but differ in one row of the feature matrix +X corresponding to the modified node ˜v. We want to train the GNN such that for all G ∼ G′, +3 + +Figure 1: Our general sampling method. Starting with a graph, we generate subgraphs by first +sampling a root node (depicted in red), and then sampling one or more random walks starting from +the root node. Every node appears in exactly one subgraph. Before every iteration, we batch m many +subgraphs, where m = 2 in this case. Root nodes are used as training nodes, while remaining nodes +are used for aggregation in the GNN only. +Dα(M(G), M(G′)) ≤ γ, where M is a randomized algorithm that returns the weights of the GNN. +To adapt DP-SGD to the graph domain, we propose to pre-process the graph into sets of +independent subgraphs that do not affect each others’ gradients, so that the sensitivity of the total +gradient on any batch depends on the gradient of one subgraph only. We summarize our training +procedure in Algorithm 1. More precisely, we pre-process the graph into a set of M disjoint +subgraphs GS = {s1, s2, . . . , sM}, i.e. subgraphs that do not have any nodes in common, using +sampling method S. Each subgraph si consists of two components: 1) one training node vi, and +2) a set of neighbors N (vi) that is used for the aggregation step of the GNN. At training time, for +every iteration t, we create a batch by sampling m subgraphs uniformly at random from the set of +subgraphs GS. We then compute the gradients ∇wtL(vj, N (vj)) on all training nodes and clip +the norm of each to a value C. We compute the total gradient by summing individual gradients and +adding Gaussian noise. Finally, we update the weights. +Due to the disjointness of subgraphs, changing one node’s features – whether it is a train- +ing node or a neighbor – will affect at most one subgraph (i.e. sample) in the batch, which reduces +the upper bound on the sensitivity of the total gradient to 2C. Since we sample subgraphs uniformly +at random, we can leverage the strong amplification by sub-sampling theorem [17], i.e. account for +the possibility of the gradient not being affected if the modified node ˜v is not part of the batch. +We generate these disjoint subgraphs via random walk sampling, which is an effective way +of training GNNs [21]. We choose random walk sampling, since it ensures that nodes form a +connected subgraph of a training node’s neighborhood, while limiting the number of nodes being +sampled from that neighborhood (i.e. from the receptive field). In the following, we propose three +different random-walk-based sampling methods, which we later compare in our experimental results. +Furthermore, we derive for each sampling method a tight upper bound on the probability of sampling +the modified node ˜v in a batch, which is required for applying the amplification by subsampling +theorem in [17]. +4.2 +Sampling methods +Our three sampling methods consist of pre-processing the graph into a set of M disjoint subgraphs +GS = {s1, s2, . . . , sM}, and then generating a batch B ⊆ GS by sampling m subgraphs uniformly +at random. An overview of our general approach is depicted in Figure 1. Given a graph with M +generated disjoint subgraphs, the true probability of sampling node ˜v is P[˜v] = +1 +M , since we know +that a node is in exactly one of the M subgraphs. However, to ensure differential privacy, we require +a bound that holds for all possible graphs and any run of the sampling procedure. Thus, we use the +upper bound P[˜v] = +1 +M ≤ +1 +Mmin where Mmin is the minimum number of subgraphs that can be +generated in any graph of N nodes. Then, the probability of sampling ˜v in a batch of m subgraphs +using sampling mechanism S is at most PS[˜v] ≤ +m +Mmin . +4 + +pre-process +batchAlgorithm 1 DP-SGD with random walk sampling +Input: Graph G = {V, E}, sampling method S, loss function L, initial model weights w0, noise +standard deviation σ, gradient clipping norm C, number of iterations T, frequency at which to +re-sample subgraphs in DRW-D i +GS = S(G) +▷ Generate subgraphs from graph G using sampling method S +for t in [0, T) do +if t % i == 0 and S == DRW-D then +GS = S(G) +end if +Sample m subgraphs uniformly at random from GS to form batch B +for sj in B do +▷ sj is a subgraph +Compute ∇wtL(vj, N (vj)) +gt(vj) = clip (∇wtL (vj, N (vj)) , C) +▷ Compute and clip individual gradients in B +end for +gt(B) = +1 +|B| +��� +sj∈B gt(vj) +� ++ N(0, σ2) +� +▷ Add noise to the gradients +wt+1 = update(wt, gt(B)) +▷ Update weights based on optimizer being used +end for +Disjoint random walks +The first sampling method we propose is called Disjoint Random Walks +(DRW). We pre-process the graph once before training and then generate batches at every iteration +using the same set of subgraphs. Each subgraph consists of one random walk of length L (refer to +Appendix B for a pseudo-code). A random walk of length L contains at most L + 1 nodes, and +generating random walks that all have maximal length would result in the minimum number of +random walks, since a node can only appear in one random walk. Therefore, we get Mmin = ⌈ N +L+1⌉ +and P[˜v] ≤ +1 +⌈ +N +L+1 ⌉. Finally, the upper bound probability of sampling a node ˜v is PDRW[˜v] ≤ +m +⌈ +N +L+1 ⌉. +Disjoint random walks with restarts +To create better subgraphs that contain more nodes for +aggregation, we also propose Disjoint Random Walks with Restarts (DRW-R). Similary to DRW, +this sampling method generates subgraphs once before training by using random walks, but instead +of sampling one random walk per training node we sample R of them (refer to Appendix B for a +pseudo-code). Given a random walk length of L and R restarts, the minimum number of subgraphs +is Mmin = ⌈ +N +1+R×L⌉ where 1 + R × L is the maximum size of one subgraph when all random +walks have length L, and the probability of sampling node u in a batch of size m is therefore +PDRW-R[u] ≤ +m +⌈ +N +1+R×L ⌉. +Disjoint random walks with dynamic re-sampling +Finally, we propose a third sampling method +in which we pre-process the graph into disjoint subgraphs every ith iteration instead of once before +training, where i is a hyper-parameter that is chosen based on the cost of the sampling procedure +on each dataset. This allows us to increase the diversity of subgraphs used for training, and prevent +overfitting on the subgraphs generated in one run of the sampling procedure. We call this procedure +DRW-D, where D stands for Dynamically re-sampling random walks. The probability of sampling +node ˜v is the same as in DRW, namely PDRW-D[˜v] = PDRW[˜v] ≤ +m +⌈ +N +L+1 ⌉. Note that this method +consists simply of re-running the subgraph generation process DRW at every ith iteration instead of +once before training, which is reflected in Algorithm 1. +5 +Experimental results +Experimental setup +We report our results on seven datasets, both in the transductive and the +inductive settings. The dataset sizes in terms of total nodes range from small (Cora [22], Citeseer +[22]) to medium (PPI [21], Pubmed [22]) to large (Flickr [21], Arxiv [21], Reddit [21]), or in number +of training nodes from small (Pubmed, Citeseer, Cora) to medium (PPI) to large (Flickr, Arxiv, +Reddit). We report the exact number of nodes as well as some additional dataset characteristics +in Appendix C. We focus on the node classification task, and report our results in terms of F1 +Micro score, a metric equivalent to accuracy except on PPI which is a multi-label classification task. +Following prior work, we report our privacy budget using ϵ and a fixed δ per dataset (see Appendix +5 + +Table 1: Comparison between the F1 Micro score (%) achieved by a basic GCN and MLP, the FDP +baseline, and our proposed method with multiple sampling methods. All DP methods are trained with +a target budget of ϵ ≤ 8. +Layers +Width +Dataset +Cora +CiteSeer +PPI +PubMed +Flickr +Arxiv +Reddit +GCN (non-DP) +1 +- +69.8 +59.5 +46.2 +68.7 +45.6 +59.7 +92.5 +2 +256 +77.3 +63.7 +58.9 +72.9 +51.3 +69.1 +94.7 +512 +76.6 +62.2 +60.7 +72.9 +51.3 +69.5 +94.7 +MLP (non-DP) +1 +- +43.0 +37.6 +45.2 +61.3 +45.7 +52.3 +67.7 +2 +256 +47.3 +36.1 +52.1 +61.5 +36.2 +52.6 +69.8 +512 +44.8 +39.3 +53.6 +63.3 +38.4 +52.0 +69.7 +FDP (DP) +1 +- +17.1 +17.5 +38.4 +39.6 +33.6 +43.8 +56.7 +2 +256 +17.6 +21.5 +40.7 +41.4 +42.5 +31.9 +43.7 +512 +23.2 +22.1 +40.0 +41.2 +42.4 +30.2 +42.3 +Ours +DRW (DP) +1 +- +19.9 +20.6 +40.2 +41.7 +42.1 +59.2 +81.4 +2 +256 +17.2 +20.9 +38.7 +40.3 +48.7 +59.6 +80.2 +512 +24.9 +21.3 +37.9 +41.1 +47.9 +59.2 +81.8 +DRW-D (DP) +1 +- +19.8 +20.6 +40.1 +41.7 +42.2 +59.2 +81.4 +2 +256 +17.2 +21.3 +38.6 +40.2 +48.5 +59.7 +80.2 +512 +25.0 +21.7 +37.9 +41.2 +47.8 +59.3 +81.5 +DRW-R (DP) +1 +- +18.3 +19.2 +40.0 +40.3 +42.3 +59.1 +82.0 +2 +256 +17.3 +20.7 +38.2 +40.4 +48.3 +59.7 +81.0 +512 +24.5 +21.3 +36.9 +40.4 +48.5 +59.4 +82.2 +C). Given a target ϵ, we keep training while tracking the (α, γt) privacy budget being spent until we +reach ϵ = fRDP→DP(α, �T ′ +t=0 γt, δ) at iteration T ′. +We compare our proposed methodology with each sampling method to three baselines: 1) A basic +GCN trained with random walk sampling; 2) A basic MLP trained with uniform node sampling; and +3) The method proposed in [9] which we call FDP for Feature-level DP. Note that while they train +their models up to an ϵ of 20, we only train them until ϵ = 8, since a very large ϵ does not have much +value in terms of privacy. +Discussion +Table 5 summarizes our results. A GCN trained without DP always outperforms the +ones trained with DP, which is expected since clipping gradients and especially adding Gaussian noise +decreases the utility of the final model. However, in some cases our method can almost match the +utility of the basic GCN, whereas the FDP baseline struggles. For example, DRW sampling on Flickr +can reach up to 48.7% accuracy – which corresponds to 95% of the baseline GCN’s performance +– whereas FDP reaches only 42.5% accuracy – which corresponds to 83% of the baseline GCN’s +performance. Similarly, our method achieves 87% of the GCN’s performance on the challenging +dataset Reddit, while FDP can only reach 60% of the GCN’s performance. This shows that our +sub-sampling approach is effective at solving the exponential growth of the receptive field while +approaching the utility of the non-DP GCN baseline, which makes our method attractive for real +world applications. That being said, our method uses a smaller amount of training nodes than what +is available at every iteration, even when computational complexity is not an issue (i.e. on small +graphs). The effect of this reduction in training training samples is exacerbated on small graphs that +do not require batching in non-DP training, which leads to our method performing on-par with the +FDP baseline on small datasets. +Comparison with variable privacy budget +Finally, in Figure 2 we expand on our previous results +by reporting the accuracy at various ϵ checkpoints during training. We report the best results that +our method achieved across all sampling methods and compare to the FDP baseline. On all datasets, +our method largely outperforms FDP across multiple epsilon values. Moreover, FDP cannot achieve +an epsilon lower than 2, whereas our method does while sometimes outperforming FDP at higher +privacy budgets. +6 + +(a) +(b) +(c) +Figure 2: F1 Micro Score vs. epsilon achieved by FDP and our best sampling method for a) Flickr, b) +Arxiv and c) Reddit datasets. +6 +Conclusion +We proposed a novel way of training differentially private graph neural networks. Since graphs +consist of inter-connected nodes that influence each other’s gradients during training, naively adapting +traditional DP methods to graph neural networks can result in unnecessarily large amounts of noise +being added to the model during training, which in turn leads to poor utility of the model. We +proposed an adapted version of DP-SGD that uses random-walk based sub-sampling to overcome +this problem and introduced three sampling methods that generate disjoint subgraphs. For each +sampling method, we derived an upper bound on the probability of sampling a modified node in +a batch to apply the amplification by sub-sampling theorem and obtain tighter privacy guarantees. +Our method achieves a better privacy-utility trade-off compared to the state-of-the-art baseline FDP +across multiple datasets, especially for large datasets. A necessary future work direction in this field +is to attempt to solve the performance issue on small datasets, which is especially exacerbated on +GNNs. For example, pre-training the models on public datasets [2] or using variable signal-to-noise +ratios during training are ways of improving the utility in DP. Moreover, different sampling methods +that do not necessarily focus on random walks can be explored. +References +[1] Matt Fredrikson, Somesh Jha, and Thomas Ristenpart. Model inversion attacks that exploit +confidence information and basic countermeasures. In Proceedings of the 22nd ACM SIGSAC +conference on computer and communications security, pages 1322–1333, 2015. +[2] Xuechen Li, Florian Tramer, Percy Liang, and Tatsunori Hashimoto. Large language mod- +els can be strong differentially private learners. In International Conference on Learning +Representations, 2021. +[3] Rohan Anil, Badih Ghazi, Vineet Gupta, Ravi Kumar, and Pasin Manurangsi. Large-scale +differentially private bert. arXiv preprint arXiv:2108.01624, 2021. +[4] Fan Wu, Yunhui Long, Ce Zhang, and Bo Li. Linkteller: Recovering private edges from graph +neural networks via influence analysis. arXiv preprint arXiv:2108.06504, 2021. +[5] Iyiola E Olatunji, Wolfgang Nejdl, and Megha Khosla. Membership inference attack on graph +neural networks. In 2021 Third IEEE International Conference on Trust, Privacy and Security +in Intelligent Systems and Applications (TPS-ISA), pages 11–20. IEEE, 2021. +[6] Zaixi Zhang, Qi Liu, Zhenya Huang, Hao Wang, Chengqiang Lu, Chuanren Liu, and Enhong +Chen. Graphmi: Extracting private graph data from graph neural networks. arXiv preprint +arXiv:2106.02820, 2021. +[7] Cynthia Dwork, Aaron Roth, et al. The algorithmic foundations of differential privacy. Founda- +tions and Trends® in Theoretical Computer Science, 9(3–4):211–407, 2014. +[8] Martin Abadi, Andy Chu, Ian Goodfellow, H Brendan McMahan, Ilya Mironov, Kunal Talwar, +and Li Zhang. Deep learning with differential privacy. In Proceedings of the 2016 ACM SIGSAC +conference on computer and communications security, pages 308–318, 2016. +7 + +0.48 +FDP +Ours +0.46 +F1 Micro Score +0.44 +0.42 +0.40 +0.38 +0.36 +0.34 +1 +2 +3 +4 +5 +6 +7 +8 +Epsilon0.60 +0.55 +0.50 +Micro Score +0.45 +0.40 +0.35 +F1 +0.30 +FDP +0.25 +Ours +0.20 +1 +2 +3 +4 +5 +6 +7 +8 +Epsilon0.8 +FDP +Ours +0.7 +Fl Micro Score +0.6 +0.5 +0.4 +0.3 +1 +2 +3 +4 +5 +6 +7 +8 +Epsilon[9] Ameya Daigavane, Gagan Madan, Aditya Sinha, Abhradeep Guha Thakurta, Gaurav Aggarwal, +and Prateek Jain. Node-level differentially private graph neural networks. In ICLR 2022 +Workshop on PAIR, 2022. +[10] Sina Sajadmanesh, Ali Shahin Shamsabadi, Aurélien Bellet, and Daniel Gatica-Perez. Gap: +Differentially private graph neural networks with aggregation perturbation. arXiv preprint +arXiv:2203.00949, 2022. +[11] Ilya Mironov. Rényi differential privacy. In 2017 IEEE 30th computer security foundations +symposium (CSF), pages 263–275. IEEE, 2017. +[12] Borja Balle, Gilles Barthe, Marco Gaboardi, Justin Hsu, and Tetsuya Sato. Hypothesis test- +ing interpretations and renyi differential privacy. In International Conference on Artificial +Intelligence and Statistics, pages 2496–2506. PMLR, 2020. +[13] Shuang Song, Kamalika Chaudhuri, and Anand D Sarwate. Stochastic gradient descent with +differentially private updates. In 2013 IEEE global conference on signal and information +processing, pages 245–248. IEEE, 2013. +[14] Raef Bassily, Adam Smith, and Abhradeep Thakurta. Private empirical risk minimization: +Efficient algorithms and tight error bounds. In 2014 IEEE 55th annual symposium on foundations +of computer science, pages 464–473. IEEE, 2014. +[15] Timour Igamberdiev and Ivan Habernal. Privacy-Preserving Graph Convolutional Networks for +Text Classification. In Proceedings of the 13th Language Resources and Evaluation Conference, +page (to appear), Marseille, France, 2022. European Language Resources Association. +[16] Borja Balle, Gilles Barthe, and Marco Gaboardi. Privacy amplification by subsampling: Tight +analyses via couplings and divergences. Advances in Neural Information Processing Systems, +31, 2018. +[17] Yu-Xiang Wang, Borja Balle, and Shiva Prasad Kasiviswanathan. Subsampled rényi differential +privacy and analytical moments accountant. In The 22nd International Conference on Artificial +Intelligence and Statistics, pages 1226–1235. PMLR, 2019. +[18] Ameya Daigavane, Gagan Madan, Aditya Sinha, Abhradeep Guha Thakurta, Gaurav Aggarwal, +and Prateek Jain. Node-level differentially private graph neural networks. arXiv preprint +arXiv:2111.15521, 2021. +[19] Sina Sajadmanesh and Daniel Gatica-Perez. Locally private graph neural networks. In Proceed- +ings of the 2021 ACM SIGSAC Conference on Computer and Communications Security, pages +2130–2145, 2021. +[20] Tamara T Mueller, Dmitrii Usynin, Johannes C Paetzold, Daniel Rueckert, and Georgios Kaissis. +Sok: Differential privacy on graph-structured data. arXiv preprint arXiv:2203.09205, 2022. +[21] Hanqing Zeng, Hongkuan Zhou, Ajitesh Srivastava, Rajgopal Kannan, and Viktor Prasanna. +Graphsaint: Graph sampling based inductive learning method. arXiv preprint arXiv:1907.04931, +2019. +[22] Prithviraj Sen, Galileo Mark Namata, Mustafa Bilgic, Lise Getoor, Brian Gallagher, and Tina +Eliassi-Rad. Collective classification in network data. AI Magazine, 29(3):93–106, 2008. +A +Upper Bound on Gradient Sensitivity +We show how to derive the upper bound on the sensitivity of the total gradient on a batch, where gt +is the function that takes a batch B as input and returns the gradients at iteration t, B and B′ are +neighboring batches that differ by one sample ˜v, Lv is the loss function on a node v, ∇wLv is the +8 + +gradient of the loss on v with respect to the weights of the model, and I[˜v ∈ B] is the indicator +function which is 1 if ˜v is in the batch and 0 otherwise. +∆2gt = ∥gt(B) − gt(B′)∥2 += ∥ +� +v∈B +∇wLv − +� +v∈B′ +∇wLv∥2 += ∥(∇wL˜v + +� +u∈RF (˜v)\{˜v} +∇wLu)I[˜v ∈ B] − (∇wL˜v′ + +� +u∈RF (˜v′)\{˜v′} +∇wLu)I[˜v′ ∈ B′]∥2 +≤ ∥(∇wL˜v + +� +u∈RF (˜v)\{˜v} +∇wLu)I[˜v ∈ B]∥2 + ∥(∇wL˜v′ + +� +u∈RF (˜v′)\{˜v′} +∇wL˜v′)I[˜v′ ∈ B′]∥2 +≤ ∥∇wL˜v + +� +u∈RF (˜v)\{˜v} +∇wLu∥2 + ∥∇wL˜v′ + +� +u∈RF (˜v′)\{˜v′} +∇wLu∥2 +≤ ∥∇wL˜v∥2 + +� +u∈RF (˜v)\{˜v} +∥∇wLu∥2 + ∥∇wL˜v′∥2 + +� +u∈RF (˜v′)\{˜v′} +∥∇wLu∥2 +≤ 2|RF(˜v)|C +≤ 2KL+1 − 1 +K − 1 +C +(1) +9 + +B +Algorithms +B.1 +DRW Sampler +The following algorithm shows how to generate disjoint subgraphs using the Disjoint Random Walks +(DRW) sampling method (see Section 4.2). To generate a subgraph, we first sample a node v from +the set of remaining nodes, then remove it from this set. We then construct the set of valid neighbors +of v, which consists of all nodes that have not been already sampled. We sample the next node v in +the subgraph from the set of valid neighbors, and repeat the process until we get a random walk of +length L. We iterate this process until all nodes are included in one subgraph. +Algorithm 2 DRW Sampler +Input: Graph G = {V, E}, random walk length L. +Output: Set of all disjoint subgraphs = () +remaining_nodes = {v1, v2, . . . , vN} +while len(remaining_nodes) != 0 do +subgraph = [] +v = sample(remaining_nodes, 1) +▷ uniformly sample over non-sampled nodes +subgraph.append(v) +remaining_nodes.remove(v) +l = 0 +while l < L do +valid_neighbors = Neighbors(v) +▷ Neighbors returns all neighbors of a node +for u in valid_neighbors do +if u not in remaining_nodes then +valid_neighbors.remove(u) +end if +end for +if len(valid_neighbors) != 0 then +v = sample(valid_neighbors, 1) +▷ uniformly sample a neighbor of v +else +break +end if +random_walk.append(v) +remaining_nodes.remove(v) +l = l + 1 +end while +subgraphs.add(subgraph) +end while +B.2 +DRW-R Sampler +The following algorithm shows how to generate disjoint subgraphs using the Disjoint Random Walks +with restarts (DRW-R) sampling method (see 4.2). The main difference to the DRW sampler is that, +instead of stopping the subgraph generation after one random walk, we sample multiple random +walks rooted at the same node by re-initializing the starting node of the random walk to the same root +node of the subgraph R times. +10 + +Algorithm 3 DRW-R Sampler +Input: Graph G = {V, E}, random walk length L. +Output: Set of all disjoint subgraphs = () +remaining_nodes = {v1, v2, . . . , vN} +while len(remaining_nodes) != 0 do +subgraph = [] +root = sample(remaining_nodes, 1) +▷ uniformly sample over non-sampled nodes +subgraph.append(root) +remaining_nodes.remove(root) +for r in range(R) do +v = root +l = 0 +while l < L do +valid_neighbors = Neighbors(v) +▷ Neighbors returns all neighbors of a node +for u in valid_neighbors do +if u not in remaining_nodes then +valid_neighbors.remove(u) +end if +end for +if len(valid_neighbors) != 0 then +v = sample(valid_neighbors, 1) +▷ uniformly sample a neighbor of v +else +break +end if +subgraph.append(v) +remaining_nodes.remove(v) +l = l + 1 +end while +subgraphs.add(subgraph) +end for +end while +11 + +C +Training Hyperparameters +We run all experiments with three different seeds, Adam optimizer and ReLU activation. We +summarize the number of roots used for different sampling scenarios in Table 4. For the non-DP +trainings, we fix the learning rate to 0.01. We perform a grid hyper-parameter search for the trainings +on all datasets. We experiment with the following hyper-parameters for both DP and non-DP trainings: +• Number of layers in {1, 2} +• Width of hidden layers in {256, 512} +• Maximum graph degree in {2 , 4} for the FDP baseline +We use the follow hyper-parameters for the DP specific trainings: +• Learning rate in {0.01, 0.1, 0.2} +• Clip norm percentage C% in {0.001, 0.01, 0.1}. +• Noise multiplier λ in {1, 2, 4, 8}. The noise multiplier is the ratio of the standard deviation +σ of the Gaussian noise added to the gradients to the sensitivity ∆2f of the function f. +Instead of tuning σ, we tune λ, then fix σ = λ × ∆2f. +• Delta value δ: we summarize the values used in Table 3 +Table 2: Characteristics of the datasets that we use in our experiments. (s) indicates a single-label +classification problem, and (m) a multi-label one. +Nodes +Feature Size +Classes +Training Nodes +Type +Cora +2,708 +1,433 +7 (s) +140 +Transductive +Citeseer +3,327 +3,703 +6 (s) +120 +Transductive +PPI +14,755 +50 +121 (m) +9,716 +Inductive +Pubmed +19,717 +500 +3 (s) +60 +Transductive +Flickr +89,250 +500 +7 (s) +44,625 +Inductive +Arxiv +169,343 +128 +40 (s) +90,941 +Inductive +Reddit +232,965 +602 +41 (s) +153,932 +Inductive +Table 3: δ value used for each dataset. +Dataset +Cora +Citeseer +PPI +Pubmed +Flickr +Arxiv +Reddit +δ +1e-5 +1e-5 +1e-5 +1e-6 +1e-6 +1e-7 +1e-7 +Table 4: Batch sizes used for training based on the sampler, depth of the model, and dataset. Note +that as a general rule, we used around 20% of total number of training nodes for the large datasets, +and 50% for the small datasets. +Sampler +Depth +Dataset +Cora +CiteSeer +PPI +PubMed +Flickr +Arxiv +Reddit +RW, uniform, PreDRW, +1 +70 +60 +2,000 +30 +10,000 +20,000 +30,000 +PreDRW-D, DynDRW +2 +46 +40 +2,000 +20 +10,000 +20,000 +30,000 +PreDRW-R +1 +46 +40 +2,000 +20 +10,000 +20,000 +30,000 +2 +28 +24 +1,800 +12 +8,000 +18,000 +30,000 +12 + diff --git a/2dAyT4oBgHgl3EQf1fnx/content/tmp_files/load_file.txt b/2dAyT4oBgHgl3EQf1fnx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d5639b8e01ba5d02c9e694dcc0aaa2d614d23189 --- /dev/null +++ b/2dAyT4oBgHgl3EQf1fnx/content/tmp_files/load_file.txt @@ -0,0 +1,551 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf,len=550 +page_content='Training Differentially Private Graph Neural Networks with Random Walk Sampling Morgane Ayle Technical University of Munich morgane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='ayle@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='de Jan Schurchardt Technical University of Munich j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='schuchardt@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='de Lukas Gosch Technical University of Munich l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='gosch@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='de Daniel Zügner Technical University of Munich zuegnerd@in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='de Stephan Günnemann Technical University of Munich s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='guennemann@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='de Abstract Deep learning models are known to put the privacy of their training data at risk, which poses challenges for their safe and ethical release to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Differentially private stochastic gradient descent is the de facto standard for training neural networks without leaking sensitive information about the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' However, applying it to models for graph-structured data poses a novel challenge: unlike with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' data, sensitive information about a node in a graph cannot only leak through its gradients, but also through the gradients of all nodes within a larger neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' In practice, this limits privacy-preserving deep learning on graphs to very shallow graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We propose to solve this issue by training graph neural networks on disjoint subgraphs of a given training graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We develop three random-walk-based methods for generating such disjoint subgraphs and perform a careful analysis of the data-generating distributions to provide strong privacy guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Through extensive experiments, we show that our method greatly outperforms the state-of-the-art baseline on three large graphs, and matches or outperforms it on four smaller ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' 1 Introduction The introduction of Graph Neural Networks (GNNs) has enabled the training of Deep Learning (DL) models on graph-structured data and for various tasks such as node classification, link prediction or graph classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' However, similar to DL models trained on image [1] or text data [2, 3], GNNs leak information about their training data [4–6], such as the features of a node, or which nodes are connected by an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' In this paper, we analyze the privacy of GNNs under the lens of Differential Privacy (DP) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' In particular, we ensure the privacy of all nodes’ features in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' While DP-SGD [8] is the de facto standard for training DL models with DP, its transfer to GNNs is not straightforward given the non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' nature of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Indeed, since an L-layer GNN typically uses the L-hop neighborhood of a node during the forward pass, the gradient of a node does not depend on that node alone, but on all nodes in its neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' While some works [9, 10] have attempted to apply DP to GNNs, most of them focus on edge-level DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Methods that can be applied to feature-level DP suffer from 2022 Trustworthy and Socially Responsible Machine Learning (TSRML 2022) co-located with NeurIPS 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='00738v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='LG] 2 Jan 2023 loose privacy guarantees [9], or rely on custom GNN architectures [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We propose an adaptation of DP-SGD to train GNNs with feature-level DP while attenuating the aforementioned problem and preserving a high model utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We experimentally demonstrate that our method can offer significantly stronger privacy guarantees than prior work, particularly on large graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' 2 Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='1 Differential privacy (ϵ, δ)-DP Differential Privacy (DP) [7] is a notion of privacy that allows data analysts to extract useful statistics from a dataset, without leaking too much information about the samples in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' More formally, given two neighboring datasets D and D′ – denoted D ∼ D′ – that differ by one sample (either by deleting, adding or modifying a sample), a randomized algorithm M with co-domain Y is (ϵ, δ)-DP if for all O ⊆ Y , and for all D ∼ D′, Pr[M(D) ∈ O] ≤ exp(ϵ)Pr[M(D′) ∈ O] + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' The parameters ϵ and δ are the privacy budget parameters: the smaller their values, the better the privacy guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' (α, γ)-RDP An alternative definition of DP is Rényi Differential Privacy (RDP) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' A randomized algorithm M is said to be γ-RDP of order α – or (α, γ)-RDP – if for any D ∼ D′ it holds that Dα(M(D), M(D′)) ≤ γ, where Dα = 1 α−1 log Ex∼Q � P (x) Q(x) �α is the Rényi divergence of order α which measures the similarity of the distributions P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Note that if M is (α, γ)-RDP, then it is also (ϵ, δ)-DP for any 0 < δ < 1 where ϵ = fRDP→DP(α, γ, δ) = γ + log( α−1 α ) − log δ+log α α−1 [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We rely on (α, γ)-RDP during our analysis, but report our results in terms of (ϵ, δ)-DP following prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' The Gaussian mechanism Given an algorithm A with real-valued output space A : ND → Rd, the Gaussian mechanism privatizes the algorithm by adding Gaussian noise to the outputs of A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' M = Gσ (A (D)) = A(D) + N(0, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Given that the ℓ2 sensitivity of A is ∆2A(D) = maxD∼D′ ∥A(D) − A(D′)∥2, the mechanism satisfies (α, γ(α))-RDP, with γ(α) = α(∆2A)2 2σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Intuitively, this indicates that the larger the sensitivity of the function, the more noise needs to be added to obtain a small privacy budget, and therefore the worse the final performance will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' A small sensitivity is therefore desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Amplification by sub-sampling A useful property of DP (and RDP) is that, given a mechanism S that samples a sub-set of the dataset D, applying a private mechanism to S(D) leads to better privacy guarantees than applying it to the entire dataset D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Intuitively, this is due to the fact that subsampling introduces a non-zero chance of an added or modified sample to not be processed by the randomized algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Typically, S is assumed to be a Poisson or uniform sampling over the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Poisson sampling is typically used when the neighboring datasets differ in size, while uniform sampling is used otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' In this paper, we rely on uniform sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='2 Differential privacy in deep learning Differentially Private Stochastic Gradient Descent (DP-SGD) [13, 14, 8] is the foundation of many works [9, 2, 15] that apply DP to deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' It privatizes the weights of a model with respect to the input dataset at every iteration of training, and then accumulates the privacy budget being spent over all iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' One private training iteration consists of batching a set of samples, computing the gradient on each sample independently, clipping the norm of each gradient vector to a maximum norm C, calculating the entire gradient by adding calibrated Gaussian noise, and finally performing an update step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' The clipping step is used to bound the sensitivity of the gradients to changes in the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Then, assuming that two neighboring datasets D and D′ differ in the features of one sample, the sensitivity of the total gradient on a batch of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' samples is bounded by 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Through batching (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' sub-sampling the dataset using a sampling mechanism S), amplification by sub-sampling theorems [16, 17] can be exploited to get better privacy guarantees at every iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Finally, assuming each iteration t is (α, γt)-RDP, the overall training is then (α, �T t=0 γt)-RDP [11] where T is the total number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='3 Graph neural networks Definition In the following, we define a graph as G = {X, A}, where X ∈ RN×d is the feature matrix in which each row corresponds to one node’s feature vector, and A ∈ {0, 1}N×N is the adjacency matrix in which Aij is 1 if there exists an edge between nodes i and j and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Note that we only consider undirected graphs, therefore A = AT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Graph Neural Networks (GNNs) are a class of models that learn a mapping f : G → Z ∈ RN×d′, where Z is an updated feature matrix of G that can be used for various downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Each layer of a GNN typically consists of two steps: 1) in the aggregation step, information about the neighborhood of every node is gathered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' 2) in the update step, the feature vector of every node is updated based on its current feature vector and the aggregated neighborhood information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' The receptive field The receptive field of a node in a GNN is defined as the region in the input graph that influences the GNN’s predictions for that specific node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' For a GNN with L layers, the receptive field of a node v is the L-hop neighborhood of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Thus, for a graph with maximum node degree K, the largest possible receptive field size of any node v is RF(v) = �L l=0 Kl = KL+1−1 K−1 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' the receptive field grows exponentially with the number of layers of the GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='4 Differential privacy in graph neural networks Given that graphs contain two types of attributes – node features and edges – multiple levels of DP [18, 9, 10] can be considered: edge-level DP, where the edges between nodes are private;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' feature-level DP, where the features of nodes are private;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' and node-level DP, where both the features and edges of nodes are private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' In this work, we focus on feature-level DP using DP-SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Contrary to traditional i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' datasets, samples in a graph (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' nodes) are not independent: changing the features of one node affects the gradients of all nodes within the receptive field of the modified node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' In fact, the sensitivity of the total gradient on a graph is bounded by 2 KL+1−1 K−1 C (see Appendix A), which grows exponentially with the number of layers L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Given that the Gaussian mechanism adds noise proportional to the sensitivity of the total gradient, this can lead to large amounts of noise being added during training, which in turn leads to poor final model utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' 3 Related work In [19], a node-level differentially private GNN is trained by perturbing features and edges locally before sending them to a global server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' This setup is called local DP, and differs from our notion of DP where a central learner is trusted with the real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' The authors in [15] propose to split the graph into disjoint sub-graphs using uniform node sampling, then treat each sub-graph as an independent sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Note that, contrary to our method which considers privacy at the individual node feature level, their approach treats the entire graph as a datapoint to privatize, rather than providing privacy for the individual nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' The method in [10] privatizes GNNs at both the node-level and edge-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' However, their approach only applies to the GNN architecture they propose and not to arbitrary GNNs, unlike our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Furthermore, it does not resolve the issue of exponentially growing sensitivity in transductive learning scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' For a survey on DP on graph data, refer to [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Finally, the authors of [9] propose to reduce the sensitivity of a GNN’s gradients by bounding the maximum degree K of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' However, this does not resolve the exponential growth with the number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Therefore, they still obtain loose privacy guarantees (ϵ = 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Since this method is the closest to our setup, we compare our approach to theirs in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' 4 Methodology 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='1 Approach We propose to adapt DP-SGD to the graph domain to ensure that the weights of a GNN are private with respect to the nodes’ features, while overcoming the problem of requiring exponentially more noise with a growing network depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' In the following, we define two graphs G and G′ as neighbors if they share the same structure A and number of nodes N but differ in one row of the feature matrix X corresponding to the modified node ˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We want to train the GNN such that for all G ∼ G′, 3 Figure 1: Our general sampling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Starting with a graph, we generate subgraphs by first sampling a root node (depicted in red), and then sampling one or more random walks starting from the root node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Every node appears in exactly one subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Before every iteration, we batch m many subgraphs, where m = 2 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Root nodes are used as training nodes, while remaining nodes are used for aggregation in the GNN only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Dα(M(G), M(G′)) ≤ γ, where M is a randomized algorithm that returns the weights of the GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' To adapt DP-SGD to the graph domain, we propose to pre-process the graph into sets of independent subgraphs that do not affect each others’ gradients, so that the sensitivity of the total gradient on any batch depends on the gradient of one subgraph only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We summarize our training procedure in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' More precisely, we pre-process the graph into a set of M disjoint subgraphs GS = {s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' , sM}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' subgraphs that do not have any nodes in common, using sampling method S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Each subgraph si consists of two components: 1) one training node vi, and 2) a set of neighbors N (vi) that is used for the aggregation step of the GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' At training time, for every iteration t, we create a batch by sampling m subgraphs uniformly at random from the set of subgraphs GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We then compute the gradients ∇wtL(vj, N (vj)) on all training nodes and clip the norm of each to a value C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We compute the total gradient by summing individual gradients and adding Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Finally, we update the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Due to the disjointness of subgraphs, changing one node’s features – whether it is a train- ing node or a neighbor – will affect at most one subgraph (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' sample) in the batch, which reduces the upper bound on the sensitivity of the total gradient to 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Since we sample subgraphs uniformly at random, we can leverage the strong amplification by sub-sampling theorem [17], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' account for the possibility of the gradient not being affected if the modified node ˜v is not part of the batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We generate these disjoint subgraphs via random walk sampling, which is an effective way of training GNNs [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We choose random walk sampling, since it ensures that nodes form a connected subgraph of a training node’s neighborhood, while limiting the number of nodes being sampled from that neighborhood (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' from the receptive field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' In the following, we propose three different random-walk-based sampling methods, which we later compare in our experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Furthermore, we derive for each sampling method a tight upper bound on the probability of sampling the modified node ˜v in a batch, which is required for applying the amplification by subsampling theorem in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='2 Sampling methods Our three sampling methods consist of pre-processing the graph into a set of M disjoint subgraphs GS = {s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' , sM}, and then generating a batch B ⊆ GS by sampling m subgraphs uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' An overview of our general approach is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Given a graph with M generated disjoint subgraphs, the true probability of sampling node ˜v is P[˜v] = 1 M , since we know that a node is in exactly one of the M subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' However, to ensure differential privacy, we require a bound that holds for all possible graphs and any run of the sampling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Thus, we use the upper bound P[˜v] = 1 M ≤ 1 Mmin where Mmin is the minimum number of subgraphs that can be generated in any graph of N nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Then, the probability of sampling ˜v in a batch of m subgraphs using sampling mechanism S is at most PS[˜v] ≤ m Mmin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' 4 pre-process batchAlgorithm 1 DP-SGD with random walk sampling Input: Graph G = {V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' E},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' sampling method S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' loss function L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' initial model weights w0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' noise standard deviation σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' gradient clipping norm C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' number of iterations T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' frequency at which to re-sample subgraphs in DRW-D i GS = S(G) ▷ Generate subgraphs from graph G using sampling method S for t in [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' T) do if t % i == 0 and S == DRW-D then GS = S(G) end if Sample m subgraphs uniformly at random from GS to form batch B for sj in B do ▷ sj is a subgraph Compute ∇wtL(vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' N (vj)) gt(vj) = clip (∇wtL (vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' N (vj)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' C) ▷ Compute and clip individual gradients in B end for gt(B) = 1 |B| ��� sj∈B gt(vj) � + N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' σ2) � ▷ Add noise to the gradients wt+1 = update(wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' gt(B)) ▷ Update weights based on optimizer being used end for Disjoint random walks The first sampling method we propose is called Disjoint Random Walks (DRW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We pre-process the graph once before training and then generate batches at every iteration using the same set of subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Each subgraph consists of one random walk of length L (refer to Appendix B for a pseudo-code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' A random walk of length L contains at most L + 1 nodes, and generating random walks that all have maximal length would result in the minimum number of random walks, since a node can only appear in one random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Therefore, we get Mmin = ⌈ N L+1⌉ and P[˜v] ≤ 1 ⌈ N L+1 ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Finally, the upper bound probability of sampling a node ˜v is PDRW[˜v] ≤ m ⌈ N L+1 ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Disjoint random walks with restarts To create better subgraphs that contain more nodes for aggregation, we also propose Disjoint Random Walks with Restarts (DRW-R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Similary to DRW, this sampling method generates subgraphs once before training by using random walks, but instead of sampling one random walk per training node we sample R of them (refer to Appendix B for a pseudo-code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Given a random walk length of L and R restarts, the minimum number of subgraphs is Mmin = ⌈ N 1+R×L⌉ where 1 + R × L is the maximum size of one subgraph when all random walks have length L, and the probability of sampling node u in a batch of size m is therefore PDRW-R[u] ≤ m ⌈ N 1+R×L ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Disjoint random walks with dynamic re-sampling Finally, we propose a third sampling method in which we pre-process the graph into disjoint subgraphs every ith iteration instead of once before training, where i is a hyper-parameter that is chosen based on the cost of the sampling procedure on each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' This allows us to increase the diversity of subgraphs used for training, and prevent overfitting on the subgraphs generated in one run of the sampling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We call this procedure DRW-D, where D stands for Dynamically re-sampling random walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' The probability of sampling node ˜v is the same as in DRW, namely PDRW-D[˜v] = PDRW[˜v] ≤ m ⌈ N L+1 ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Note that this method consists simply of re-running the subgraph generation process DRW at every ith iteration instead of once before training, which is reflected in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' 5 Experimental results Experimental setup We report our results on seven datasets, both in the transductive and the inductive settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' The dataset sizes in terms of total nodes range from small (Cora [22], Citeseer [22]) to medium (PPI [21], Pubmed [22]) to large (Flickr [21], Arxiv [21], Reddit [21]), or in number of training nodes from small (Pubmed, Citeseer, Cora) to medium (PPI) to large (Flickr, Arxiv, Reddit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We report the exact number of nodes as well as some additional dataset characteristics in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We focus on the node classification task, and report our results in terms of F1 Micro score, a metric equivalent to accuracy except on PPI which is a multi-label classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Following prior work, we report our privacy budget using ϵ and a fixed δ per dataset (see Appendix 5 Table 1: Comparison between the F1 Micro score (%) achieved by a basic GCN and MLP, the FDP baseline, and our proposed method with multiple sampling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' All DP methods are trained with a target budget of ϵ ≤ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Layers Width Dataset Cora CiteSeer PPI PubMed Flickr Arxiv Reddit GCN (non-DP) 1 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='8 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='5 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='2 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='7 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='6 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='7 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='5 2 256 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='3 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='9 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='9 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='3 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='7 512 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='6 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='7 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='9 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='3 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='5 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='7 MLP (non-DP) 1 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='6 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='3 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='7 52.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='4 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='2 C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Given a target ϵ, we keep training while tracking the (α, γt) privacy budget being spent until we reach ϵ = fRDP→DP(α, �T ′ t=0 γt, δ) at iteration T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We compare our proposed methodology with each sampling method to three baselines: 1) A basic GCN trained with random walk sampling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' 2) A basic MLP trained with uniform node sampling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' and 3) The method proposed in [9] which we call FDP for Feature-level DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Note that while they train their models up to an ϵ of 20, we only train them until ϵ = 8, since a very large ϵ does not have much value in terms of privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Discussion Table 5 summarizes our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' A GCN trained without DP always outperforms the ones trained with DP, which is expected since clipping gradients and especially adding Gaussian noise decreases the utility of the final model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' However, in some cases our method can almost match the utility of the basic GCN, whereas the FDP baseline struggles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' For example, DRW sampling on Flickr can reach up to 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='7% accuracy – which corresponds to 95% of the baseline GCN’s performance – whereas FDP reaches only 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='5% accuracy – which corresponds to 83% of the baseline GCN’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Similarly, our method achieves 87% of the GCN’s performance on the challenging dataset Reddit, while FDP can only reach 60% of the GCN’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' This shows that our sub-sampling approach is effective at solving the exponential growth of the receptive field while approaching the utility of the non-DP GCN baseline, which makes our method attractive for real world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' That being said, our method uses a smaller amount of training nodes than what is available at every iteration, even when computational complexity is not an issue (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' on small graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' The effect of this reduction in training training samples is exacerbated on small graphs that do not require batching in non-DP training, which leads to our method performing on-par with the FDP baseline on small datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Comparison with variable privacy budget Finally, in Figure 2 we expand on our previous results by reporting the accuracy at various ϵ checkpoints during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We report the best results that our method achieved across all sampling methods and compare to the FDP baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' On all datasets, our method largely outperforms FDP across multiple epsilon values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Moreover, FDP cannot achieve an epsilon lower than 2, whereas our method does while sometimes outperforming FDP at higher privacy budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' 6 (a) (b) (c) Figure 2: F1 Micro Score vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' epsilon achieved by FDP and our best sampling method for a) Flickr, b) Arxiv and c) Reddit datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' 6 Conclusion We proposed a novel way of training differentially private graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Since graphs consist of inter-connected nodes that influence each other’s gradients during training, naively adapting traditional DP methods to graph neural networks can result in unnecessarily large amounts of noise being added to the model during training, which in turn leads to poor utility of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We proposed an adapted version of DP-SGD that uses random-walk based sub-sampling to overcome this problem and introduced three sampling methods that generate disjoint subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' For each sampling method, we derived an upper bound on the probability of sampling a modified node in a batch to apply the amplification by sub-sampling theorem and obtain tighter privacy guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Our method achieves a better privacy-utility trade-off compared to the state-of-the-art baseline FDP across multiple datasets, especially for large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' A necessary future work direction in this field is to attempt to solve the performance issue on small datasets, which is especially exacerbated on GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' For example, pre-training the models on public datasets [2] or using variable signal-to-noise ratios during training are ways of improving the utility in DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Moreover, different sampling methods that do not necessarily focus on random walks can be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' References [1] Matt Fredrikson, Somesh Jha, and Thomas Ristenpart.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='34 1 2 3 4 5 6 7 8 Epsilon0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='50 Micro Score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='35 F1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='30 FDP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='25 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='20 1 2 3 4 5 6 7 8 Epsilon0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='8 FDP Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='7 Fl Micro Score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='3 1 2 3 4 5 6 7 8 Epsilon[9] Ameya Daigavane, Gagan Madan, Aditya Sinha, Abhradeep Guha Thakurta, Gaurav Aggarwal, and Prateek Jain.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='15521, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' [19] Sina Sajadmanesh and Daniel Gatica-Perez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Locally private graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' In Proceed- ings of the 2021 ACM SIGSAC Conference on Computer and Communications Security, pages 2130–2145, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' [20] Tamara T Mueller, Dmitrii Usynin, Johannes C Paetzold, Daniel Rueckert, and Georgios Kaissis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Sok: Differential privacy on graph-structured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='09205, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' [21] Hanqing Zeng, Hongkuan Zhou, Ajitesh Srivastava, Rajgopal Kannan, and Viktor Prasanna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Graphsaint: Graph sampling based inductive learning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' arXiv preprint arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='04931, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' [22] Prithviraj Sen, Galileo Mark Namata, Mustafa Bilgic, Lise Getoor, Brian Gallagher, and Tina Eliassi-Rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Collective classification in network data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' AI Magazine, 29(3):93–106, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' A Upper Bound on Gradient Sensitivity We show how to derive the upper bound on the sensitivity of the total gradient on a batch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' where gt is the function that takes a batch B as input and returns the gradients at iteration t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' B and B′ are neighboring batches that differ by one sample ˜v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Lv is the loss function on a node v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' ∇wLv is the 8 gradient of the loss on v with respect to the weights of the model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' and I[˜v ∈ B] is the indicator function which is 1 if ˜v is in the batch and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='∆2gt = ∥gt(B) − gt(B′)∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='= ∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='v∈B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='∇wLv − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='v∈B′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='∇wLv∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='= ∥(∇wL˜v + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='u∈RF (˜v)\\{˜v} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='∇wLu)I[˜v ∈ B] − (∇wL˜v′ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='u∈RF (˜v′)\\{˜v′} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='∇wLu)I[˜v′ ∈ B′]∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='≤ ∥(∇wL˜v + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='u∈RF (˜v)\\{˜v} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='∇wLu)I[˜v ∈ B]∥2 + ∥(∇wL˜v′ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='u∈RF (˜v′)\\{˜v′} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='∇wL˜v′)I[˜v′ ∈ B′]∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='≤ ∥∇wL˜v + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='u∈RF (˜v)\\{˜v} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='∇wLu∥2 + ∥∇wL˜v′ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='u∈RF (˜v′)\\{˜v′} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='∇wLu∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='≤ ∥∇wL˜v∥2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='u∈RF (˜v)\\{˜v} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='∥∇wLu∥2 + ∥∇wL˜v′∥2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='u∈RF (˜v′)\\{˜v′} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='∥∇wLu∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='≤ 2|RF(˜v)|C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='≤ 2KL+1 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='K − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='Algorithms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='1 DRW Sampler The following algorithm shows how to generate disjoint subgraphs using the Disjoint Random Walks (DRW) sampling method (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' To generate a subgraph, we first sample a node v from the set of remaining nodes, then remove it from this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We then construct the set of valid neighbors of v, which consists of all nodes that have not been already sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We sample the next node v in the subgraph from the set of valid neighbors, and repeat the process until we get a random walk of length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We iterate this process until all nodes are included in one subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Algorithm 2 DRW Sampler Input: Graph G = {V, E}, random walk length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Output: Set of all disjoint subgraphs = () remaining_nodes = {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' , vN} while len(remaining_nodes) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='= 0 do subgraph = [] v = sample(remaining_nodes, 1) ▷ uniformly sample over non-sampled nodes subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='append(v) remaining_nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='remove(v) l = 0 while l < L do valid_neighbors = Neighbors(v) ▷ Neighbors returns all neighbors of a node for u in valid_neighbors do if u not in remaining_nodes then valid_neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='remove(u) end if end for if len(valid_neighbors) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='= 0 then v = sample(valid_neighbors, 1) ▷ uniformly sample a neighbor of v else break end if random_walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='append(v) remaining_nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='remove(v) l = l + 1 end while subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='add(subgraph) end while B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='2 DRW-R Sampler The following algorithm shows how to generate disjoint subgraphs using the Disjoint Random Walks with restarts (DRW-R) sampling method (see 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' The main difference to the DRW sampler is that, instead of stopping the subgraph generation after one random walk, we sample multiple random walks rooted at the same node by re-initializing the starting node of the random walk to the same root node of the subgraph R times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' 10 Algorithm 3 DRW-R Sampler Input: Graph G = {V, E}, random walk length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Output: Set of all disjoint subgraphs = () remaining_nodes = {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' , vN} while len(remaining_nodes) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='= 0 do subgraph = [] root = sample(remaining_nodes, 1) ▷ uniformly sample over non-sampled nodes subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='append(root) remaining_nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='remove(root) for r in range(R) do v = root l = 0 while l < L do valid_neighbors = Neighbors(v) ▷ Neighbors returns all neighbors of a node for u in valid_neighbors do if u not in remaining_nodes then valid_neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='remove(u) end if end for if len(valid_neighbors) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='= 0 then v = sample(valid_neighbors, 1) ▷ uniformly sample a neighbor of v else break end if subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='append(v) remaining_nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='remove(v) l = l + 1 end while subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='add(subgraph) end for end while 11 C Training Hyperparameters We run all experiments with three different seeds, Adam optimizer and ReLU activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We summarize the number of roots used for different sampling scenarios in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' For the non-DP trainings, we fix the learning rate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We perform a grid hyper-parameter search for the trainings on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' We experiment with the following hyper-parameters for both DP and non-DP trainings: Number of layers in {1, 2} Width of hidden layers in {256, 512} Maximum graph degree in {2 , 4} for the FDP baseline We use the follow hyper-parameters for the DP specific trainings: Learning rate in {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='2} Clip norm percentage C% in {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content='1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Noise multiplier λ in {1, 2, 4, 8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' The noise multiplier is the ratio of the standard deviation σ of the Gaussian noise added to the gradients to the sensitivity ∆2f of the function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Instead of tuning σ, we tune λ, then fix σ = λ × ∆2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Delta value δ: we summarize the values used in Table 3 Table 2: Characteristics of the datasets that we use in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' (s) indicates a single-label classification problem, and (m) a multi-label one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Nodes Feature Size Classes Training Nodes Type Cora 2,708 1,433 7 (s) 140 Transductive Citeseer 3,327 3,703 6 (s) 120 Transductive PPI 14,755 50 121 (m) 9,716 Inductive Pubmed 19,717 500 3 (s) 60 Transductive Flickr 89,250 500 7 (s) 44,625 Inductive Arxiv 169,343 128 40 (s) 90,941 Inductive Reddit 232,965 602 41 (s) 153,932 Inductive Table 3: δ value used for each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Dataset Cora Citeseer PPI Pubmed Flickr Arxiv Reddit δ 1e-5 1e-5 1e-5 1e-6 1e-6 1e-7 1e-7 Table 4: Batch sizes used for training based on the sampler, depth of the model, and dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Note that as a general rule, we used around 20% of total number of training nodes for the large datasets, and 50% for the small datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} +page_content=' Sampler Depth Dataset Cora CiteSeer PPI PubMed Flickr Arxiv Reddit RW, uniform, PreDRW, 1 70 60 2,000 30 10,000 20,000 30,000 PreDRW-D, DynDRW 2 46 40 2,000 20 10,000 20,000 30,000 PreDRW-R 1 46 40 2,000 20 10,000 20,000 30,000 2 28 24 1,800 12 8,000 18,000 30,000 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQf1fnx/content/2301.00738v1.pdf'} diff --git a/3tE0T4oBgHgl3EQfvAFe/vector_store/index.faiss b/3tE0T4oBgHgl3EQfvAFe/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..842786bfdc713daaee4c30ce6bc822f9d6cb04af --- /dev/null +++ b/3tE0T4oBgHgl3EQfvAFe/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bea7e67c20ba3bc55372bc3eebd829cb15d32ffb4b83f54494acb00f8e29fa0c +size 3342381 diff --git a/5NAyT4oBgHgl3EQfQPZ4/content/tmp_files/2301.00041v1.pdf.txt b/5NAyT4oBgHgl3EQfQPZ4/content/tmp_files/2301.00041v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..84dbaa13b1a85fc869f5a212af11c45213831a76 --- /dev/null +++ b/5NAyT4oBgHgl3EQfQPZ4/content/tmp_files/2301.00041v1.pdf.txt @@ -0,0 +1,1329 @@ +HIP-2022-35/TH +Vector dark matter in supercooled Higgs portal models +Mads T. Frandsen∗ and Mattias E. Thing† +CP3-Origins, University of Southern Denmark, Denmark +Matti Heikinheimo‡ and Kimmo Tuominen§ +Department of Physics, University of Helsinki, +P.O.Box 64, FI-00014 University of Helsinki, Finland and +Helsinki Institute of Physics, P.O.Box 64, +FI-00014 University of Helsinki, Finland +Martin Rosenlyst¶ +Rudolf Peierls Centre for Theoretical Physics, University of Oxford, +1 Keble Road, Oxford OX1 3NP, United Kingdom +We consider extensions of the Standard Model by a hidden sector consisting of a +gauge field coupled with a scalar field. Assuming the absence of dimensionful param- +eters in the tree level potential, radiative symmetry breaking will make the hidden +sector gauge field massive and induce the electroweak scale of the Standard Model. +We consider separately dark sector gauge groups U(1)D and SU(2)D, and focus on +probing the models with a combination of direct detection experiments and gravita- +tional wave observatories. We find that recent dark matter direct detection results +significantly constrain the parameter space of the models where they can account for +the observed dark matter relic density via freeze-out. The gravitational wave signals +originating from strongly first order electroweak phase transition in these models can +be probed in future gravitational wave observatories such as LISA. We show how +the projected results compliment direct detection experiments and can help probe +parameter space near the neutrino floor of direct detection. +I. +INTRODUCTION +Despite the success of the Standard Model (SM) of particle physics, there are many phenomena +that it does not explain and that appear to require new particles and interactions. One enigmatic +phenomenon is the problem of missing mass, which emerged in a wide range of astrophysical +systems including galaxy clusters [1] and galaxies [2]. One possible solution to the missing mass +problem is cold dark matter (DM), constituted by a new stable and neutral massive particle. This +hypothesis provides an excellent parametrisation for 26% of the energy density of the universe in +addition to the components parametrised as baryonic matter and dark energy [3]. On the other +hand, the non-gravitational nature of dark matter (DM) remains unknown [4–6]. +The cosmological observations on the light element abundance and cosmic microwave back- +ground radiation spectrum imply that the Standard Model (SM) degrees of freedom must have +been in thermal equilibrium in the early universe [7–11]. Whether DM was ever part of the same +heat bath is not known. +However, assuming that this was the case, allows for the abundance of dark matter to arise as a +relic from thermal decoupling in the early universe via interactions between the DM and the SM. +∗ frandsen@cp3.sdu.dk +† thing@cp3.sdu.dk +‡ matti.heikinheimo@helsinki.fi +§ kimmo.i.tuominen@helsinki.fi +¶ martin.jorgensen@physics.ox.ac.uk +arXiv:2301.00041v1 [hep-ph] 30 Dec 2022 + +2 +Moreover, these interactions offer the prospect of detecting DM in direct detection experiments. +The most studied example of this paradigm is Weakly Interacting Massive Particle (WIMP). How- +ever, simplest WIMP models are now very strongly constrained by direct detection experiments. +It is therefore worthwhile to explore the phenomenology of different types of simple benchmark +hidden sectors instead coupled with the SM via portal interactions. +In this paper we analyze two simple models of vector DM, that feature scale invariance of the +tree-level Lagrangian and are coupled to the SM via the Higgs portal, where one scalar mass +eigenstate is SM-like, with mass 125.46 ± 0.16 GeV [12]. The other eigenstate is massless at tree +level but obtains its mass via loop corrections as an effect of radiative symmetry breaking [13]. +This framework of classically scale invariant DM models that feature radiative symmetry breaking, +mediated to the SM via the Higgs portal, has been explored in literature, see e.g. [14–24]. +In this paper we aim to clarify how simple U(1)D and SU(2)D models of this type can be tested +with a combination of direct detection and gravitational wave observations. +Direct detection +experiments have provided very stringent constraints on interactions of weak scale dark matter +with nuclei. Currently, the most stringent constraints come from the recent PandaX-4T and LZ +(2022) experiments [25, 26]. It is well known that radiative symmetry breaking in classically scale +invariant models typically results in a strongly first order electroweak phase transition (EWPT). +Such a first order EWPT could be relevant for baryogenesis and produces gravitational wave signals +which could be observable in upcoming gravitational wave experiments such as LISA [27]. +We present a careful examination of the first order phase transition using different numerical +packages in order to characterise the theoretical uncertainty in the predictions. +II. +DEFINITIONS OF THE MODELS +We consider two models where the SM is extended with a hidden sector gauge group and a +new scalar field charged under the gauge group. Spontaneous symmetry breaking of the hidden +sector gauge group via this scalar leads to new massive vector DM candidates. The first model we +consider is an U(1)D extension defined by the Langrangian [22], +LU(1)D = L0 +SM − 1 +4VµνV µν + (DµS)∗(DµS) − V (H, S), +(1) +where L0 +SM is the SM Lagrangian without the Higgs potential. +The covariant derivative is +Dµ = ∂µ + igVµ and the field strength tensor of the U(1)D vector field is Vµν = ∂µVν − ∂νVµ. +The scalar potential is given by +V (H, S) = 1 +6λH(H†H)2 + 1 +6λS(S∗S)2 + 2λHS(H†H)(S∗S). +(2) +In principle a kinetic mixing term BµνVµν could be present, but we assume this does not arise. For +example, the mixing term can be explicitly prohibited by a Z2 symmetry under which Vµ → −Vµ +and all other fields are singlets. In the unitary gauge the scalar fields are written as +H = 1 +√ +2 +� +0 +v1 + h1 +� +, +S = 1 +√ +2(v2 + h2), +(3) +and upon symmetry breaking vi, (i = 1, 2), becomes nonzero. The SM gauge boson masses are +determined by the vacuum expectation value (VEV) v1 = 246 GeV while the DM mass is related +to the VEV v2 via M 2 +V = g2v2 +2. +The second model we consider is the similar SU(2)D extension defined by the Langrangian [24] +LSU(2)D = L0 +SM − 1 +4V i +µνV µν +i ++ (DµS)†(DµS) − V (H, S), +(4) + +3 +where the DM candidate is now the SU(2)D vector triplet V i +µ. The covariant derivative and the +field strength tensor take the forms +Dµ = ∂µ + igV i +µti, +V i +µν = ∂µV i +ν − ∂νV i +µ + gϵi +jkV j +µ V k +ν , +(5) +where ti = σi/2 is the SU(2) generator. In this non-Abelian model, the kinetic mixing is forbidden +by gauge symmetry. The normalization of the scalar potential is here chosen as +V (H, S) = λH(H†H)2 + λS(S†S)2 + λHS(H†H)(S†S), +(6) +where the scalars are now both complex SU(2) doublets, and in the unitary gauge given by +H = 1 +√ +2 +� +0 +v1 + h1 +� +, +S = 1 +√ +2 +� +0 +v2 + h2 +� +. +(7) +In both of the above models the two neutral scalar states mix and the resulting mass eigenstates +are connected to the gauge eigenstates via a mixing matrix of the form +� +h +hS +� += +� +cos α − sin α +sin α +cos α +� � +h1 +h2 +� +, +(8) +where the mixing angle α describes the mixing between the SM and DM sectors. Generally, this +angle is restricted to small values, sin α ≲ 0.1. +The parameters of these models can be written in uniform notation as, +v2 = cV MV +g +, +sin α = v1 +v +(9) +λH = 3M 2 +h +v2 +1 +cos2 α, +λS = 3M 2 +h +v2 +2 +sin2 α, +λHS = − M 2 +h +2v1v2 +sin α cos α, +(10) +where MV is the mass of the DM candidate, Mh is the SM-Higgs mass and cV = 2 for the SU(2)D +model and cV = 1 for the U(1)D. We have also defined v2 = v2 +1 + v2 +2. +The tree-level potential has a flat direction along the scalon hS field direction, while the SM-like +Higgs h is perpendicular to the flat direction. We can thus consider the loop corrections in the flat +direction as per the Gildener-Weinberg formalism [13]. The first order loop corrections lead to an +effective potential of the general form, +V 1 +eff(hS) = +1 +64π2 +n +� +s=1 +gsM 4 +s +� +ln +�M 2 +s +Λ2 +� +− Ci +� +, +(11) +where Ms refers to tree level masses, gs is the degrees of freedom (with positive values for bosons +and negative for fermions), n is the number of states, and Λ is a renormalization scale. The scalon +field is massless at tree level, but obtains a mass from the loop corrections, given by +M 2 +S = +1 +8π2v2 +� +gV M 4 +V + 3M 4 +Z + 6M 4 +W + M 4 +h − 12m4 +t +� +, +(12) +where gV is the degrees of freedom for the vector boson: gV = 9 for the SU(2)D model and gV = 3 +for the U(1)D. Here MS is the scalon mass for each respective model and MV is the DM candidate. +Notice that Equation (12) relates the scalon and DM masses. In order for the scalon mass to be +non-negative, this sets a lower bound for the DM masses. The bound is MV > 240 GeV for the +SU(2)D model and MV > 185 GeV for the U(1)D model. + +4 +III. +FREEZE-OUT RELIC DENSITY +The dark matter abundance in the model is determined via the freeze-out mechanism. While +other possibilities, namely super-cool DM and filtered DM have been considered in the context of +radiative symmetry breaking models such as those under the present study [28–31], we will see +that the freeze-out mechanism is operational throughout the parameter space considered in this +work. +To see how the observed DM abundance Ωh2 = 0.120 ± 0.001 [3] is generated via the freeze-out +mechanism, we recall the basic formalism below. The present-day dark matter density is obtained +from the Boltzmann equation +dnV +dt + 3HnV = − ⟨σav⟩ +� +n2 +V − n2 +V,eq +� +, +(13) +where nV is the number density of the dark matter, which in equilibrium in the broken phase is +given as +neq +V (T) = gV +�MV T +2π +�3/2 +e− MV +T . +(14) +Here H is the Hubble parameter and ⟨σav⟩ is the thermally averaged annihilation cross section. +Equation (13) can be rewritten using entropy conservation, the yield YV = nV +s , and x = MV +T +into +the form +dYV +dx = +1 +3H +ds +dx ⟨σav⟩ +� +Y 2 +V − Y 2 +V,eq +� +, +(15) +and solving this equation we obtain the present day yield Y 0 +V that links to the abundance as +Ωh2 = MV s0Y 0 +V h2 +ρc +0 +≃ 2.755 · 108MV s0Y 0 +V GeV−1, +(16) +where +s0 = 2.8912 · 109 m−3, +ρc +0 = 10.537h2 GeVm−3 for H = h100 km/s/Mpc, +(17) +and h = 0.678. +To solve the Boltzmann equation numerically we use the micrOMEGA package [32]. This software +uses CalcHEP input files with the models Feynman rules to compute the thermally averaged cross +section, which we generate with the LanHEP package [33, 34]. The numerical results for the relic +density for both models can be seen in Figure 1. To asses the validity of the numerical results +we have compared these to the analytical result, obtained in the non-relativistic limit and under +the approximation of instantaneous freeze-out. Both of these approximations tend to overestimate +the relic density. Nevertheless, the analytical result only deviates up to around 10% for the U(1)D +model and slightly more for the SU(2)D model, considering only the leading annihilation processes +σ (V V → hShS) for the U(1)D model and σ (V iV j → hShS) plus the semi-annihilation process +σ +� +V iV j → V khS +� +for the SU(2)D model. +From Figure 1 it is evident that both models can reproduce the observed relic density. A larger +coupling g leads to more efficient annihilation of the vector DM candidate V into scalons hS and +thus the correct abundance is obtained for a correspondingly higher vector mass MV . In the non- +Abelian model the semi-annihilation process is taken into account in the analytic approximation +by defining the effective thermally averaged total annihilation cross section as +⟨σav⟩ = ⟨σannv⟩ + 1 +2 ⟨σsemi−annv⟩ , +(18) + +5 +(a) The DM relic density as a function of the mass of +the U(1)D vector DM candidate for different coupling +constants, including the Planck collaboration result. +(b) The DM relic density as a function of the mass of +the SU(2)D vector DM candidate for different coupling +constants, including the Planck collaboration result. +FIG. 1. The red line representing the Planck collaboration result of Ωh2 = 0.120 ± 0.001 is shown in red, +and both models can match it via a freeze-out relic density [3]. +where the first term is the annihilation and the second term is the semi-annihilation cross section. +The addition of the semi-annihilation generally leads to more efficient annihilation, and thus +one would expect the relic density to be lower. However, the SU(2)D result in Figure 1(b) is very +close to the U(1)D result in Figure 1(a), which indicates that there is not much difference in the +abundance for the two models considered. The origin of this is that that while the additional +degrees of freedom in the non-Abelian model increase the relic density, this is balanced by the +reducing effect of the semi-annihilations. Concretely, the semi-annihilations increases the overall +thermally averaged total annihilation cross section only by roughly 15%. +Finally, we comment on the possibility of a freeze-in origin for the DM abundance in these +models. In the freeze-in regime the DM particle V needs to be feebly coupled to the visible sector, +so that it does not reach equilibrium with the SM thermal bath in the early universe. To achieve +this, either the gauge coupling g needs to be very small so that the vector remains decoupled while +the scalar S is in equilibrium, or the portal coupling λHS can be very small, so that both the vector +and the scalar remain decoupled from the SM. +In the first scenario, the typical scale for the gauge coupling would be g ∼ O(10−7), as seen +from the approximate relation [35] +YV (T) ∼ g2Mpl +T , +(19) +where Mpl is the reduced Planck mass. Since this process is IR dominated, the dominant production +would be at the lowest kinematically allowed temperature T ∼ MV . Thus we can approximate the +abundance by the replacement T = MV in the above to obtain +Y 0 +V ∼ g2Mpl +MV +. +(20) +Consider now the relationship between the coupling and DM mass in Equations (10) and (12). +If the coupling is g ∼ O(10−7) as necessary for the freeze-in mechanism to work, the VEV, v2, + +RelicDensity of U(1)pModel +g= 0.5 +g = 0.7 +g= 0.9 +100 +Planck +10~1 +10~2 +250500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +Mv[GeV]RelicDensity of SU(2)p Model +101 +g=0.5 +g= 0.7 +g= 0.9 +Planck +100 +10-1 +10~2 +250500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +My[GeV]6 +becomes very large and the scalon mass, MS, is approximately zero. +The presence of a very +light scalar in the spectrum is potentially problematic, e.g. due to Higgs invisible decays, unless +suppressed by a small portal coupling. On the other hand, the scenario where the portal coupling +would be very small, would also require a large hidden sector VEV v2 ≫ v1. If the gauge coupling +is not very small, then this implies that the DM mass MV becomes very large. In this case the +hidden sector can only be effectively populated in the broken phase, as there is no scalar mixing in +the unbroken phase. However, in this scenario there will be large supercooling, as discussed below, +and the DM production should take place after reheating from thermal inflation. Now the scalar +VEV is mostly in the S-direction v2 ≫ v1, so that the energy stored in the inflaton field mostly +goes into S-quanta, but since these are feebly coupled to the SM, the reheating will be very slow +and the reheating temperature suppressed. Thus, the heavy DM can not be efficiently produced +after reheating, since Tr ≪ MV . While there might be some way to overcome these apparent +problems with freeze-in, we do not consider this scenario further in this work. +IV. +INFLATION, REHEATING AND SUPERCOOLING +In the previous section, we discussed the DM abundance in the standard freeze-out scenario. +The situation may however be more complicated [28, 30, 36, 37], due to a possible phase of thermal +inflation characteristic of classically scale invariant models with radiative symmetry breaking. The +thermal history in the models can be summarised in terms of the following temperature thresholds: +• TFO: The freeze-out temperature of the DM candidate defined roughly by neq +V ⟨σv⟩ = H. +• Tn: The nucleation temperature when the probability to nucleate an expanding bubble of +the broken phase vacuum inside a Hubble horizon becomes of O(1), approximately the +temperature at which the phase transition completes. +• Tinf: The temperature at the beginning of thermal inflation defined by ρV = ρrad, where +ρV is the energy density of the false unbroken vacuum (i.e. the difference in the potential +between the local minimum at V (S) = 0 and the true minimum at V (S = v)), and ρrad is +the energy density of the radiation dominated universe. When ρV begins to dominate the +energy density, inflation begins. +In the case of the two vector DM models discussed in this paper, the finite temperature potential +includes the thermal integral summing over the bosons and fermions [38], +V 1 +eff(hS, T) = +n +� +s=1 +gs +� +1 +64π2M 4 +s +� +ln +�M 2 +s +Λ2 +� +− Ci +� ++ T 4 +2π2 +� ∞ +0 +x2 ln +� +1 ∓ e−√ +x2±M2s /T 2� +dx +� +. +(21) +For some models, it might be necessary to consider the additional ring diagrams for the bosons, but +for this investigation they can be ignored as they are insignificant [39]. This thermal potential is +not amenable to an analytic solution, but can be approximated using modified Bessel functions of +the second kind [23]. We compute the freeze-out temperature, TFO, numerically with micrOMEGA, +and the nucleation temperature, Tn, numerically using CosmoTransitions and Bubbleprofiler +(for cross-checking) [32, 40, 41]. +Let us now consider the thermal history of the model depending on the order of the above three +temperature thresholds. If Tn > TFO, the phase transition completes before DM freeze-out, and +the freeze-out then takes place as usual in the broken phase. This means that we can calculate +the relic abundance as presented in the previous section. +In the opposite case, Tn < TFO there are three scenarios to consider. The filtered DM scenario +takes place for the ordering TFO > Tn > Tinf. In this situation, there is no thermal inflation, as the + +7 +phase transition completes before inflation would begin, but the DM annihilations are immediately +out of equilibrium after the phase transition, and therefore the abundance is set by the amount of +DM particles that are able to enter the boundary to the broken phase, as described in [30]. +The supercool DM scenario [28], takes place for TFO > Tinf > Tn. In this situation, there is a +period of thermal inflation, which ends at Tn. After inflation, the latent heat stored in the false +vacuum is released to reheat the universe back to temperature Tinf, under the assumption of instant +reheating, or to a lower reheating temperature for delayed reheating. However, since TFO > Tinf, +no DM is produced in reheating and the abundance is set by the amount that was present before +inflation, diluted by the expansion of the scale factor and by the filtering effect as in the above +scenario. +Finally, there is the case where Tinf > TFO. In this situation, assuming instant reheating, the +reheating will bring DM back to equilibrium, and the relic abundance is again obtained via the +usual freeze-out mechanism as presented in the previous section. +The inflation temperature is obtained by solving for Tinf from +∆V (Tinf) = V high +eff +(hS, Tinf) − V low +eff (0, Tinf) = g∗π2 +30 T 4 +inf, +(22) +where V high +eff +(hS, T) is the true vacuum and V low +eff (0, T) is the false vacuum. We find that throughout +the parameter space of interest in this work, we are either in the first or the last situation described +above, and the DM abundance is thus obtained via the usual freeze-out mechanism in both cases. +See Appendix A for more on the reheating. +V. +DIRECT DETECTION +In this section, we present the direct detection constraints on the two models. We will see +that the recent results from the LZ experiment significantly affect the SU(2)D model and that the +U(1)D model is already very constrained. +To compute the direct detection cross section, we again use the micrOMEGA package [32]. The +DM coupling to nucleons arises from the scalar mixing and is mediated via exchange of the SM-like +Higgs and the scalon in the t-channel leading to a spin-independent cross section with negligible +difference between protons and neutrons. The results of this computation for both models are +shown in Figure 2. The correct relic abundance is obtained along the red solid line. +The purple region is excluded by LHC constraints on to Higgs decays into two scalons [44, 45]. +This process becomes kinetically forbidden for larger DM mass, as larger DM mass leads to larger +scalon mass as shown in Equation (12). In the orange region, the DM-nucleon cross section is below +the neutrino floor, and the yellow regions indicate the exclusion limit due to the LZ experiment +[26], providing a significant improvement over the XENON1T experiment shown in green [42]. +Finally, the grey region shows the projected exclusion limit from XENONnT [43]. +Starting with the U(1)D model we see a small gap in the direct detection limits at the resonance +region, Mh ≃ 2MV , where the DM mass is around 0.9-1 TeV and the coupling 0.65 ≤ g ≤ 0.7. In +the middle of this range the direct detection cross section falls below the neutrino floor. Outside of +the resonance region, the model can not produce an O(1)-fraction of DM without being excluded +by direct detection, unless the DM mass is well above 10 TeV. +For the SU(2)D model the new constraints from LZ alter the picture compared to the situation +with the previous XENON1T limits: the relic abundance line above the resonance region is now +excluded for DM masses below 7.5 TeV, while prior to the LZ result there were no constraints +beyond 1 TeV. In the resonance region, we find the nucleation temperature for the phase transition +below the QCD-scale. This alters the computation for the gravitational wave signal, as the phase +transition will be completed in conjunction with the QCD phase transition, as discussed in [46, 47]. +This picture slightly changes when including additional scalar self-energy corrections for the SU(2) + +8 +(a) Constraints for the the U(1)D model. +(b) Constraints for the the SU(2)D model. +FIG. 2. The red line shows the correct relic abundance, Ωh2 = 0.12 [3]. The yellow region is excluded by +the LZ (2022) experiment [26], the green region is the XENON1T experiment [42], the purple region is +the LHC constraint for exotic Higgs decay, the orange region is the neutrino floor and the gray region is +the projected 90% CL constraint from the XENONnT experiment [43]. +model [48]. First, the scalon mass is slightly larger than in our leading order analysis, pushing +the resonance region in figure 2(b) to the right. Additionally, the correction appears to slightly +increase the nucleation temperature compared to our results. However, we find that overall the +resulting gravitational wave (GW) signal is not significantly affected, and the GW signal prediction +remains comparable to our results presented in the next section. +For DM mass above 7.5 TeV the model is again allowed by direct detection. In Figure 2 we +have marked three benchmark points allowed by direct detection with the blue, indigo, and purple +markers. These points will be used as examples for analyzing the GW signals in the next section. +VI. +GRAVITATIONAL WAVES +The strongly first order phase transition possible in classically scale invariant models is inter- +esting due to the implications for baryogenesis [49], and due to potentially observable gravitational +wave (GW) signal. +To explore the gravitational wave signals, we consider the finite temperature potential in Equa- +tion (21). This potential contains a barrier between the unbroken false vacuum and the broken +phase minimum, leading to a first order phase transition. At the nucleation temperature Tn, the +phase transition will complete via the formation of bubbles of the true vacuum. The expanding +and colliding bubbles deposit energy in the surrounding plasma, generating gravitational waves as +described in [50–52]. +For the purpose of solving Equation +(21) and obtaining the parameters that describe the +gravitational wave signal, we use the Python package CosmoTransitions[40], with custom modifi- +cations including a method of computing the β value. The relevant parameters are the latent heat +normalized with respect to the radiation energy, α, the inverse duration of the phase transition, + +Constraints ofU(1)p Model +1.0 +Qh2 = 0.12 +0.9 +0.8 +9 0.7 +LZ(2022) +0.6 +LHC +KENONIT +0.5 +XENONnT +0.4 +300 +1000 +2000 +3000 +4000 +Mv[GeV]ConstraintsofSU(2)pModel +2.0 +Qh2 = 0.12 +1.8 +1.6 +1.4 +1.0 +XENONIT +0.8 +0.6 +0.4 +300 +1000 +2000 +4000 +8000 +My[GeV]9 +Model +Benchmark point Parameter CosmoTransitions BubbleProfiler +U(1)D +g = 0.66 +MV = 911 GeV +Tc = 303 GeV +α +20740 +92180 +β +23.8 +39.2 +Tn +7.04 GeV +4.78 GeV +U(1)D +g = 0.7 +MV = 1028 GeV +Tc = 336 GeV +α +1497 +4597 +β +36.8 +49.5 +Tn +15.3 GeV +11.4 GeV +SU(2)D +g = 2.0 +MV = 7530 GeV +Tc = 2345 GeV +α +0.16 +0.22 +β +289 +301 +Tn +1430 GeV +1446 GeV +. +FIG. 3. Table with benchmark points used for the discussion of gravitaitonal wave signals. The two first +benchmark points are from the U(1)D model and the last is from the SU(2)D model. +β, and the nucleation temperature, Tn, defined as [24, 53], +α ≡ 1 +ρ +� +∆V − T +4 +d∆V +dT +� ���� +Tn +, +β +H ≡ T d(S/T) +dT +���� +Tn +, +(23) +where, +∆V = V high +eff +(hS, T) − V low +eff (hS, T), +ρ = geπ2 +30 T 4 +n, +(24) +where the ge ≈ 103 is the number of effective degrees of freedom during the nucleation at the +temperature Tn. Finally the Euclidean action is defined as, +S = 4π +� ∞ +0 +r2 +� +1 +2 +�dhφ/S +dr +�2 ++ Veff(hφ/S) +� +dr, +(25) +where r is the radial distance from the center of the true vacuum bubble. +In order to assess the reliability of the results, we make use of two different numerical tools +for computing the nucleation temperature and the β and α parameters. The parameters α and β +depend heavily on the nucleation temperature, Tn, so that possible errors on Tn will propagate to +α and β. For the computation we use CosmoTransitions and BubbleProfiler[40, 41]. As shown +in the appendix, we obtain a smaller numerical error with CosmoTransitions, but the results of +both numerical computations agree within uncertainty. In general, we find that for sub-TeV DM +masses the nucleation temperature in the BubbleProfiler implementation tends to be smaller +than in CosmoTransitions. +In Figure 2, we identify three benchmark points allowed by all constraints. These benchmark +points are shown in 3 corresponding to the indigo diamond, blue square and purple hexagon shown +in Figure 2. +Notice that the first point is below one TeV, the trend we observed regarding the performance of +the two simulation tools is noticeable, and the BubbleProfiler nucleation temperature is signifi- +cantly below the value obtained from CosmoTransitions, affecting also the α and β parameters. +At this point, the critical temperature is Tc = 303. +In summary, both CosmoTransitions and BubbleProfiler show similar behavior for both +models and are in reasonable agreement. For high masses the latter tool yields slightly higher +nucleation temperatures and therefore α is also a bit lower and β as indicated by Equation 23. + +10 +Having computed the relevant parameters for calculating gravitational waves (GW) spectra, we +can consider the following equation for computing the total signal, +Ωtoth2 = Ωcolh2 + Ωswh2 + Ωturbh2, +(26) +where the first term is the collision term, the second term is the sound wave term, and the last +term is the turbulence term. The collisions from the bubbles themselves contribute to the GW +spectra, but they do not give the most significant contribution. The collisions also produce bulk +motion in the fluid. This causes sound waves that result in the primary contribution to the GW +spectra. Finally, there is also some turbulence caused by the collisions which contribute to the +GW spectra [23, 52, 54]. The relevant equations for computing the collision term are, +Ωcolh2(f) = 1.67 · 10−5 +� +α +1 + α +�2 H2 +β2 +� ge +100 +�− 1 +3 0.11κ2 +vv3 +b +0.42 + v2 +b +Scol +Scol(f) = +3.8 +� +f +fcol +�2.8 +2.8 +� +f +fcol +�3.8 ++ 1 +fcol = 16.5 µHz +0.62 +v2 +b − 0.1vb + 1.8 +β +H +Tn +100 GeV +� ge +100 +� 1 +6 , +(27) +similarly, the equations for the sound wave term are +Ωswh2(f) = 2.65 · 10−6 +� +α +1 + α +�2 H +β +� ge +100 +�− 1 +3 κ2 +vvbSsw +Ssw(f) = +� f +fsw +�3 +� +� +� +7 +3 +� +f +fsw +�2 ++ 4 +� +� +� +3.5 +fsw = 19 µHz 1 +vb +β +H +Tn +100 GeV +� ge +100 +� 1 +6 , +(28) +and lastly, the equations for the turbulence term are +Ωturbh2(f) = 3.35 · 10−4 +�κturbα +1 + α +� 3 +2 H +β +� ge +100 +�− 1 +3 vbSturb +Sturb(f) = +� +f +fturb +�3 +� +1 + 8πf +h∗ +� � +1 + +f +fturb +� 11 +3 +fturb = 22.7 µHz 1 +vb +β +H +Tn +100 GeV +� ge +100 +� 1 +6 , +(29) +where the inverse Hubble time, h∗, red-shifted to today, at the GW production is given as +h∗ = 1.65 · 10−5 +Tn +100 GeV +� ge +100 +� 1 +6 , +(30) + +11 +FIG. 4. The GW spectra for two different sets of transition parameters for the U(1)D model and one +for the SU(2)D model (g = 2.0, MV = 7530) computed with CosmoTransitions, dashed lines, and +BubbleProfiler, dotted lines. The sensitivity curves (C1-C4) of the LISA detector are also shown [27]. +According to this result, the signals from this model are strong enough for LISA to detect the GW signal +from the phase transition. +and the two modified efficiency factors can be written as, +κv = +α +0.73 + 0.083√α + α, +κturb = 0.05κv. +(31) +The result of this computation can be seen in Figure 4 for the three benchmark points, two +from the U(1)D model and one from the SU(2)D. +The marker shape indicates the parameter as shown in Figure 2. The diamond and square +shapes are from the U(1)D model. For SU(2)D model we have the high mass case marked by +the hexagon shape. The projected sensitivity curves (for the configurations C1-C4) for the LISA +detector are also shown [27], and one can see that for the U(1)D model the signal should be +detectable by three out of four configurations, but for the SU(2)D model the mass becomes too +high and we need other future experiments to detect such high DM mass models such as the +proposed TianQin detector [55]. +VII. +DISCUSSION AND CONCLUSIONS +We have investigated two vector DM models in light of existing DM direct detection experiments +and future GW experiments. Both of the models investigated in this work are already strongly +constrained by direct detection. For the SU(2)D model this is in particular due to recent results + +Gravitational wavespectra of selectedpoints +10-5 +10 +10~9 +yo +10-11 +10-13 +C1 +g = 0.66 My= 911 [GeV] +10-15 +C2 +g= 0.70 My = 1028 [GeV] +C3 +g = 2.0 My = 7530 [GeV] +C4 +10~17 +10-4 +10-3 +10-2 +10-1 +f[Hz]12 +from the LZ experiment which has ruled out most of parameter space consistent with a full relic +abundance from freeze-out in the range MV ∈ (1 − 10) TeV and with XENONnT the DM will +either be detected or the entire parameter space above the neutrino floor will be ruled out as shown +in Figure 2. +GW signals in both models have been discussed in earlier literature. In our analysis we find +that results differ significantly between different numerical implementations. Recently, the SU(2)D +model was discussed in [48], and we find that their results for the α and Tn parameters agree with +our findings. +Regarding the U(1)D model, it was previously suggested that GW signals could be used to probe +the model in case the direct detection cross section remains below the neutrino floor [23]. We agree +with this conclusion, but numerically we find differences to [23] in the GW parameters. While we +can reproduce the critical temperature reported, the nucleation temperature and the α and β +parameters differ from those reported in [23]. Their results were obtained with the AnyBubble +package [56], for which we failed to obtain results in agreement with the other two numerical +implementations used in this work. +This raises the question of comparability between the phase transition parameters obtained via +the various numerical implementations available. This issue has been investigated in [41], where +a fairly good agreement between BubbleProfiler and CosmoTransitions is observed. This is +compatible with our findings. +The finite temperature potential in both cases leads to a strong first order electroweak phase +transition. The U(1)D model can produce significant GW signals, which can be detected by LISA +[27] and future experiments would be able to test the SU(2)D model also in the high DM mass +regime. +ACKNOWLEDGMENTS +The financial support from Academy of Finland, project #342777, is gratefully acknowledged. +MTF and MR acknowledge partial funding from The Independent Research Fund Denmark, grant +numbers DFF 6108-00623 and DFF 1056-00027B, respectively. MET acknowledges funding from +Augustinus Fonden, application #22−19584, to cover part of the expenses associated with visiting +the University of Helsinki for half a year. +Appendix A: Supercooling, inflation and reheating +The investigation of the GW spectra leads to the discussion of supercooling in the models +presented. As shown in the GW section there are orders of magnitude in the difference between +the critical and nucleation temperature at the low mass scale. As discussed in the other papers, this +can lead to different kinds of phenomena including inflation, filtering and, reheating [28, 30, 37]. +These effects are expected to affect the GW signal for low masses, and it might effects some of +the results even presented in Figure 4, but it is beyond this paper to look at the details of this. +As discussed in a recent paper, the universe could escape inflation via bubble nucleation or via +quantum tunneling, two different scenarios leading to different GW signals [36]. +We would however like to highlight the fact that strong supercooling from hundreds of GeV to +the QCD scale might not be a big issue for the models. The bigger the supercooling the greater +the inflation as the scalon Higgs field will be stuck in a false vacuum acting like a cosmological +constant. The main constraint for any possible is lover than the max number of e-folds, +Nmax = 23.8 + ln TR +TeV, +(A1) + +13 +where TR is the reheating temperature after the inflationary epoch and one finds that this limits the +temperature to TR < 6.6· 1015 GeV [36]. To compute the reheating temperature, we are interested +in computing the decay of the inflaton-like field which in this case is the scalon Higgs field S for the +U(1)D model. Due to the mass constraints, only the scalon Higgs is kinetically allowed to decay +as Γ(hS → h, h), but this requires a DM mas of MV > 1 TeV. From the Lagrangian, we find that +the Feynman rule for this vertex and this yields the decay, +Γ(hS → 2h) = +� +M 2 +S − 4M 2 +h +32πM 2 +S +|M|2, +(A2) +where, +|M|2 = +�M 2 +h +4v1 +(5 + 3 cos(4α)) sin(α) +�2 +. +(A3) +We can furthermore include decays into quarks and leptons, +Γ(hS → f ¯f/ℓ¯ℓ) = NC +8π +m2 +b +v2 +1 +MS +� +1 − 4m2 +b +M 2 +φ +sin(α)2. +(A4) +where NC = 3 for fermions and NC = 1 for leptons. Using the decays one can calculate the +reheating temperature, TR, using the following equation [57], +TR ≈ 0.2 +�200 +g∗ +�1/4 � +ΓMpl, +(A5) +where Mpl is the reduced Planck mass and g∗ = 103. +Considering a rather low mixing value +0 ≤ α ≤ +π +64 and a mass range of 250 GeV ≤ MV ≤ 2500 GeV the reheating temperature is +somewhere around 0.1-1.6 PeV yielding mass of the scalon field around 1 GeV < MS < 200 GeV. +This is so hot that the universe will reheat back to a temperature much hotter than the scales +of freeze-out. It also satisfies the constraint from Equation (A1), thus it is not too hot and not +causing too much inflation. One can repeat this exercise for the SU(2)D, but the result is roughly +the same with the main differences being a slightly heavier scalon mass, 1 GeV < MS < 350 GeV, +and higher reheating temperature 0.1-2 PeV. Conclusively, dark matter production can take place +via freeze-out as the universe subsequently cools down again. +Appendix B: Model implementation in CosmoTransitions +For the implementation of the model in CosmoTransitions we feed in the tree level potentials +as shown in Equation (2) and (6). Then we manually implement the mass matrix with the massive +SM bosons, plus the new bosons, and the top quark, +M 2 +W = g2 +W +4 h2 +1, +M 2 +Z = g2 +W + g2 +Z +4 +h2 +1, +m2 +t = λ2 +t +2 h2 +1, +(B1) +where the Yukawa coupling of the top quark is λt = 1. The DM candidate have their respective +implementations for each model where, +M 2 +V = cV g2h2 +2, +(B2) + +14 +with cV = 1 (1/4) for the U(1)D (SU(2)D) model, and then the scalar mass matrices yield, +M 2 +h,S(U(1)D) = 1 +4 +� +h2 +1 (λh + 2λφh) + h2 +2(λφ + 2λφh) +± +� +h4 +1 (λh − 2λφh)2 + h4 +2 (λφ − 2λφh)2 + 2h2 +1h2 +2 +� +2λhλφh + 28λ2 +φh − λhλφ + 2λφλφh +� +� +(B3) +M 2 +h,S(SU(2)D) = 1 +4 +� +h2 +1 (6λH + λHS) + h2 +2(6λS + λHS) +± +� +h4 +1 (λHS − 6λH)2 + h4 +2 (λHS − 6λS)2 + 2h2 +1h2 +2 (6λH(λHS − 6λHS) + λHS(7λHS + 6λS)) +� +, (B4) +A custom solution is made for computing the beta value. This is done by simply calculating the +action divided by the temperature around the point of the nucleation temperature, making a fit +to those plots, and taken the derivative etc. Some tweaks have been done to the source code to +make this work, and also to improve the precision at low nucleation temperatures. +Appendix C: Model implementation in BubbleProfiler +For this package, we give it the full thermal potential in Equation (21), but instead of evaluating +the thermal integral an approximation is made using Bessel function[23]. Specifically we use the +modified Bessel functions, K2(kx), as follows +� ∞ +0 +x2 ln +� +1 ∓ e−√ +x2±M2s /T 2� +dx = − +3 +� +k=1 +x2 +k2K2(kx) − +2 +� +k=1 +(−1)kx2 +k2 +K2(kx). +(C1) +The BubbleProfiler is written in C++, but comes with a command line interface (CLI). Using +this one can implement simple potentials like polynomials. In order to avoid implementing all +these functions, we created a python interface where we implement the model in python. Then +we create a higher order polynomial fit to the full potential. +This polynomial is then fed to +BubbleProfiler via the CLI together with other relevant parameters. We compute several points +around the nucleation temperature and make a fit to that and from there we determine the β and +Tn value, and the latter is then used to find α. +Appendix D: Computing parameters of the EWPT +The essential computation for the phase transition is finding the relationship between the action +and temperature. The nucleation temperature condition is defined as +S(T) +T +���� +Tn +≈ 140, +(D1) +thus when the action divided by the temperature is equal to 140. 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Reheating and preheating after inflation : an introduction, 2010. +https://www.desy.de/~westphal/workshop_seminar_fall_2010/reheating.pdf. + diff --git a/5NAyT4oBgHgl3EQfQPZ4/content/tmp_files/load_file.txt b/5NAyT4oBgHgl3EQfQPZ4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa057be82a4096de15ab72fe55b2ef449427683d --- /dev/null +++ b/5NAyT4oBgHgl3EQfQPZ4/content/tmp_files/load_file.txt @@ -0,0 +1,750 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf,len=749 +page_content='HIP-2022-35/TH Vector dark matter in supercooled Higgs portal models Mads T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Frandsen∗ and Mattias E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Thing† CP3-Origins, University of Southern Denmark, Denmark Matti Heikinheimo‡ and Kimmo Tuominen§ Department of Physics, University of Helsinki, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='Box 64, FI-00014 University of Helsinki, Finland and Helsinki Institute of Physics, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='Box 64, FI-00014 University of Helsinki, Finland Martin Rosenlyst¶ Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom We consider extensions of the Standard Model by a hidden sector consisting of a gauge field coupled with a scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Assuming the absence of dimensionful param- eters in the tree level potential, radiative symmetry breaking will make the hidden sector gauge field massive and induce the electroweak scale of the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' We consider separately dark sector gauge groups U(1)D and SU(2)D, and focus on probing the models with a combination of direct detection experiments and gravita- tional wave observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' We find that recent dark matter direct detection results significantly constrain the parameter space of the models where they can account for the observed dark matter relic density via freeze-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The gravitational wave signals originating from strongly first order electroweak phase transition in these models can be probed in future gravitational wave observatories such as LISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' We show how the projected results compliment direct detection experiments and can help probe parameter space near the neutrino floor of direct detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' INTRODUCTION Despite the success of the Standard Model (SM) of particle physics, there are many phenomena that it does not explain and that appear to require new particles and interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' One enigmatic phenomenon is the problem of missing mass, which emerged in a wide range of astrophysical systems including galaxy clusters [1] and galaxies [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' One possible solution to the missing mass problem is cold dark matter (DM), constituted by a new stable and neutral massive particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' This hypothesis provides an excellent parametrisation for 26% of the energy density of the universe in addition to the components parametrised as baryonic matter and dark energy [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' On the other hand, the non-gravitational nature of dark matter (DM) remains unknown [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The cosmological observations on the light element abundance and cosmic microwave back- ground radiation spectrum imply that the Standard Model (SM) degrees of freedom must have been in thermal equilibrium in the early universe [7–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Whether DM was ever part of the same heat bath is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' However, assuming that this was the case, allows for the abundance of dark matter to arise as a relic from thermal decoupling in the early universe via interactions between the DM and the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' ∗ frandsen@cp3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='dk † thing@cp3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='dk ‡ matti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='heikinheimo@helsinki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='fi § kimmo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='tuominen@helsinki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='fi ¶ martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='jorgensen@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='uk arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='00041v1 [hep-ph] 30 Dec 2022 2 Moreover, these interactions offer the prospect of detecting DM in direct detection experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The most studied example of this paradigm is Weakly Interacting Massive Particle (WIMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' How- ever, simplest WIMP models are now very strongly constrained by direct detection experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' It is therefore worthwhile to explore the phenomenology of different types of simple benchmark hidden sectors instead coupled with the SM via portal interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In this paper we analyze two simple models of vector DM, that feature scale invariance of the tree-level Lagrangian and are coupled to the SM via the Higgs portal, where one scalar mass eigenstate is SM-like, with mass 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='16 GeV [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The other eigenstate is massless at tree level but obtains its mass via loop corrections as an effect of radiative symmetry breaking [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' This framework of classically scale invariant DM models that feature radiative symmetry breaking, mediated to the SM via the Higgs portal, has been explored in literature, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' [14–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In this paper we aim to clarify how simple U(1)D and SU(2)D models of this type can be tested with a combination of direct detection and gravitational wave observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Direct detection experiments have provided very stringent constraints on interactions of weak scale dark matter with nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Currently, the most stringent constraints come from the recent PandaX-4T and LZ (2022) experiments [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' It is well known that radiative symmetry breaking in classically scale invariant models typically results in a strongly first order electroweak phase transition (EWPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Such a first order EWPT could be relevant for baryogenesis and produces gravitational wave signals which could be observable in upcoming gravitational wave experiments such as LISA [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' We present a careful examination of the first order phase transition using different numerical packages in order to characterise the theoretical uncertainty in the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' DEFINITIONS OF THE MODELS We consider two models where the SM is extended with a hidden sector gauge group and a new scalar field charged under the gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Spontaneous symmetry breaking of the hidden sector gauge group via this scalar leads to new massive vector DM candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The first model we consider is an U(1)D extension defined by the Langrangian [22], LU(1)D = L0 SM − 1 4VµνV µν + (DµS)∗(DµS) − V (H, S), (1) where L0 SM is the SM Lagrangian without the Higgs potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The covariant derivative is Dµ = ∂µ + igVµ and the field strength tensor of the U(1)D vector field is Vµν = ∂µVν − ∂νVµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The scalar potential is given by V (H, S) = 1 6λH(H†H)2 + 1 6λS(S∗S)2 + 2λHS(H†H)(S∗S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' (2) In principle a kinetic mixing term BµνVµν could be present, but we assume this does not arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' For example, the mixing term can be explicitly prohibited by a Z2 symmetry under which Vµ → −Vµ and all other fields are singlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In the unitary gauge the scalar fields are written as H = 1 √ 2 � 0 v1 + h1 � , S = 1 √ 2(v2 + h2), (3) and upon symmetry breaking vi, (i = 1, 2), becomes nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The SM gauge boson masses are determined by the vacuum expectation value (VEV) v1 = 246 GeV while the DM mass is related to the VEV v2 via M 2 V = g2v2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The second model we consider is the similar SU(2)D extension defined by the Langrangian [24] LSU(2)D = L0 SM − 1 4V i µνV µν i + (DµS)†(DµS) − V (H, S), (4) 3 where the DM candidate is now the SU(2)D vector triplet V i µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The covariant derivative and the field strength tensor take the forms Dµ = ∂µ + igV i µti, V i µν = ∂µV i ν − ∂νV i µ + gϵi jkV j µ V k ν , (5) where ti = σi/2 is the SU(2) generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In this non-Abelian model, the kinetic mixing is forbidden by gauge symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The normalization of the scalar potential is here chosen as V (H, S) = λH(H†H)2 + λS(S†S)2 + λHS(H†H)(S†S), (6) where the scalars are now both complex SU(2) doublets, and in the unitary gauge given by H = 1 √ 2 � 0 v1 + h1 � , S = 1 √ 2 � 0 v2 + h2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' (7) In both of the above models the two neutral scalar states mix and the resulting mass eigenstates are connected to the gauge eigenstates via a mixing matrix of the form � h hS � = � cos α − sin α sin α cos α � � h1 h2 � , (8) where the mixing angle α describes the mixing between the SM and DM sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Generally, this angle is restricted to small values, sin α ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The parameters of these models can be written in uniform notation as, v2 = cV MV g , sin α = v1 v (9) λH = 3M 2 h v2 1 cos2 α, λS = 3M 2 h v2 2 sin2 α, λHS = − M 2 h 2v1v2 sin α cos α, (10) where MV is the mass of the DM candidate, Mh is the SM-Higgs mass and cV = 2 for the SU(2)D model and cV = 1 for the U(1)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' We have also defined v2 = v2 1 + v2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The tree-level potential has a flat direction along the scalon hS field direction, while the SM-like Higgs h is perpendicular to the flat direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' We can thus consider the loop corrections in the flat direction as per the Gildener-Weinberg formalism [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The first order loop corrections lead to an effective potential of the general form, V 1 eff(hS) = 1 64π2 n � s=1 gsM 4 s � ln �M 2 s Λ2 � − Ci � , (11) where Ms refers to tree level masses, gs is the degrees of freedom (with positive values for bosons and negative for fermions), n is the number of states, and Λ is a renormalization scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The scalon field is massless at tree level, but obtains a mass from the loop corrections, given by M 2 S = 1 8π2v2 � gV M 4 V + 3M 4 Z + 6M 4 W + M 4 h − 12m4 t � , (12) where gV is the degrees of freedom for the vector boson: gV = 9 for the SU(2)D model and gV = 3 for the U(1)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Here MS is the scalon mass for each respective model and MV is the DM candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Notice that Equation (12) relates the scalon and DM masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In order for the scalon mass to be non-negative, this sets a lower bound for the DM masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The bound is MV > 240 GeV for the SU(2)D model and MV > 185 GeV for the U(1)D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' 4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' FREEZE-OUT RELIC DENSITY The dark matter abundance in the model is determined via the freeze-out mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' While other possibilities, namely super-cool DM and filtered DM have been considered in the context of radiative symmetry breaking models such as those under the present study [28–31], we will see that the freeze-out mechanism is operational throughout the parameter space considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' To see how the observed DM abundance Ωh2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='120 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='001 [3] is generated via the freeze-out mechanism, we recall the basic formalism below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The present-day dark matter density is obtained from the Boltzmann equation dnV dt + 3HnV = − ⟨σav⟩ � n2 V − n2 V,eq � , (13) where nV is the number density of the dark matter, which in equilibrium in the broken phase is given as neq V (T) = gV �MV T 2π �3/2 e− MV T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' (14) Here H is the Hubble parameter and ⟨σav⟩ is the thermally averaged annihilation cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Equation (13) can be rewritten using entropy conservation, the yield YV = nV s , and x = MV T into the form dYV dx = 1 3H ds dx ⟨σav⟩ � Y 2 V − Y 2 V,eq � , (15) and solving this equation we obtain the present day yield Y 0 V that links to the abundance as Ωh2 = MV s0Y 0 V h2 ρc 0 ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='755 · 108MV s0Y 0 V GeV−1, (16) where s0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='8912 · 109 m−3, ρc 0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='537h2 GeVm−3 for H = h100 km/s/Mpc, (17) and h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='678.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' To solve the Boltzmann equation numerically we use the micrOMEGA package [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' This software uses CalcHEP input files with the models Feynman rules to compute the thermally averaged cross section, which we generate with the LanHEP package [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The numerical results for the relic density for both models can be seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' To asses the validity of the numerical results we have compared these to the analytical result, obtained in the non-relativistic limit and under the approximation of instantaneous freeze-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Both of these approximations tend to overestimate the relic density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Nevertheless, the analytical result only deviates up to around 10% for the U(1)D model and slightly more for the SU(2)D model, considering only the leading annihilation processes σ (V V → hShS) for the U(1)D model and σ (V iV j → hShS) plus the semi-annihilation process σ � V iV j → V khS � for the SU(2)D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' From Figure 1 it is evident that both models can reproduce the observed relic density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' A larger coupling g leads to more efficient annihilation of the vector DM candidate V into scalons hS and thus the correct abundance is obtained for a correspondingly higher vector mass MV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In the non- Abelian model the semi-annihilation process is taken into account in the analytic approximation by defining the effective thermally averaged total annihilation cross section as ⟨σav⟩ = ⟨σannv⟩ + 1 2 ⟨σsemi−annv⟩ , (18) 5 (a) The DM relic density as a function of the mass of the U(1)D vector DM candidate for different coupling constants, including the Planck collaboration result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' (b) The DM relic density as a function of the mass of the SU(2)D vector DM candidate for different coupling constants, including the Planck collaboration result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The red line representing the Planck collaboration result of Ωh2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='120 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='001 is shown in red, and both models can match it via a freeze-out relic density [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' where the first term is the annihilation and the second term is the semi-annihilation cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The addition of the semi-annihilation generally leads to more efficient annihilation, and thus one would expect the relic density to be lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' However, the SU(2)D result in Figure 1(b) is very close to the U(1)D result in Figure 1(a), which indicates that there is not much difference in the abundance for the two models considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The origin of this is that that while the additional degrees of freedom in the non-Abelian model increase the relic density, this is balanced by the reducing effect of the semi-annihilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Concretely, the semi-annihilations increases the overall thermally averaged total annihilation cross section only by roughly 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Finally, we comment on the possibility of a freeze-in origin for the DM abundance in these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In the freeze-in regime the DM particle V needs to be feebly coupled to the visible sector, so that it does not reach equilibrium with the SM thermal bath in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' To achieve this, either the gauge coupling g needs to be very small so that the vector remains decoupled while the scalar S is in equilibrium, or the portal coupling λHS can be very small, so that both the vector and the scalar remain decoupled from the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In the first scenario, the typical scale for the gauge coupling would be g ∼ O(10−7), as seen from the approximate relation [35] YV (T) ∼ g2Mpl T , (19) where Mpl is the reduced Planck mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Since this process is IR dominated, the dominant production would be at the lowest kinematically allowed temperature T ∼ MV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Thus we can approximate the abundance by the replacement T = MV in the above to obtain Y 0 V ∼ g2Mpl MV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' (20) Consider now the relationship between the coupling and DM mass in Equations (10) and (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' If the coupling is g ∼ O(10−7) as necessary for the freeze-in mechanism to work, the VEV, v2, RelicDensity of U(1)pModel g= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='5 g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='7 g= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='9 100 Planck 10~1 10~2 250500 1000 1500 2000 2500 3000 3500 4000 Mv[GeV]RelicDensity of SU(2)p Model 101 g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='5 g= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='7 g= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='9 Planck 100 10-1 10~2 250500 1000 1500 2000 2500 3000 3500 4000 My[GeV]6 becomes very large and the scalon mass, MS, is approximately zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The presence of a very light scalar in the spectrum is potentially problematic, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' due to Higgs invisible decays, unless suppressed by a small portal coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' On the other hand, the scenario where the portal coupling would be very small, would also require a large hidden sector VEV v2 ≫ v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' If the gauge coupling is not very small, then this implies that the DM mass MV becomes very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In this case the hidden sector can only be effectively populated in the broken phase, as there is no scalar mixing in the unbroken phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' However, in this scenario there will be large supercooling, as discussed below, and the DM production should take place after reheating from thermal inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Now the scalar VEV is mostly in the S-direction v2 ≫ v1, so that the energy stored in the inflaton field mostly goes into S-quanta, but since these are feebly coupled to the SM, the reheating will be very slow and the reheating temperature suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Thus, the heavy DM can not be efficiently produced after reheating, since Tr ≪ MV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' While there might be some way to overcome these apparent problems with freeze-in, we do not consider this scenario further in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' INFLATION, REHEATING AND SUPERCOOLING In the previous section, we discussed the DM abundance in the standard freeze-out scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The situation may however be more complicated [28, 30, 36, 37], due to a possible phase of thermal inflation characteristic of classically scale invariant models with radiative symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The thermal history in the models can be summarised in terms of the following temperature thresholds: TFO: The freeze-out temperature of the DM candidate defined roughly by neq V ⟨σv⟩ = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Tn: The nucleation temperature when the probability to nucleate an expanding bubble of the broken phase vacuum inside a Hubble horizon becomes of O(1), approximately the temperature at which the phase transition completes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Tinf: The temperature at the beginning of thermal inflation defined by ρV = ρrad, where ρV is the energy density of the false unbroken vacuum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' the difference in the potential between the local minimum at V (S) = 0 and the true minimum at V (S = v)), and ρrad is the energy density of the radiation dominated universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' When ρV begins to dominate the energy density, inflation begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In the case of the two vector DM models discussed in this paper, the finite temperature potential includes the thermal integral summing over the bosons and fermions [38], V 1 eff(hS, T) = n � s=1 gs � 1 64π2M 4 s � ln �M 2 s Λ2 � − Ci � + T 4 2π2 � ∞ 0 x2 ln � 1 ∓ e−√ x2±M2s /T 2� dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' (21) For some models, it might be necessary to consider the additional ring diagrams for the bosons, but for this investigation they can be ignored as they are insignificant [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' This thermal potential is not amenable to an analytic solution, but can be approximated using modified Bessel functions of the second kind [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' We compute the freeze-out temperature, TFO, numerically with micrOMEGA, and the nucleation temperature, Tn, numerically using CosmoTransitions and Bubbleprofiler (for cross-checking) [32, 40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Let us now consider the thermal history of the model depending on the order of the above three temperature thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' If Tn > TFO, the phase transition completes before DM freeze-out, and the freeze-out then takes place as usual in the broken phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' This means that we can calculate the relic abundance as presented in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In the opposite case, Tn < TFO there are three scenarios to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The filtered DM scenario takes place for the ordering TFO > Tn > Tinf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In this situation, there is no thermal inflation, as the 7 phase transition completes before inflation would begin, but the DM annihilations are immediately out of equilibrium after the phase transition, and therefore the abundance is set by the amount of DM particles that are able to enter the boundary to the broken phase, as described in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The supercool DM scenario [28], takes place for TFO > Tinf > Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In this situation, there is a period of thermal inflation, which ends at Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' After inflation, the latent heat stored in the false vacuum is released to reheat the universe back to temperature Tinf, under the assumption of instant reheating, or to a lower reheating temperature for delayed reheating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' However, since TFO > Tinf, no DM is produced in reheating and the abundance is set by the amount that was present before inflation, diluted by the expansion of the scale factor and by the filtering effect as in the above scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Finally, there is the case where Tinf > TFO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In this situation, assuming instant reheating, the reheating will bring DM back to equilibrium, and the relic abundance is again obtained via the usual freeze-out mechanism as presented in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The inflation temperature is obtained by solving for Tinf from ∆V (Tinf) = V high eff (hS, Tinf) − V low eff (0, Tinf) = g∗π2 30 T 4 inf, (22) where V high eff (hS, T) is the true vacuum and V low eff (0, T) is the false vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' We find that throughout the parameter space of interest in this work, we are either in the first or the last situation described above, and the DM abundance is thus obtained via the usual freeze-out mechanism in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' See Appendix A for more on the reheating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' DIRECT DETECTION In this section, we present the direct detection constraints on the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' We will see that the recent results from the LZ experiment significantly affect the SU(2)D model and that the U(1)D model is already very constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' To compute the direct detection cross section, we again use the micrOMEGA package [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The DM coupling to nucleons arises from the scalar mixing and is mediated via exchange of the SM-like Higgs and the scalon in the t-channel leading to a spin-independent cross section with negligible difference between protons and neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The results of this computation for both models are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The correct relic abundance is obtained along the red solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The purple region is excluded by LHC constraints on to Higgs decays into two scalons [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' This process becomes kinetically forbidden for larger DM mass, as larger DM mass leads to larger scalon mass as shown in Equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In the orange region, the DM-nucleon cross section is below the neutrino floor, and the yellow regions indicate the exclusion limit due to the LZ experiment [26], providing a significant improvement over the XENON1T experiment shown in green [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Finally, the grey region shows the projected exclusion limit from XENONnT [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Starting with the U(1)D model we see a small gap in the direct detection limits at the resonance region, Mh ≃ 2MV , where the DM mass is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='9-1 TeV and the coupling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='65 ≤ g ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In the middle of this range the direct detection cross section falls below the neutrino floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Outside of the resonance region, the model can not produce an O(1)-fraction of DM without being excluded by direct detection, unless the DM mass is well above 10 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' For the SU(2)D model the new constraints from LZ alter the picture compared to the situation with the previous XENON1T limits: the relic abundance line above the resonance region is now excluded for DM masses below 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='5 TeV, while prior to the LZ result there were no constraints beyond 1 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In the resonance region, we find the nucleation temperature for the phase transition below the QCD-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' This alters the computation for the gravitational wave signal, as the phase transition will be completed in conjunction with the QCD phase transition, as discussed in [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' This picture slightly changes when including additional scalar self-energy corrections for the SU(2) 8 (a) Constraints for the the U(1)D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' (b) Constraints for the the SU(2)D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The red line shows the correct relic abundance, Ωh2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='12 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The yellow region is excluded by the LZ (2022) experiment [26], the green region is the XENON1T experiment [42], the purple region is the LHC constraint for exotic Higgs decay, the orange region is the neutrino floor and the gray region is the projected 90% CL constraint from the XENONnT experiment [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' model [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' First, the scalon mass is slightly larger than in our leading order analysis, pushing the resonance region in figure 2(b) to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Additionally, the correction appears to slightly increase the nucleation temperature compared to our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' However, we find that overall the resulting gravitational wave (GW) signal is not significantly affected, and the GW signal prediction remains comparable to our results presented in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' For DM mass above 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='5 TeV the model is again allowed by direct detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In Figure 2 we have marked three benchmark points allowed by direct detection with the blue, indigo, and purple markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' These points will be used as examples for analyzing the GW signals in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' GRAVITATIONAL WAVES The strongly first order phase transition possible in classically scale invariant models is inter- esting due to the implications for baryogenesis [49], and due to potentially observable gravitational wave (GW) signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' To explore the gravitational wave signals, we consider the finite temperature potential in Equa- tion (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' This potential contains a barrier between the unbroken false vacuum and the broken phase minimum, leading to a first order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' At the nucleation temperature Tn, the phase transition will complete via the formation of bubbles of the true vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The expanding and colliding bubbles deposit energy in the surrounding plasma, generating gravitational waves as described in [50–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' For the purpose of solving Equation (21) and obtaining the parameters that describe the gravitational wave signal, we use the Python package CosmoTransitions[40], with custom modifi- cations including a method of computing the β value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The relevant parameters are the latent heat normalized with respect to the radiation energy, α, the inverse duration of the phase transition, Constraints ofU(1)p Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='0 Qh2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='8 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='7 LZ(2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='6 LHC KENONIT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='5 XENONnT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='4 300 1000 2000 3000 4000 Mv[GeV]ConstraintsofSU(2)pModel 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='0 Qh2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='0 XENONIT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='4 300 1000 2000 4000 8000 My[GeV]9 Model Benchmark point Parameter CosmoTransitions BubbleProfiler U(1)D g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='66 MV = 911 GeV Tc = 303 GeV α 20740 92180 β 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='8 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='2 Tn 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='04 GeV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='78 GeV U(1)D g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='7 MV = 1028 GeV Tc = 336 GeV α 1497 4597 β 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='5 Tn 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='3 GeV 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='4 GeV SU(2)D g = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='0 MV = 7530 GeV Tc = 2345 GeV α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='22 β 289 301 Tn 1430 GeV 1446 GeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Table with benchmark points used for the discussion of gravitaitonal wave signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The two first benchmark points are from the U(1)D model and the last is from the SU(2)D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' β, and the nucleation temperature, Tn, defined as [24, 53], α ≡ 1 ρ � ∆V − T 4 d∆V dT � ���� Tn , β H ≡ T d(S/T) dT ���� Tn , (23) where, ∆V = V high eff (hS, T) − V low eff (hS, T), ρ = geπ2 30 T 4 n, (24) where the ge ≈ 103 is the number of effective degrees of freedom during the nucleation at the temperature Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Finally the Euclidean action is defined as, S = 4π � ∞ 0 r2 � 1 2 �dhφ/S dr �2 + Veff(hφ/S) � dr, (25) where r is the radial distance from the center of the true vacuum bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In order to assess the reliability of the results, we make use of two different numerical tools for computing the nucleation temperature and the β and α parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The parameters α and β depend heavily on the nucleation temperature, Tn, so that possible errors on Tn will propagate to α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' For the computation we use CosmoTransitions and BubbleProfiler[40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' As shown in the appendix, we obtain a smaller numerical error with CosmoTransitions, but the results of both numerical computations agree within uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In general, we find that for sub-TeV DM masses the nucleation temperature in the BubbleProfiler implementation tends to be smaller than in CosmoTransitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In Figure 2, we identify three benchmark points allowed by all constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' These benchmark points are shown in 3 corresponding to the indigo diamond, blue square and purple hexagon shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Notice that the first point is below one TeV, the trend we observed regarding the performance of the two simulation tools is noticeable, and the BubbleProfiler nucleation temperature is signifi- cantly below the value obtained from CosmoTransitions, affecting also the α and β parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' At this point, the critical temperature is Tc = 303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In summary, both CosmoTransitions and BubbleProfiler show similar behavior for both models and are in reasonable agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' For high masses the latter tool yields slightly higher nucleation temperatures and therefore α is also a bit lower and β as indicated by Equation 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' 10 Having computed the relevant parameters for calculating gravitational waves (GW) spectra, we can consider the following equation for computing the total signal, Ωtoth2 = Ωcolh2 + Ωswh2 + Ωturbh2, (26) where the first term is the collision term, the second term is the sound wave term, and the last term is the turbulence term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The collisions from the bubbles themselves contribute to the GW spectra, but they do not give the most significant contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The collisions also produce bulk motion in the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' This causes sound waves that result in the primary contribution to the GW spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Finally, there is also some turbulence caused by the collisions which contribute to the GW spectra [23, 52, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The relevant equations for computing the collision term are, Ωcolh2(f) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='67 · 10−5 � α 1 + α �2 H2 β2 � ge 100 �− 1 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='11κ2 vv3 b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='42 + v2 b Scol Scol(f) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='8 � f fcol �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='8 � f fcol �3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='8 + 1 fcol = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='5 µHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='62 v2 b − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='1vb + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='8 β H Tn 100 GeV � ge 100 � 1 6 , (27) similarly, the equations for the sound wave term are Ωswh2(f) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='65 · 10−6 � α 1 + α �2 H β � ge 100 �− 1 3 κ2 vvbSsw Ssw(f) = � f fsw �3 � � � 7 3 � f fsw �2 + 4 � � � 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='5 fsw = 19 µHz 1 vb β H Tn 100 GeV � ge 100 � 1 6 , (28) and lastly, the equations for the turbulence term are Ωturbh2(f) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='35 · 10−4 �κturbα 1 + α � 3 2 H β � ge 100 �− 1 3 vbSturb Sturb(f) = � f fturb �3 � 1 + 8πf h∗ � � 1 + f fturb � 11 3 fturb = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='7 µHz 1 vb β H Tn 100 GeV � ge 100 � 1 6 , (29) where the inverse Hubble time, h∗, red-shifted to today, at the GW production is given as h∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='65 · 10−5 Tn 100 GeV � ge 100 � 1 6 , (30) 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The GW spectra for two different sets of transition parameters for the U(1)D model and one for the SU(2)D model (g = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='0, MV = 7530) computed with CosmoTransitions, dashed lines, and BubbleProfiler, dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The sensitivity curves (C1-C4) of the LISA detector are also shown [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' According to this result, the signals from this model are strong enough for LISA to detect the GW signal from the phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' and the two modified efficiency factors can be written as, κv = α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='73 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='083√α + α, κturb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='05κv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' (31) The result of this computation can be seen in Figure 4 for the three benchmark points, two from the U(1)D model and one from the SU(2)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The marker shape indicates the parameter as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The diamond and square shapes are from the U(1)D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' For SU(2)D model we have the high mass case marked by the hexagon shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The projected sensitivity curves (for the configurations C1-C4) for the LISA detector are also shown [27], and one can see that for the U(1)D model the signal should be detectable by three out of four configurations, but for the SU(2)D model the mass becomes too high and we need other future experiments to detect such high DM mass models such as the proposed TianQin detector [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' DISCUSSION AND CONCLUSIONS We have investigated two vector DM models in light of existing DM direct detection experiments and future GW experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Both of the models investigated in this work are already strongly constrained by direct detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' For the SU(2)D model this is in particular due to recent results Gravitational wavespectra of selectedpoints 10-5 10 10~9 yo 10-11 10-13 C1 g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='66 My= 911 [GeV] 10-15 C2 g= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='70 My = 1028 [GeV] C3 g = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='0 My = 7530 [GeV] C4 10~17 10-4 10-3 10-2 10-1 f[Hz]12 from the LZ experiment which has ruled out most of parameter space consistent with a full relic abundance from freeze-out in the range MV ∈ (1 − 10) TeV and with XENONnT the DM will either be detected or the entire parameter space above the neutrino floor will be ruled out as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' GW signals in both models have been discussed in earlier literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In our analysis we find that results differ significantly between different numerical implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Recently, the SU(2)D model was discussed in [48], and we find that their results for the α and Tn parameters agree with our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Regarding the U(1)D model, it was previously suggested that GW signals could be used to probe the model in case the direct detection cross section remains below the neutrino floor [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' We agree with this conclusion, but numerically we find differences to [23] in the GW parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' While we can reproduce the critical temperature reported, the nucleation temperature and the α and β parameters differ from those reported in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Their results were obtained with the AnyBubble package [56], for which we failed to obtain results in agreement with the other two numerical implementations used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' This raises the question of comparability between the phase transition parameters obtained via the various numerical implementations available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' This issue has been investigated in [41], where a fairly good agreement between BubbleProfiler and CosmoTransitions is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' This is compatible with our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The finite temperature potential in both cases leads to a strong first order electroweak phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The U(1)D model can produce significant GW signals, which can be detected by LISA [27] and future experiments would be able to test the SU(2)D model also in the high DM mass regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' ACKNOWLEDGMENTS The financial support from Academy of Finland, project #342777, is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' MTF and MR acknowledge partial funding from The Independent Research Fund Denmark, grant numbers DFF 6108-00623 and DFF 1056-00027B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' MET acknowledges funding from Augustinus Fonden, application #22−19584, to cover part of the expenses associated with visiting the University of Helsinki for half a year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Appendix A: Supercooling, inflation and reheating The investigation of the GW spectra leads to the discussion of supercooling in the models presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' As shown in the GW section there are orders of magnitude in the difference between the critical and nucleation temperature at the low mass scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' As discussed in the other papers, this can lead to different kinds of phenomena including inflation, filtering and, reheating [28, 30, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' These effects are expected to affect the GW signal for low masses, and it might effects some of the results even presented in Figure 4, but it is beyond this paper to look at the details of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' As discussed in a recent paper, the universe could escape inflation via bubble nucleation or via quantum tunneling, two different scenarios leading to different GW signals [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' We would however like to highlight the fact that strong supercooling from hundreds of GeV to the QCD scale might not be a big issue for the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The bigger the supercooling the greater the inflation as the scalon Higgs field will be stuck in a false vacuum acting like a cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The main constraint for any possible is lover than the max number of e-folds, Nmax = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='8 + ln TR TeV, (A1) 13 where TR is the reheating temperature after the inflationary epoch and one finds that this limits the temperature to TR < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='6· 1015 GeV [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' To compute the reheating temperature, we are interested in computing the decay of the inflaton-like field which in this case is the scalon Higgs field S for the U(1)D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Due to the mass constraints, only the scalon Higgs is kinetically allowed to decay as Γ(hS → h, h), but this requires a DM mas of MV > 1 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' From the Lagrangian, we find that the Feynman rule for this vertex and this yields the decay, Γ(hS → 2h) = � M 2 S − 4M 2 h 32πM 2 S |M|2, (A2) where, |M|2 = �M 2 h 4v1 (5 + 3 cos(4α)) sin(α) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' (A3) We can furthermore include decays into quarks and leptons, Γ(hS → f ¯f/ℓ¯ℓ) = NC 8π m2 b v2 1 MS � 1 − 4m2 b M 2 φ sin(α)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' (A4) where NC = 3 for fermions and NC = 1 for leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Using the decays one can calculate the reheating temperature, TR, using the following equation [57], TR ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='2 �200 g∗ �1/4 � ΓMpl, (A5) where Mpl is the reduced Planck mass and g∗ = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Considering a rather low mixing value 0 ≤ α ≤ π 64 and a mass range of 250 GeV ≤ MV ≤ 2500 GeV the reheating temperature is somewhere around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='6 PeV yielding mass of the scalon field around 1 GeV < MS < 200 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' This is so hot that the universe will reheat back to a temperature much hotter than the scales of freeze-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' It also satisfies the constraint from Equation (A1), thus it is not too hot and not causing too much inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' One can repeat this exercise for the SU(2)D, but the result is roughly the same with the main differences being a slightly heavier scalon mass, 1 GeV < MS < 350 GeV, and higher reheating temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='1-2 PeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Conclusively, dark matter production can take place via freeze-out as the universe subsequently cools down again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Appendix B: Model implementation in CosmoTransitions For the implementation of the model in CosmoTransitions we feed in the tree level potentials as shown in Equation (2) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Then we manually implement the mass matrix with the massive SM bosons, plus the new bosons, and the top quark, M 2 W = g2 W 4 h2 1, M 2 Z = g2 W + g2 Z 4 h2 1, m2 t = λ2 t 2 h2 1, (B1) where the Yukawa coupling of the top quark is λt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The DM candidate have their respective implementations for each model where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' M 2 V = cV g2h2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' (B2) 14 with cV = 1 (1/4) for the U(1)D (SU(2)D) model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' and then the scalar mass matrices yield,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' M 2 h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='S(U(1)D) = 1 4 � h2 1 (λh + 2λφh) + h2 2(λφ + 2λφh) ± � h4 1 (λh − 2λφh)2 + h4 2 (λφ − 2λφh)2 + 2h2 1h2 2 � 2λhλφh + 28λ2 φh − λhλφ + 2λφλφh � � (B3) M 2 h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='S(SU(2)D) = 1 4 � h2 1 (6λH + λHS) + h2 2(6λS + λHS) ± � h4 1 (λHS − 6λH)2 + h4 2 (λHS − 6λS)2 + 2h2 1h2 2 (6λH(λHS − 6λHS) + λHS(7λHS + 6λS)) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' (B4) A custom solution is made for computing the beta value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' This is done by simply calculating the action divided by the temperature around the point of the nucleation temperature, making a fit to those plots, and taken the derivative etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Some tweaks have been done to the source code to make this work, and also to improve the precision at low nucleation temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Appendix C: Model implementation in BubbleProfiler For this package, we give it the full thermal potential in Equation (21), but instead of evaluating the thermal integral an approximation is made using Bessel function[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Specifically we use the modified Bessel functions, K2(kx), as follows � ∞ 0 x2 ln � 1 ∓ e−√ x2±M2s /T 2� dx = − 3 � k=1 x2 k2K2(kx) − 2 � k=1 (−1)kx2 k2 K2(kx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' (C1) The BubbleProfiler is written in C++, but comes with a command line interface (CLI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Using this one can implement simple potentials like polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' In order to avoid implementing all these functions, we created a python interface where we implement the model in python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Then we create a higher order polynomial fit to the full potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' This polynomial is then fed to BubbleProfiler via the CLI together with other relevant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' We compute several points around the nucleation temperature and make a fit to that and from there we determine the β and Tn value, and the latter is then used to find α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Appendix D: Computing parameters of the EWPT The essential computation for the phase transition is finding the relationship between the action and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The nucleation temperature condition is defined as S(T) T ���� Tn ≈ 140, (D1) thus when the action divided by the temperature is equal to 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' We can compute the action and by dividing by the temperature a plot of this relationship can be obtained as seen in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Given some data points, it is possible to make a fit, and from that read off the Tn value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Furthermore, the fit is also a function of S(T)/T, which can thus be used to compute β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Now 15 (a) Using BubbleProfiler for computing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' (b) Using CosmoTransitions for computing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' A comparison of the apparent error when computing the β and Tn parameter when computing the EWPT parameters in U(1)D model for g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='75 and MV = 1184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Note the temperature range is different for the implementation, thus the range is different in the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' recall that α is evaluated at the nucleation temperature and α ∝ 1/T 4 n, thus, the value of α is also highly dependent on the nucleation temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Since our BubbleProfiler result in general yields a slightly higher nucleation temperature we get a lower value of α as discussed in the GW section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' For lower masses the BubbleProfiler result yield significantly lower nucleation temperatures suggesting that our implementation might not be as good in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Looking at Figure 5, we see that the apparent error of the BubbleProfiler is significantly higher than the error from the CosmoTransitions result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' This may be attributed to the fact that we used an approximated potential via our custom Python interface instead of implementing the model using C++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' This leads us to consider the CosmoTransitions as the better result in this paper even though BubbleProfiler is claimed to be more accurate [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' [1] Fritz Zwicky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The redshift of extragalactic nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Helvetica Physica Acta, 1933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' [2] Vera C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Rubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' The rotation of spiral galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Science, 1983, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='org/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='4604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='1339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' org/doi/abs/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='4604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='1339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' [3] The Planck Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Planck 2018 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Astronomy & Astrophysics, Sep 2020, 1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='06209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' ISSN 1432-0746.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' [4] Lars Bergstr¨om.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Nonbaryonic dark matter: Observational evidence and detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=', 2000, hep-ph/0002126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' [5] Gianfranco Bertone and Dan Hooper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' History of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Phys.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='09642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' ISSN 1475-7516.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' [54] Tommi Alanne, Thomas Hugle, Moritz Platscher, and Kai Schmitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' A fresh look at the gravitational- wave signal from cosmological phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Journal of High Energy Physics, Mar 2020, 1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='02076.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' [56] Ali Masoumi, Ken D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Olum, and Jeremy M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Wachter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Approximating tunneling rates in multi- dimensional field spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Journal of Cosmology and Astroparticle Physics, Oct 2017, 1702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='00356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' ISSN 1475-7516.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' [57] Kai Schmitz and Gilles Vertongen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' Reheating and preheating after inflation : an introduction, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='desy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='de/~westphal/workshop_seminar_fall_2010/reheating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfQPZ4/content/2301.00041v1.pdf'} diff --git a/5dAzT4oBgHgl3EQfu_2Z/content/tmp_files/2301.01700v1.pdf.txt b/5dAzT4oBgHgl3EQfu_2Z/content/tmp_files/2301.01700v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9f0a8ccbe22e151813187ec08ab44ccee19e920f --- /dev/null +++ b/5dAzT4oBgHgl3EQfu_2Z/content/tmp_files/2301.01700v1.pdf.txt @@ -0,0 +1,1572 @@ +arXiv:2301.01700v1 [cs.GT] 4 Jan 2023 +Non-Adaptive Matroid Prophet Inequalities +Kanstantsin Pashkovich, Alice Sayutina +University of Waterloo +Department of Combinatorics & Optimization +200 University Avenue West +Waterloo, ON, Canada +N2L 3G1 +Abstract +We consider the matroid prophet inequality problem. This problem +has been extensively studied in the case of adaptive mechanisms. In par- +ticular, there is a tight 2-competitive mechanism for all matroids [KW12]. +However, it is not known what classes of matroids admit non-adaptive +mechanisms with constant guarantee. +Recently, in [CGKM20] it was +shown that there are constant-competitive non-adaptive mechanisms for +graphic matroids. In this work, we show that various known classes of +matroids admit constant-competitive non-adaptive mechanisms. +1 +Introduction +Let us consider the classical prophet inequality problem [KS77]. +A gambler +observes a sequence of non-negative independent random variables X1, X2, . . . , +Xn, which correspond to a sequence of values for n items. The gambler knows +the distributions of X1, X2, . . . , Xn. The gambler is allowed to accept at most +one item; and the gambler is interested in maximizing the value of the accepted +item. However, the gambler cannot simply select an item of the maximum value, +because the values of the n items are revealed to the gambler one by one; and +each time a value of the current item is revealed the gambler has to make an +irrevocable choice whether to accept the current item or not. +What stopping rule the gambler should use to maximize the expected value of +the item they accept? The gambler knows only the distributions of X1, X2, . . . , +Xn while a prophet knows the realization of X1, X2, . . . , Xn. Thus, in contrast +to the gambler the prophet can always obtain the maximum item’s value. The +seminal result of Krengel and Sucheston [KS77] showed that the gambler can +obtain at least a half of the expected value obtained by the prophet. +The classical prophet inequality problem led to a series of works on different +variants of the problem. A natural variant of the problem is the generalization +1 + +of the problem where a gambler can buy more than one item, but the set of +bought items should satisfy a known feasibility constraint. Formally, let us be +given a collection S ⊆ 2[n] of item sets. Then both gambler and prophet can +select any item set S from S. So S defines a feasibility constraint for selecting +items. In most standard examples of feasibility constraints, S can be defined as +a collection of all item sets with cardinality at most k for some natural number k. +More generally S can be defined as a collection of all independent sets in some +matroid, in this case we speak about the matroid prophet inequality problem. +The result in [SC84] showed that in the single-item setting a gambler can +obtain at least half of the prophet value by using the following threshold-rule: +determine a constant T as a function of known distributions and accept the first +item exceeding T . This rule results in a 2-competitive mechanism, similar to +the adaptive approach of [KS77]. Note, that this approximation guarantee is +known to be tight. There is also another method to set a threshold presented in +[KW12], which also results in a 2-competitive mechanism. This was extended +by Chawla et al. in [CHMS10] and [CGKM20] to the setting of several items. +The results presented in [KW12] further extend to the matroid prophet in- +equalities, where accepted items need to form an independent set in a known +matroid. It leads to a 2-competitive mechanism for every matroid, matching +the single-item setting result. However, unlike the mechanism in the single-item +setting, the mechanism for matroids is adaptive: the thresholds for items are +computed based on the previously accepted items. By [KW12], there also exists +a constant-competitive adaptive mechanism for feasibility constraints defined as +an intersection of constant number of matroids. The mechanism by Kleinberg +and Weinberg was further extended to a 2-competitive mechanism for polyma- +troids by Dütting and Kleinberg in [DK15]. +Gravin and Wang [GW19] studied the bipartite matching version of this +problem: in their version, the arriving items are the edges of the (known) bi- +partite graph. Gravin and Wang obtained a 3-competitive non-adaptive mech- +anism, which assigns thresholds to each vertex in the graph and an edge is +accepted only if its weight is at least the sum of the thresholds associated with +its endpoints. +Feldman, Svensson and Zenklusen [FSZ16] studied online item selection +mechanisms called “online contention resolution schemes" (OCRS). They showed +that given special properties, OCRS translate directly into a constant-competitive +prophet inequality for the same problem against almighty adversary, i.e. an ad- +versary which knows in advance realizations of all the items and the random +bits generated by an algorithm. As a result, they develop a constant-competitive +mechanism for prophet inequalities of the intersection of a constant number of +matroids, knapsack and matching constraints. Those mechanisms are “almost” +non-adaptive in a sense that they fix thresholds for all items, however their mech- +anisms also impose a subconstraint: an item cannot be accepted if together with +previously accepted items it forms one of the fixed forbidden sets. +Finally, in a later version of their paper [FSZ21], they prove that pure non- +adaptive mechanisms cannot achieve a constant-competitive approximation even +against a “normal” adversary. They construct a family of gammoid matroids +2 + +showing a lower bound of Ω(log n/ log log n) for a guarantee of non-adaptive +mechanisms on gammoids with n elements. +There have been works studying similar setups with other goals. Chawla +et al. [CHMS10] studied a Bayesian item selection process in a fixed item ar- +rival order or against an adversary in control of the order. They studied it +from a perspective of the revenue maximization for the auctioneer. The per- +formance is constant-competitive compared to the well-known Myerson mech- +anism +[Mye81], which achieves the largest possible expected revenue among +truthful mechanisms. The mechanism by Chawla et al. [CHMS10] has an ad- +vantage that it determines static thresholds together with a subconstraint so +that each agent can be offered take-it-or-leave-it prices in an online fashion. +Recently, Chawla et al. [CGKM20] developed a 32-competitive non-adaptive +mechanism for graphic matroids against adversary item ordering. +1.1 +Our results +First, we list the known results for non-adaptive mechanism that were mentioned +in the previous section. +Theorem 1 (Uniform Rank 1 Matroid [SC84]). There exists a 2-competitive +non-adaptive mechanism for single-item setting. +Theorem 2 (Graphic Matroid [CGKM20]). There exists a 32-competitive +non-adaptive mechanism for graphic matroids. +Now let us list our results. In case of a simple graph, i.e. a graph with no +parallel edges or loops, we can slightly improve the above theorem by considering +essentially the same mechanism as [CGKM20] but considering a different scaling +of a point from the matroid polytope. We provide this result for the sake of +completeness. +Theorem 3. There exists a 16-competitive non-adaptive mechanism for graphic +matroids in the case of simple graphs. +Furthermore, the mechanism [CGKM20] can be generalized to the setting of +k-column sparse matroids. This result we need later to obtain Theorem 8. +Theorem 4 (k-Column Sparse Matroids). There exists a (2k+2k)-competitive +non-adaptive mechanism for k-column sparse matroids. +Note, that Theorem 2 of [CGKM20] follows from Theorem 4, since a graphic +matroid is also a 2-column sparse matroid over F2. +Using analogous approach to the one in [Sot13], we also develop a mechanism +for cographic matroids. +Theorem 5 (Cographic Matroids). There exists a 6-competitive non-adaptive +mechanism for cographic matroids. +The approach in [Sot13] immediately leads to the following result for γ-sparse +matroids. +3 + +Theorem 6 (γ-Sparse Matroids). There exists a γ-competitive non-adaptive +mechanism for γ-sparse matroids. +Combining the above results and using classic Seymour’s decomposition re- +sults we obtain the following theorem. +Theorem 7 (Regular Matroids). There exists a 256-competitive non-adaptive +mechanism for regular matroids. +Subject to the Structural Hypothesis 1 due to Geelen, Gerards and Whittle, +which is stated later, we can also derive the following result. +Theorem 8. Subject to the Structural Hypothesis 1, for every prime number p +there exists a constant-competitive mechanism for every proper minor-closed +class of matroids representable over Fp. +We also would like to observe that some of the recent results on “single +sample prophet inequalities” (SSPI) lead to non-adaptive constant-competitive +mechanisms. For this, the single sample required by the gambler in SSPI can +be directly sampled by our gambler from the available distributions. In partic- +ular, the results in [AKW19] and [CFPP21] on laminar matroids and truncated +partition matroids inspired by the mechanism in [MTW16] lead to non-adaptive +mechanisms for prophet inequalities. To obtain these results, it is crucial that +the mechanism in [MTW16] does not involve subconstraints, i.e. each item is +accepted as long as the item is not in the “observation phase”, the item passes +its threshold based only on the “observation phase” and the item forms an in- +dependent set with previously accepted items. In comparison, it is not clear +how from the results on regular matroids in [AKW19] based on the mechanism +in [DK14] one can obtain non-adaptive mechanisms. +So the following results can be directly obtained from [AKW19] and [CFPP21], +respectively. +Theorem 9 (Laminar Matroid). There exists a 9.6-competitive non-adaptive +mechanism for laminar matroids. +Theorem 10 (Truncated Partition Matroid). There exists an 8-competitive +non-adaptive mechanism for truncated partition matroids. +2 +Comparison to known results +Our results for cographic matroids and k-column sparse matroids are obtained +through modifications of the arguments in [Sot13] and [CGKM20], respectively. +The results on regular matroids and minor-closed families of matroids follow the +approach outlined in [HN20] for the secretary problem. As necessary building +blocks we use our results for cographic and 2-column sparse matroids. Note that +a biggest challenge for us is the compatibility of non-adaptive thresholds with +contractions. Indeed, standard tools for deriving mechanisms for contraction +4 + +minors need subconstraints, while subconstraints are not permitted in non- +adaptive mechanisms. To obtain our results, we resolve this issue only in the +context of matroids representable over finite fields, see arguments in Lemma 8. +It would be interesting to see whether analogous results for contraction minors +hold with no assumption about representability over finite fields. +3 +Preliminaries +In this paper, we consider the matroid prophet inequality problem, where items +arrive online in adversarial order. Here, the adversary knows the distributions +of all X1, X2, . . . , Xn and knows the gambler’s mechanism, but the realization +of X1, X2, . . . , Xn is not known to the adversary. +Based on the available +information, the adversary can decide on the order in which items and their +values are observed by the gambler. +3.1 +Prophet inequality +Definition 1. Let M be a matroid on the ground set [n] := {1, . . . , n}, where +[n] corresponds to n items. Let ⃗X := (X1, . . . , Xn) be non-negative independent +random variables representing the values of these n items. +• For every subset of items S ⊆ [n] we define its weight as follows +w(S) := +� +i∈S +Xi. +• Let PROPHM be the random variable corresponding to the value obtained +by the prophet +PROPHM := +max +S∈I(M) w(S) , +where I(M) is a collection of independent sets for M. +• Let EPROPHM be the expectation of the value obtained by prophet +EPROPHM := E[PROPHM] . +Definition 2. Let us be given a number α > 0. +• We call a mechanism α-competitive (alternatively, we say that the mech- +anism guarantees an α-approximation) on the matroid M if the expected +value obtained by the gambler via this mechanism is at least 1 +αEPROPHM. +• We call a mechanism α-competitive (alternatively, we say that the mech- +anism guarantees an α-approximation) on the matroid class M if this +mechanism is α-competitive for every matroid M ∈ M. +5 + +3.2 +Non-adaptive mechanism +We say that a mechanism is non-adaptive if it has the following structure: +• Given the distributions of ⃗X = (X1, . . . , Xn), the mechanism determines +the values of thresholds ⃗T = (T1, . . . , Tn), where each Ti, i ∈ [n] is a real +number or +∞. +• If the value of item i ∈ [n] is observed, the gambler accepts the item i if +and only both conditions hold: +1. the observed value of Xi is at least Ti +2. the item i together with all previously selected items forms an inde- +pendent set with respect to the matroid M. +Note, that a non-adaptive mechanism does not change thresholds during +its course. +So, none of the thresholds depends on the realization of ⃗X = +(X1, . . . , Xn). +Another crucial feature of a non-adaptive mechanism is that the mechanism +works only with the original matroid M. A non-adaptive mechanism does not +allow us to define a new matroid M ′, such that a set of items is independent in +M ′ only if it is independent in M, and modify the condition (2) based on M ′. +In this work, we focus on non-adaptive mechanisms. From here and later we +use the term mechanism to refer to non-adaptive mechanisms exclusively. +Remark 1. In this work, non-adaptive mechanisms are allowed to make non- +deterministic decisions. Hence, we allow a non-adaptive mechanism to construct +the thresholds ⃗T = (T1, . . . , Tn) non-deterministically. +To measure the performance of such a mechanism we use the expected total +value, where the expectation is taken not only with respect to ⃗X = (X1, . . . , Xn) +but also with respect to ⃗T = (T1, . . . , Tn). +3.3 +Matroids +We provide a review of matroids here. +Experienced readers should consider +skipping or skimming this section. For further results about matroids, consider +consulting [Oxl06]. +A matroid M = (E, S) is a pair of a finite ground set E and a collection S ⊆ +2E of independent sets. The collection S ⊆ 2E of subsets of E satisfies the +following conditions: +(i) Empty set is an independent set, so ∅ ∈ S. +(ii) The collection S is closed with respect to taking subsets, so for all A ⊆ +B ⊆ E if B is in S then A is in S. +(iii) The collection S satisfies so called augmenation property. In other words, +for all A, B ⊆ E such that A, B ∈ S and |A| > |B|, there exists c ∈ A \ B +such that B ∪ {c} ∈ S. +6 + +A subset of E is called dependent if it is not in S. The inclusion-maximal +independent sets are called bases and the inclusion-minimal dependent sets are +called circuits. +For every two bases, their cardinalities are equal: for every +bases A and B of M we have |A| = |B|. A rank function for the matroid M is +a function rM : 2E → N such that for every A ⊆ E the value rM(A) equals the +cardinality of an inclusion-maximal independent subset of A. In the cases when +the choice of the matroid is clear from the context, we write r instead of rM. +Given a matroid M, we can define the dual matroid M ∗ over the same ground +set E. A set A is independent for matroid M ∗ if and only if E \ A contains a +basis of M. An element c ∈ E is called a loop in M if rM(c) = 0. An element +c ∈ E is called a free element in M if rM∗(c) = 0. To put it another way, an +element c is free, if and only if for every set A, which is independent in M, +A∪{c} is also independent in M. We say that elements c and d ∈ E are parallel +in matroid M, denoted by c ∥ d, if rM(c) = rM(d) = rM({c, d}) = 1. One +can show that “being parallel” defines an equivalence relation on the non-loop +elements of M. A matroid is called simple if it has no loops and no parallel +edges. +Let M = (E, S) be a matroid and A ⊆ E. The contraction of M by A, de- +noted as M/A, is a matroid over ground set E\A with the following independent +sets +{S ⊆ E \ A : S ∪ A′ ∈ S} , +where A′ is an inclusion-maximal independent subset of A. +The restriction of M to A, denoted as M |A or M \ A, is a matroid over +the ground set A where a set S ⊆ A is independent in M |A if and only if it is +independent in M. +A matroid M ′ is called a simple version of M if M ′ is obtained from M by +deleting all loops and contracting every parallel class of elements into a single +element. +For matroids M, N, we say that N is a minor of M = (E, S) if N is +isomorphic to M/A\B for some disjoint sets A, B ⊆ E. A matroid class M is +called minor-closed if for any M ∈ M every minor of M is also in M. +Let us now list some of the classical examples of matroids, which were ex- +tensively studied in the context of various mathematical fields. +• A uniform matroid M = (E, S) of rank k is matroid where +S := {A ⊆ E : |A| ≤ k} . +When |E| = n, we denote the uniform matroid of rank k as Uk,n. +• A graphic matroid over graph G = (V, E) is a matroid M = (E, S), where +S := {A ⊆ E : A is acyclic} . +The graphic matroid over graph G is denoted as M(G). +7 + +• A cographic matroid over graph G = (V, E) is a dual matroid M = (E, S) +to the graphic matroid over the same graph G. In this case we have +S := {A ⊆ E : (V, E\A) has the same number of components as (V, E)} . +• A vector matroid M = (E, S) is a matroid such that there is a vector +space V and a map φ : E → V satisfying +S := {A ⊆ E : multiset φ(A) is linearly independent} . +Given a field F, we say that M is representable over field F if M is iso- +morphic to the vector matroid where V is a vector space over field F. +A matroid is called regular if it is representable over every field. A matroid +is called binary if it is representable over F2. +• A k-column sparse matroid M = (E, S) is a matroid such that there is a +field F and dimension m and a map φ : E → Fm such that +S := {A ⊆ E : multiset φ(A) is linearly independent over F} ; +and moreover φ(c) ∈ Fm has at most k nonzero coordinates for every +c ∈ E. +• A γ-sparse matroid M = (E, S) is a matroid such that the inequality +|S| ⩽ γrM(S) holds for every S ⊆ E. +• A laminar matroid M = (E, S) is a matroid such that there exists a +laminar family F over the ground set E and there are numbers cF ∈ N, +F ∈ F such that +S := {A ⊆ E : |A ∩ F| ≤ cF for every F ∈ F} . +Moreover, if F = {E, E1, . . . , Ek}, where E1, . . . , Ek form a partition of +the ground set E, then M is called a truncated partition matroid. Recall, +that a family F is called laminar if for every A, B ∈ F we have A ⊆ B or +B ⊆ A or A ∩ B = ∅. +Given a matroid M = (E, S) we can define the corresponding polytope +PM ⊆ RE as the convex hull of points corresponding to the characteristic vectors +of independent sets. The polytope PM is known to admit the following outer +description [Sch03]. +PM = {x ∈ RE : x ≥ 0 and +x(S) ≤ rM(S) for every S ⊆ E} , +where x(S) stands for � +c∈S xc. +For a matroid M = (E, S) and a set A ⊆ E we can define the closure of A +as the following set +clM(A) := {c ∈ E | rM(A ∪ {c}) = rM(A)} . +8 + +For a matroid M = (E, S), we call the following function ⊓M : E × E → Z +a local connectivity function +⊓M(X, Y ) = r(X) + r(Y ) − r(X ∪ Y ) . +The following function λM : E → Z⩾0 is called a connectivity function +λM(X) := ⊓M(X, E \ X) = r(X) + r(E \ X) − r(E) . +Informally, connectivity functions measure dependence with respect to the +matroid between parts of the ground set. To illustrate it, let us consider the +connectivity function for vector matroids. +Suppose M = (E, S) is a vector +matroid defined by a vector space V and a map φ : E → V . Then we have +λM(S) =r(S) + r(E \ S) − r(E) = +dim(span φ(S)) + dim(span φ(E \ S)) − dim(φ(E)) = +dim ((span φ(S)) ∩ (span φ(E \ S))) . +3.4 +Ex-ante relaxation to the matroid polytope +The goal of ex-ante relaxation [FSZ16] or [CGKM20] is to reduce the origi- +nal problem to the problem where item values are distributed as independent +Bernoulli random variables. Note, that both problems are using the same ma- +troid. +In the original problem item values ⃗X = (X1, . . . , Xn) are independent ran- +dom variables with known distributions. For i ∈ [n] let Fi be the cumulative +distribution function of Xi. The reduction of the original problem to a new +problem is done using a point p in the matroid polytope PM. Let us first show +that there is a point p ∈ PM with properties that prove to be desirable later +following the argumentation in [CGKM20]. +Lemma 1. Given a matroid M over the ground set [n] and random variables +⃗X = (X1, . . . , Xn), there exists p ∈ PM such that +EPROPHM ⩽ +n +� +i=1 +piti , +where ti := E[Xi | Xi ⩾ F −1 +i +(1 − pi)] for every i ∈ [n]1. +Proof. Let Iopt be a random variable indicating an optimal independent set +in M with respect to ⃗X = (X1, . . . , Xn). In case when for some realization +of ⃗X = (X1, . . . , Xn) there are several optimal independent sets, Iopt can be +selected as any of these sets. For i ∈ [n], let pi be the probability that element +1Here, we assume that for every i ∈ [n] the event Xi = F −1 +i +(1 − pi) happens with the zero +probability, which is true for all continuous distributions. In case of discrete distributions one +needs to introduce appropriate tie-breaking. +9 + +i is in Iopt. Note that p = (p1, . . . , pm) is a convex combination of independent +sets of M, and so lies in PM. +Due to EPROPHM = E[� +i∈Iopt Xi], it remains to show that +E[ +� +i∈Iopt +Xi] ⩽ +n +� +i=1 +piti . +We have +E[ +� +i∈Iopt +Xi] = +n +� +i=1 +P[i ∈ Iopt]E[Xi | i ∈ Iopt] = +n +� +i=1 +piE[Xi | i ∈ Iopt] . +For every i ∈ [n] we have that ti and E[Xi | i ∈ Iopt] are expectations of the +same random variable Xi but conditioned on the event Xi ⩾ F −1 +i +(1−pi) and on +the event i ∈ Iopt, respectively. Note, that the probability of both these events +equals pi. However, the expectation of Xi conditioned on Xi ⩾ F −1 +i +(1 − pi) is +the “largest” conditional expectation of Xi on an event of probability pi. Thus, +we have piE[Xi | i ∈ Iopt] ⩽ piti for every i ∈ [n] and so we get the desired +inequality +n +� +i=1 +piE[Xi | i ∈ Iopt] ⩽ +n +� +i=1 +piti . +Let us show how one can use the point p = (p1, . . . , pn) guaranteed by +Lemma 1 to reduce the original problem. Let us define independent Bernoulli +random variables ⃗X′ = (X′ +1, . . . , X′ +n) as follows, for each i ∈ [n] +X′ +i = +� +ti +with probability pi +0 +with probability 1 − pi , +where ti := E[Xi | Xi ⩾ F −1 +i +(1 − pi)]. +Let us assume that we have a non-adaptive mechanism for the original ma- +troid M and item values ⃗X′ = (X′ +1, . . . , X′ +n), which sets nonnegative thresholds +⃗T ′ = (T ′ +1, . . . , T ′ +n). By definition of ⃗X′ = (X′ +1, . . . , X′ +n), for every i ∈ [n] the +exact value of T ′ +i is not relevant per se, but it is crucial whether ti ≥ T ′ +i or +ti < T ′ +i. If for some i ∈ [n] we have T ′ +i > ti then this item i is “inactive” and so +is never selected by the gambler working with M and ⃗X′ = (X′ +1, . . . , X′ +n). +The key is to construct a non-adaptive mechanism for the original matroid M +and item values ⃗X′ = (X′ +1, . . . , X′ +n) with positive thresholds ⃗T ′ = (T ′ +1, . . . , T ′ +n) +such that for each item i ∈ [n] the probability that i is selected by the gambler +is at least αpi. Now we can use such a non-adaptive mechanism for the original +matroid M and item values ⃗X′ = (X′ +1, . . . , X′ +n) to construct a non-adaptive +α-competitive mechanism for the same matroid M and random variables ⃗X = +10 + +(X1, . . . , Xn). Let us define the thresholds ⃗T = (T1, . . . , Tn) as follows, for every +i ∈ [n] +Ti := +� ++∞ +if ti < T ′ +i +F −1 +i +(1 − pi) +otherwise . +To see that the thresholds ⃗T = (T1, . . . , Tn) lead to an α-competitive mech- +anism for M and ⃗X = (X1, . . . , Xn), let us couple random variables X′ +i with +random variables Xi as follows +X′ +i := +� +ti +if Xi ≥ F −1 +i +(1 − pi) +0 +otherwise. +Note that ⃗X′ = (X′ +1, . . . , X′ +n) are independent Bernoulli random variables, +where for each i ∈ [n] the variable X′ +i equals ti with probability pi and equals 0 +with probability 1 − pi. When ⃗X′ are coupled with ⃗X this way, Xi and X′ +i have +the same expected value when conditioned on X′ +i being ti. The mechanism with +thresholds ⃗T selects an item i ∈ [n] when run for ⃗X only if the mechanism with +thresholds ⃗T ′ selects the item i when run for ⃗X′. Moreover, for both of these +algorithms, conditionally on the event that the item i is selected the expected +value of i equals ti. Now, α-competitiveness guarantee of the thresholds ⃗T for +M and ⃗X follows from Lemma 1. +4 +Graphic and k-column sparse matroids +First, we construct a 16-competitive non-adaptive mechanism for graphic ma- +troids without parallel edges. Our construction is done through the ex-ante re- +laxation to the matroid polytope, following the works in [FSZ16] or [CGKM20]. +Later, we present a constant-competitive non-adaptive mechanism for k-column +sparse matroids whenever k is constant. +4.1 +Graphic matroids +Now we are ready to provide a 16-competitive non-adaptive mechanism for +graphic matroid. The provided mechanism is essentially the one constructed +in [CGKM20] but with saving a factor of 2 in the guarantee, which is achieved +by rescaling the point from the matroid polytope by 2 and not by 4. +Let us be given a simple graph G = (V, E) and let us consider the corre- +sponding graphic matroid M over the ground set E. Recall that a subset of E +is independent with respect to M if and only if it is acyclic in G. Let us also +assume that the graph G has n edges and so E = {e1, e2, . . . , en}. +Lemma 2. Let p = (p1, . . . , pn) be a point in the polytope PM. Thus we assume +that for every i ∈ [n] the coordinate pi of p corresponds to the edge ei. Then +there exists an orientation of edges E = {e1, e2, . . . , en} in the graph G = (V, E) +such that for every vertex v ∈ V we have � +i∈[n]:ei∈δ−(v) pi ≤ 2. +11 + +Proof. Observe that the average degree of a vertex in a forest on |V | vertices is +at most (2|V | − 2)/|V | = 2 − 1/|V | ⩽ 2. +Let us use this fact to prove the desired statement by induction on the +number of vertices in the graph G. +If the graph G has at most two vertices then the orientation is trivial. Other- +wise, since p is a convex combination of points corresponding to forests in G, we +have that the average of the value � +i∈[n]:ei∈δ(v) pi over all vertices v ∈ V is at +most 2. Thus there exists a vertex v ∈ V such that we have � +i∈[n]:ei∈δ(v) pi ≤ 2. +We orient all edges incident to v as edges in δ−(v), so these edges are incoming +with respect to v. Then we remove the vertex v and all edges incident to it +and orient the remaining edges according to the orientation guaranteed by the +inductive hypothesis. +Now we present an algorithm for graphic matroids of simple graphs. +Algorithm 1 A non-adaptive 16-competitive mechanisms for graphic matroids +of a simple graph +1: Let p be a point in the polytope PM so that the statement of Lemma 1 is +satisfied. +2: Let the edges of the original graph G = (V, E) be oriented so that the +statement of Lemma 2 is satisfied. +3: For every edge ei ∈ E, i ∈ [n], mark the edge ei as “discarded" independently +at random with probability 1/2. +4: Select a cut S ⊆ V uniformly at random, mark all edges not in [S; S] as +“discarded". Here, [S; S] stands for the set of edges which are oriented such +that their tail is in S and their head is in S. +5: Set thresholds ⃗T = (T1, . . . , Tn) as follows, for each i ∈ [n] +Ti := +� ++∞ +if ei is “discarded” +F −1 +i +(1 − pi) +otherwise . +Lemma 3. For every i ∈ [n], we have +P[ei is selected | Xi ≥ Ti and ei is not “discarded”] ≥ 1/2 . +Proof. Let us assume that the vertex v is the head of the oriented edge ei. Let +us also assume that ei is not marked as “discarded” and Xi ≥ Ti. +Since the edge ei is not “discarded”, the edge ei is in the selected set [S; S]. +Hence, every not “discarded” edge incident to v has the vertex v as its head. +Thus, as long as no other edge with the head at the vertex v is selected by +the gambler, the gambler has to select ei. We claim, that with probability at +least 1/2 no other edge with the head at v was selected by the gambler. +Let I be the event indicating that "the gambler selected an edge ej, j ̸= i +such that v is the head of ej", in other words “there is j ∈ [n], j ̸= i such that +12 + +v is the head of ej and Xj ≥ Tj and ej is not “discarded”". Let J indicate the +event that "ei is not marked as “discarded” after the selection of the cut", in +other words, "the head of ei is in S and the tail of ei is in S". +Let us show +P[I | J] ≤ 1/2 . +By the union bound, we have +P[I | J] ⩽ +� +j∈[n]\{i}:ej∈δ−(v) +P[Xj ≥ Tj and ej is not “discarded” | J] +Note that for each edge ej ∈ δ−(v) we have P[Xj ≥ Tj|J] = pj and we also +have P[ej is not “discarded”|J] = 1/4. Note that any edge is not “discarded” +in Step 3 of Algorithm 1 with probability 1/2, and not “discarded” in Step 4 +of Algorithm 1 with probability 1/4. However, since the probabilities are with +respect to the edge ej ∈ δ−(v) and are counted conditioned on the event J, +the conditioned probability of not being “discarded” in Step 4 of Algorithm 1 +is 1/2. Moreover, even conditioned on J the events "Xj ≥ Tj" and "ej is not +“discarded”" are independent events. Thus we have +� +j∈[n]\{i}:ej∈δ−(v) +P[Xj ≥ Tj and ej is not “discarded” | J] ≤ +� +j∈[n]\{i}:ej∈δ−(v) +pj/4 ⩽ 1/2 , +where the last inequality follows from the orientation. +We are ready to prove Theorem 3 by showing that Algorithm 1 is a 16- +competitive for graphic matroids without parallel edges. +Proof of Theorem 3. By Lemma 3 for every i ∈ [n] the probability of edge ei +being accepted conditional on Xi ≥ Ti and being not “discarded” is at least 1/2. +Overall, the probability of edge ei being accepted is at least +1 +16pi. Thus +mechanism guarantees at least �n +i=1 +1 +16piti of the expected total value. +By +Lemma 1, we have � +i∈[n] +1 +16piti ⩾ +1 +16EPROPHM, finishing the proof. +4.2 +k-column sparse matroids +There are known constant-competitive mechanisms for k-column sparse ma- +troids in the context of the secretary problem [Sot13]. However they do not +immediately lead to a non-adaptive mechanism of constant competitiveness +guarantee. The reason for that are not the updated thresholds but implicit +changes to the considered matroid. +Here, we present a constant competitive mechanism for k-column sparse +matroid class for each constant k. +Note, graphic matroids form a subclass +of 2-column sparse matroids . Because of their significance, 2-column sparse +matroids are also known in literature as represented frame matroids. Later, we +use 2-column sparse matroids to prove results in Section 6.4. +13 + +Suppose M is a k-column sparse matroid over field F. In this section, we +prove that there exists a (2k+2k)-competitive mechanism for M. +Suppose a k-sparse representation of M = (E, S) is defined by a map φ : +E → Fd. Note, if for some element t ∈ E the vector φ(t) is a zero vector then c +is a loop and therefore can be removed from consideration. +Now we consider an undirected hyper-multigraph G with vertex set [d]. Each +matroid element t ∈ E induces a hyperedge et in this graph between non-zero +coordinates of φ(t). Formally, the hyperedge et is defined as follows et := {i ∈ +[d] : φ(t)i ̸= 0}. We say that a vertex i ∈ [d] of the hyper-multigraph G is +incident to every edge e of G such that i ∈ e. For a vertex i ∈ [d] we denote +the collection of incident hyperedges by δ(i). The degree of a vertex i in the +hyper-multigraph G equals |δ(i)|. +Claim 1. Suppose I is an independent set of the matroid M. Then the average +degree of a vertex is at most k when one considers the hyper-multigraph with +vertices [d] and hyperedges {et : t ∈ I}. +Proof. Observe that |I| ⩽ d because having more than d vectors in d-dimensional +vector space Fd leads to a a linear dependency. +Since M is k-column sparse, we have that every edge in {et : t ∈ I} is +incident to at most k vertices in [d]. Hence, the total degree is at most kd and +thus the average degree of a vertex is at most k. +Now we consider orientations of the graph G. An orientation of the graph +G is a function ϕ which maps every edge et into one vertex of G incident to et. +We call ϕ(et) to be the head of the edge et, and all other vertices, if any, to +be tails. For every vertex i ∈ [d] we denote the set of incoming edges by δ−(i), +formally δ−(i) = {et : ϕ(et) = i, t ∈ E}. +Lemma 4. Let p be a point in the polytope PM. We assume that for every +t ∈ E, the coordinate pt of p corresponds to the element t. Then there exists +an orientation ϕ of hyperedges in the hyper-mulrigraph G such that for every +vertex i ∈ [d] we have � +t∈E:et∈δ−(i) pt ⩽ k. +The proof of Lemma 4 is analogous to the proof of Lemma 2. Now let us +describe an algorithm for k-column sparse matroids. +Lemma 5. For every t ∈ E we have +P[t is selected | Xt ≥ Tt and t is not “discarded”] ≥ 1/2 . +Proof. Note that item t ∈ E is accepted whenever Xt ≥ Tt and no other item +was selected from non-discarded edges in δ−(ϕ(t)). By the union bound, for +every event J we can upper bound the probability that +P[there j ∈ E \ {t} such that j is selected and ej ∈ δ−(ϕ(t)) | J] ⩽ +� +j∈E\{t}:ej∈δ−(ϕ(t)) +P[ej is not “discarded” and Xj ≥ Tj | J] . +14 + +Algorithm 2 A non-adaptive 2k+2k-competitive mechanisms for k-column +sparse matroids +1: Let p be a point in the polytope PM so that the statement of Lemma 1 is +satisfied. +2: Let the edges of the hyper-multigraph G be oriented so that the statement +of Lemma 4 is satisfied. +3: For every edge ei ∈ E, i ∈ [n], mark the edge ei as “discarded" independently +at random with probability 1 − +1 +2k. +4: Select a cut S ⊆ [d] uniformly at random, mark all edges not in [S; S] as +“discarded”. Here, [S; S] stands for the set of edges which are oriented such +that all their tails are in S and their head is in S. In particular, for t ∈ E we +say that et lies in a cut [S; S] with respect to the orientation ϕ if ϕ(et) ∈ S +and for every i ∈ et \ {ϕ(et)} we have i ∈ S. +5: Set thresholds {Tt : t ∈ E} as follows, for each t ∈ E +Tt := +� ++∞ +if t is “discarded” +F −1 +t +(1 − pt) +otherwise . +Let J indicate the event that "et is not marked as “discarded” after the selection +of the cut". Then for each j ∈ E \{t} we have P[ej is not “discarded” and Xj ≥ +Tj | J] ≤ +1 +2kpj. By Lemma 4, we have � +j∈E:ej∈δ−(ϕ(t)) pj ⩽ k, leading to the +desired inequality. +Note that the proof of Lemma 5 is analogous to the proof of Lemma 3. We are +ready to prove Theorem by showing that the Algorithm 2 is a 2k+2k-competitive +for k-column sparse matroids. +Proof of Theorem 4. For every item t ∈ E we have P[Xt ≥ Tt] = pt and +P[t is not “discarded”] ≥ +1 +2k+1k. By Lemma 5, we have that with probability +at least 1/2 the item t is selected when it is not “discarded” and Xt ≥ Tt. Thus +the expected total value of Algorithm 2 is at least � +j∈E +1 +2k+2kpjtj which is at +least +1 +2k+2kEPROPHM by Lemma 1. +5 +Cographic and gamma-sparse matroids +5.1 +Cographic matroids +Let us revisit a mechanism of Soto [Sot13] for the cographic matroid secretary +problem which is based on the following corollary of Edmond’s matroid parti- +tioning theorem [Edm65]. This mechanism leads to a non-adaptive mechanism +for cographic matroids. +15 + +Proposition 1. Let G = (V, E) be a three edge-connected graph. Then there +exist spanning trees H1, H2, H3 in G such that the union of their complements +contains all the edges E, i.e. E = (E \ H1) ∪ (E \ H2) ∪ (E \ H3). +Algorithm 3 A non-adaptive 3-competitive mechanisms for cographic matroids +in the case of three edge-connectivity +1: Let H1, H2 and H3 be the spanning trees as in Proposition 6. +2: Uniformly at random select a spanning tree H∗ from H1, H2 and H3. Set +thresholds {Te : e ∈ E} as follows, for each e ∈ E +Te := +� ++∞ +if e is not in H∗ +0 +otherwise . +Lemma 6. Let G = (V, E) be a three edge-connected graph and let M be the +cographic matroid over G. Then Algorithm 3 is a 3-competitive non-adaptive +mechanism for the matroid M. +Proof. The expected total value of the mechanism provided by Algorithm 3 +equals E[� +e∈E\H∗ Xe] which can be estimated as follows +E[ +� +e∈E\H∗ +Xe] = 1 +3E[ +� +i∈[3] +� +e∈E\Hi +Xe] ≥ 1 +3E[ +� +e∈E +Xe] ≥ 1 +3EPROPHM . +The next theorem provides a proof for Theorem 5. +Theorem 11. Let G = (V, E) be a graph and let M be the cographic matroid +over G. Then Algorithm 4 is a 6-competitive non-adaptive mechanism for the +matroid M. +Proof. We can assume that G does not have bridges, because every such bridge +is a loop in M. Thus these edges can be selected neither by the gambler nor by +the prophet. So we can assume G = G′ and M = M ′. +In the case when each connected component of G is three edge-connected, +then Algorithm 4 runs Algorithm 3 for each component to obtain a 3-competitive +non-adaptive mechanism. +Otherwise, there is one or more pairs of edges e,e′ such that {e, e′} corre- +sponds to a cut in G. In this case, the edges e,e′ correspond to parallel elements +of the cographic matroid M. +Algorithm 4 considers the partition of E into classes of parallel elements C1, +C2, . . . , Ck. Let us construct the matroid M ′′ from M by contracting all but +one edge in each class C1, C2, . . . , Ck. Note, that the ground set of M ′′ has k +elements. Abusing the notation we refer to these elements of the ground set as +16 + +Algorithm 4 A non-adaptive 6-competitive mechanisms for cographic matroids +1: Delete all loops of M to obtain a matroid M ′. Remove all bridges from +G = (V, E) and obtain a graph G′ = (V ′, E′). +2: Let C1,. . . , Ck be equivalence classes of M ′ with respect to the relation of +being parallel. Construct the matroid M ′′ from M ′ by contracting all but +one edge in each class C1, C2, . . . , Ck. Note, that the ground set of M ′′ +has k elements and matroid M ′′ is the cographic matroid over a graph G′′, +where each connected component of G′′ is three edge-connected. Abusing +the notation we refer to the elements of the ground set of M ′′ as C1, C2, +. . . , Ck. +3: Let H1, H2 and H3 be forests in G′′ such that the restriction of H1, H2 +and H3 to each connected component of G′′ satisfies Proposition 6 for the +respective connected component. +4: Uniformly at random select a forest H∗ from H1, H2 and H3. +5: For each i ∈ [k] select thresholds T e, e ∈ Ci according to Theorem 1 when +the gambler is allowed to accept only one item of Ci and the distributions +of Xe, e ∈ Ci are the same as original distributions of values for e ∈ Ci. +6: Set thresholds {Te : e ∈ E} as follows, for each e ∈ E +Te := +� +T e +if e ∈ Ci and Ci ∈ H∗ for some i ∈ [k] ++∞ +otherwise . +C1, C2, . . . , Ck. The matroid M ′′ is isomorphic to the cographic matroid over +a graph G′′, where each connected component of G′′ is three edge-connected. +Following Lemma 6, Algorithm 4 constructs forests H1, H2, H3 for the graph G′′. +So Algorithm 4 leads us to a 6-competitive mechanism. Indeed, the prophet +with M and with the original distributions of Xe, e ∈ E performs exactly +as the prophet with M ′′ and with the corresponding distributions of X′′ +i := +maxe∈Ci Xe, i ∈ [k]. By selecting forests in Algorithm 4 the gambler acheives +in expectation E[� +i∈[k] X′′ +i ]/3 when all classes C1, C2, . . . , Ck are singletons. +However, for classes that are not singletons we need to take into account an- +other 2 approximation factor with respect to the prophet, who can achieve the +expected value E[X′′ +i ] for each i ∈ [k], while the gambler is guaranteed in ex- +pectation to achieve only E[X′′ +i ]/2 for each i ∈ [k]. +5.2 +Gamma-sparse matroids +Let us also revisit a mechanism of Soto [Sot13] for γ-sparse matroids to verify +that it directly leads to a non-adaptive mechanism. +Theorem 12. Let M = (E, S) be a γ-sparse matroid. +There exists a γ- +competitive non-adaptive mechanism for M. +Proof. First observe that the point x := 1/γ lies in the matroid polytope PM. +17 + +Indeed, it is non-negative and for every set S ⊆ E(M) we have x(S) = |S|/γ ⩽ +rM(S). +Then x can be expressed as a convex combination of indicator variables +corresponding to the independent sets of M. +In other words, we have x = +� +S∈S αS1S for some α ⩾ 0, � +S∈S αS = 1, where 1S refers to the characteristic +vector of S. +Now sample an independent set S in matroid M randomly with probabil- +ity αS. Let the gambler select all items in S and let the gambler leave all the +items not in S unselected. +If Xe is the random variable corresponding to the weight of element e ∈ +E(M), then this mechanism results in a total expected value as follows +� +S∈S +αS +� +e∈S +E[Xe] = +� +e∈E +(1/γ)E[Xe] = E[ +� +e∈E +Xe]/γ ⩾ EPROPH/γ , +finishing the proof. +Observe that Proposition 1 implies that for a three edge-connected graph G, +the cographic matroid of G is 3-sparse. Thus Lemma 6 is a corollary of Theo- +rem 12. +Similarly, for a planar graph G the graphic matroid is 3-sparse, leading us +to the following corollary. +Corollary 1. Let G is a planar graph and let M be the corresponding graphic +matroid. There is a 3-competitive non-adaptive mechanism for M. +6 +Representable matroids +Many results in the theory of matroids make use of minors coming from re- +strictions and contractions. To get access to the toolbox provided by matroid +theory, we need to understand how prophet inequality guarantees change when +we consider minors. +6.1 +Preliminaries +Lemma 7. Let M be a matroid and let matroid N be a restriction of the ma- +troid M. If there exists an α-competitive non-adaptive mechanism on M, then +there is an α-competitive non-adaptive mechanism for N. +Proof. To obtain a mechanism for the matroid N, we can impose thresholds +∞ +for the items that were removed from the ground set to obtain the restriction N +from the matroid M. The remaining items are assigned the same thresholds in +both mechanisms. +A similar result for contractions is harder to obtain in the case of non- +adaptive mechanisms. +Indeed, a straightforward approach would require us +to impose the thresholds +∞ for the contracted items, while using the given +18 + +mechanism on the remaining items. Unfortunately, this would also require us +to “change" the underlying matroid, in other words a gambler might be forced +to reject an item even though its value is over the assigned threshold and its +addition to the currently selected items keeps the selected set independent with +respect to M. +Because of this difficulty, in this work we provide a matching result for +contractions only for matroids representable over a finite field. This result is +sufficient for the purpose of this work. +Lemma 8. Let M = (E, S) be a matroid representable over the field Fp for +some p. Let T ⊆ E be a subset of the ground set such that λM(T ) ⩽ k for +some k. +Then there exists S ⊆ T so that every set that is independent in M |S is also +independent in M/T and +EPROPHM|S ⩾ +1 +pk+1 EPROPHM/T . +Recall that T stands for the complement of T with respect to the ground set E. +Proof. Consider the representation of the matroid M over Fp. Let φ : E → Fm +p +be the map describing the representation of M. Thus, for every S ⊆ E we have +that the set φ(S) = {φ(e) ∈ Fm +p : e ∈ S} is independent over the field Fp if and +only if S is an independent set for the matroid M. +Since λM(T ) ⩽ k holds, by definition of λM we have +rM(T ) + rM(T ) − rM(E) ⩽ k . +We have rM(R) = dim span(φ(R)) for every R ⊆ E. Thus, we have +dim span φ(E) = dim span φ(T )+dim span φ(T )−dim +� +(span φ(T )) ∩ (span φ(T)) +� +. +and so +dim +� +(span φ(T )) ∩ (span φ(T )) +� +⩽ k . +Since we are working over the field Fp, the linear space L := (span φ(T )) ∩ +(span φ(T )) has at most pk vectors. Let C be the orthogonal complement of +the linear space L in the space span φ(T ). Thus, we can represent span φ(T ) as +L ⊕ C. For every vector v ∈ span φ(T ) we denote v orthogonal projection to L +and C by v |L and v |C, respectively. +For each vector a ∈ L, define the set Ta := {t ∈ T : φ(t) |L= a, φ(t) ̸= a}. +Note that by definition for every a ∈ L we have Ta ∩ L = ∅. Now let us select +a uniformly at random from L. +Claim 2. Ea[EPROPHM|Ta ] ≥ +1 +pk EPROPHM/T . +Proof. To prove the desired inequality, we prove the corresponding inequality +for any realization of item values. From now on we consider the realization of +item values fixed and thus we prove the following inequality +Ea[PROPHM|Ta] ≥ 1 +pk PROPHM/T +19 + +Let us consider the set Iopt on which the prophet achieves PROPHM/T . +Note that the set Iopt does not contain any item e such that φ(e) is in L, because +every such an item e is a loop in M/T . Thus, the set Iopt can be partitioned +into sets Iopt,a, a ∈ L where Iopt,a is a subset of Ta. +The set Iopt is independent in M/T and so Iopt is also independent in M. +Hence the sets Iopt,a, a ∈ L are also independent in M. Thus for every a ∈ L, +PROPHM|Ta ⩾ w(Iopt,a). Then we have +Ea[PROPHM|Ta] ⩾ +� +a∈L w(Iopt,a) +|L| += 1 +|L|w(Iopt) ⩾ 1 +pk PROPHM/T , +finishing the proof of the claim. +Let us now select a∗ ∈ L such that EPROPHM|Ta is maximized. By the +previous claim, we have +PROPHM|Ta∗ ⩾ 1 +pk PROPHM/T . +Now for every c ∈ C define set Hc := {t ∈ Ta∗ : (φ(t) |C) · c = 1}. Now let us +select c uniformly at random from C. +Claim 3. Ec[EPROPHM|Hc ] ≥ 1 +pEPROPHM|T a∗ . +Proof. To prove the desired inequality, we prove the corresponding inequality +for any realization of item values. From now on we consider the realization of +item values fixed and thus we prove the following inequality +Ec[PROPHM|Hc ] ≥ 1 +pPROPHM|T a∗ . +Let Iopt be the set corresponding to PROPHM|Ta∗ . Thus, we have that for +every e ∈ Iopt, φ(e) is not in L and hence φ(e) |C is not the zero vector. Due to +Pc[c · t = 1] = 1/p, for every t ∈ Ta∗, we have +Ec[w(Iopt∩Hc)] = +� +t∈Iopt +Pc[c·t = 1]w(t) = 1 +p +� +t∈Iopt +w(t) = 1 +pw(Iopt) = PROPHM|Ta∗ . +Finally, since Iopt is independent in M so is Iopt ∩ Hc. Thus, we have +Ec[PROPHM|Hc ] ≥ 1 +pPROPHM|T a∗ , +finishing the proof of the claim. +Now let us select c∗ so that EPROPHM|Hc is maximized and let S∗ := Hc∗. +Then we have EPROPH(M |S∗) ⩾ +1 +pk+1 EPROPH(M/T ). +Finally, we need to show that every set independent in M |S∗ is an indepen- +dent set in M/T . Suppose the contrary, i.e. there exists a set that is independent +20 + +in M |S∗ but is not an independent set in M/T . Then span S∗ has a non-trivial +intersection with span T, suppose x ∈ (span φ(S∗)) ∪ (span φ(T )). Let us show +that x is a zero vector. Since x ∈ span S∗, we have x = � +s∈S∗ αsφ(s) for some +αs ∈ Fp, s ∈ S∗. +Let us consider the projections of x on C and L. Since x ∈ span φ(T ) we have +that x lies in L and so x |C is the zero vector. Thus x |C= � +s∈S∗ αs(φ(s) |C) +is the zero vector. +Note that by definition, φ(s) |L= a∗ and c∗ · (φ(s) |C) = 1 hold for every s ∈ +S∗ . Thus over the field Fp we have +� +s∈S∗ +αs = +� +s∈S∗ +αs(c∗ · (φ(s) |C)) = c∗ · +� � +s∈S∗ +αs(φ(s) |C) +� += +c∗ · (x |C) = 0 . +Now let us consider x |L. We have +x |L= +� +s∈S∗ +αs(φ(s) |L) = +� � +s∈S∗ +αs +� +a∗ , +where the last expression equals the zero vector since � +s∈S∗ αs = 0. Thus we +have a vector x ∈ L ⊕ C such that both projections x |L and x |C are the zero +vector. Hence, the vector x is the zero vector, finishing the proof. +6.2 +Tree Decompositions +Similarly to the approach [HN20] for the matroid secretary problem, we exten- +sively use the tree decomposition of matroids. A tree decomposition of bounded +thickness allows us to construct non-adaptive mechanisms with good approxi- +mation ratios. Before proceeding with these constructions, let us introduce tree +decompositions. +A tree decomposition of a matroid M = (E, S) is a pair (T, X) where T is +a tree and X = {Xv ⊆ E : v ∈ V (T )}, where sets in X form a partition of +E. Here, we refer to the vertex and edge sets of the tree T as V (T ) and E(T ), +respectively. +Given an edge e = {v1, v2} ∈ E(T ) of the tree T , let T1 and T2 be two +connected components of T − e, in other word the removal of the edge e from +T leads to two connected components T1 and T2. The thickness of the edge +e = (v1, v2) is denoted as λ(e) and is defined as follows +λ(e) := λM(∪v∈V (T1)Xv) . +The thickness of the tree decomposition is the maximum thickness of the edge e +in E(T ). +Let M be a family of matroids, M be a matroid and (T, X) be a tree decom- +position of M. We say that tree decomposition (T, X) is M-tree decomposition +21 + +if M |clM(Xv)∈ M holds for every v ∈ V (T ). Let tk(M) be a set of matroids +which have M-tree decomposition of thickness at most k. +Theorem 13. Let Mα,p be the family of matroids which admit α-competitive +non-adaptive mechanisms and are representable over the finite field Fp. Then +for every natural number k and every matroid M in tk(Mα,p), the matroid M +has an (αpk+1)-competitive non-adaptive mechanism. +Proof. For a natural number m, let tk,m(Mα,p) be the set of matroids which +have an Mα,p-tree decomposition (T, X) of thickness at most k satisfying |V (T )| = +m. +Let us prove the statement of the lemma by induction on m. +The base +case follows from the definition of the family Mα,p and the fact that Mα,p = +tk,1(Mα,p). +Let us now show how to do the inductive step. Let us assume m ≥ 2 and +consider a matroid M = (E, S) in tk,m(Mα,p) with its Mα,p-tree decomposition +(T, X) of thickness at most k and with |V (T )| = m. Let ℓ be a leaf of the tree T +and let u be the neighbour of the vertex ℓ in the tree T . +Observe that the tree (V (T ) \ {ℓ}, E(T )\ {ℓu}) together with the subfamily +{Xw : w ∈ V (T ) \ {ℓ}} defines an Mα,p-tree decomposition of the matroid +M \ Xℓ. Thus the matroid M \ Xℓ is in M ∈ tk,m−1(Mα,p). Hence, by the +inductive hypothesis there are thresholds T ′ +e, e ∈ E \ Xℓ guaranteeing αpk+1- +competitiveness of the gambler in comparison to the prophet on the matroid M \ +Xℓ. +Claim 4. There are thresholds T ′′ +e , e ∈ Xℓ leading to an (α · pk+1)-competitive +non-adaptive mechanism for matroid M |Xℓ, such that the gambler always selects +a set that is independent in M/Xℓ. +Proof. By Lemma 8 there exists a set S ⊆ Xl such that every set independent +in M |S is also independent in the matroid M/Xℓ and +EPROPHM|S ⩾ +1 +pk+1 EPROPHM/Xℓ . +By definition of Mα,p and the appearance of Xℓ in the tree decomposition, we +have that M |Xℓ is in the family Mα,p. By Lemma 7, since S is a subset of Xℓ +the matroid M |S is also in the family Mα,p. Thus, there are thresholds T ′′ +e , +e ∈ S that lead to an α-competitive non-adaptive mechanism on M |S. The +thresholds T ′′ +e , e ∈ Xℓ \ S can be defined as +∞, finishing the proof of the +claim. +Now we can define thresholds Te, e ∈ E for all elements of the matroid M +as follows +Te := +� +T ′ +e +if e ̸∈ Xℓ +T ′′ +e +otherwise. +Let us now demonstrate that such thresholds Te, e ∈ E lead to an (αpk+1)- +competitive non-adaptive mechanism for M. +22 + +First, by the above claim the selected items from Xℓ always form an inde- +pendent set in M/Xℓ when used with the thresholds Te, e ∈ Xℓ on the matroid +M |Xℓ. Thus the definition of the thresholds guarantees that in expectation the +value of selected items from Xℓ is at least EPROPHM|Xℓ/(αpk+1); and in ex- +pectation the value of selected items from E\Xℓ is at least EPROPHM\Xℓ/(αpk+1). +To finish the proof, note that we have +PROPHM|Xℓ + PROPHM\Xℓ ≥ PROPHM +and so +EPROPHM|Xℓ + EPROPHM\Xℓ ≥ EPROPHM . +6.3 +Regular matroids +In this section, we prove Theorem 7. Before we proceed to the proof, let us +define key notions related to regular matroids. +A subset of the matroid’s ground set is called a circuit, if it is an inclusion- +minimal dependent set. A cycle is a subset of the ground set which can be +partitioned into a disjoint union of circuits. +Let M1 = (E1, S1), M2 = (E2, S2) be two binary matroids. Then the matroid +sum M1△M2 has the ground set E1△E2 and the cycles of M1△M2 are all sets +of the form C1△C2, where C1 is a cycle for M1 and C2 is a cycle for M2. +Definition 3. Consider two binary matroids M1 = (E1, S1), M2 = (E2, S2) +and M = M1△M2. +1. If |E1 ∩ E2| = 0, and E1 ̸= ∅, E2 ̸= ∅, M is called a 1-sum of M1 and +M2. +2. If |E1 ∩ E2| = 1, |E1| ≥ 3, |E2| ≥ 3 and E1 ∩ E2 is not a loop of M1 or +M2 or their dual matroids, M is called a 2-sum of M1 and M2. +3. If |E1 ∩ E2| = 3, |E1| ≥ 7, |E2| ≥ 7 and E1 ∩ E2 is a circuit in both M1 +and M2, and E1 ∩ E2 does not contain a circuit in their dual matroids, +then M is called a 3-sum of M1 and M2. +Proof of Theorem 7. By Seymour’s regular matroid decomposition theorem [Sey80], +every regular matroid M can be obtained from graphic, cographic or a special +matroid R10 through a sequence of 1-sums, 2-sums or 3-sums. +This gives a tree decomposition (T, X) of thickness at most 2 so that each M |Xv, +v ∈ V (T ) is either a graphic, cographic or a special matroid R10. +By performing parallel extensions of the elements to be deleted before each +2-sum and 3-sum, we construct a matroid M ′, so that M is a restriction of M ′ +and M ′ has a tree decomposition (T, X ′) so that each M ′ |clM′ (X′v), v ∈ V (T ) +is either graphic, cographic or a parallel extension of R10. +23 + +By Theorem 2, every graphic matroid has a 32-competitive non-adaptive +mechanism. By Theorem 5, every cographic matroid has a 6-competitive non- +adaptive mechanism. Since matroid R10 has ground set of size 10, by Theorem 1 +every parallel extension of R10 has a 20-competitive non-adaptive mechanism. +Note that by definition every regular matroid is representable over finite +field F2. Thus, by Theorem 13 with p = 2, k = 2 and α = 32 there is a 256- +competitive non-adaptive mechanism for matroid M ′. Since M is a restriction +of M ′, by Lemma 7, there is a 256-competitive non-adaptive mechanism for M, +finishing the proof. +6.4 +Minor-closed representable matroid families +In this section we show that every minor-clossed subclass of matroids repre- +sentable over Fp has a constant-competitive non-adaptive mechanism, where +the constant is a function only of p. The proof of this fact is analogous to the +proof in [HN20]. +Theorem 14 (Theorem 4.3 in [Gee11]). Given natural numbers q ⩾ 2 and +n ⩾ 1, let M = (E, S) be a matroid with no U2,q+2 or M(Kn) minors. Then +we have |E| ≤ qq3nrM(E). +Corollary 2. Given natural numbers q ⩾ 2 and n ⩾ 1, let M = (E, S) be a +matroid with no U2,q+2 or M(Kn) minors. Then there exists a qq3n-competitive +non-adaptive mechanism for M. +Proof. If M has no U2,q+2 or M(Kn) minors, then every restriction of M also +has no U2,q+2 or M(Kn) minors. +Thus for every X ⊆ E we have |X| ⩽ +qq3nrM(X). So, M is a qq3n-sparse matroid and by Theorem 12 there exists +a qq3n-competitive non-adaptive mechanism for M. +6.4.1 +Projections and lifts +Let M be a matroid and x be an element of the ground set, which is a not a +loop and not a free element of the matroid M. Then M/x is called a projection +of M \ x; M \ x is called a lift of M/x. Note that here and later we write M/x +and M \ x instead of M/{x} and M \ {x}, repsectively. +Let M and N be two matroids with the same ground set. We say that the +distance between M and N is t, denoted by dist(M, N) = t if t is the smallest +integer such that there exists a sequence of matroids P0, P1, . . . , Pt where +P0 = M and Pt = N and for every i ∈ [t] the matroid Pi is either a lift or a +projection of Pi−1. +Lemma 9. Let N be a lift of the matroid M. If there is an α-competitive non- +adaptive mechanism for M then there exists a (2α+2)-competitive non-adaptive +mechanism for N. +24 + +Proof. Since N is a lift of M, there exists a matroid L = (E, S) and an element +x of its ground set, such that M = L/x, N = L\x. Here, x is not a loop and +not a free element of L. +Let P be the set of elements in L that are parallel to x, in other words P := +{x′ ∈ E : x′ ∥ x}. Note that N |P \{x} is a uniform matroid of rank 1. Note also +that elements in P \{x} are loops in M and so EPROPHM = EPROPHM\P . +Let T ′ +e, e ∈ E \ {x} be the thresholds imposed by an α-competitive non- +adaptive mechanism for the matroid M. +Let T ′′ +e , e ∈ P be the thresholds +guaranteeing 2-competitive non-adaptive mechanism as in Theorem 1 for the +uniform matroid of rank 1 on the ground set P \{x}; and let T ′′ +e , e ∈ E\(P ∪{x}) +be +∞. We select one of these two sets of thresholds for the matroid N as +described below. The constructed mechanism for the matroid N selects one +of those two sets at random, where first set of thresholds T ′ +e, e ∈ E \ {x} is +selected with probability γ := α/(α + 1) and the second set T ′′ +e , e ∈ E \ {x} +with probability 1 − γ = 1/(α + 1). +Next part is dedicated to the analysis of how thresholds T ′ +e, e ∈ E \ {x} +perform on the matroid N. +Note, that these thresholds are coming from a +mechanism for the matroid M, while they are used for the matroid N with +probability γ. We show that the total expected value achieved by thresholds +T ′ +e, e ∈ E \ {x} on N is at least the total expected value achieved by these +thresholds on M. For this we can assume that for every realization of item +values, the orders of items in matroid N and M are the same. To see that +this assumption is valid, we can assume that the order for N is chosen in an +adversarial way and is used also as the items order for M. +Claim 5. Let us assume that the items order for M and N is the same for +a given realization of item values. Let us also assume that for every item e ∈ +E \ {x} the threshold T ′ +e is used. Then the gambler with matroid N selects all +items that the gambler with matroid M selects. +Proof. We fix the item values realization and items order. Let e1, e2,. . . , ek be +the items with their values being at least their threshold and with the corre- +sponding order. +Now we need to show that if the gambler with matroid N selects items +greedily from e1, e2,. . . , ek starting from e1, then the set of selected items is a +superset of the items greedily selected by the gambler with matroid M. If both +gamblers end up selecting exactly the same set of items, then proof of the claim +is complete. Otherwise consider the first index i ∈ [k] such that the item ei is +selected by exactly one of the two gamblers. Since N = L\x and M = L/x we +have that it is only possible if ei is selected by the gambler with the matroid N +and rejected by the gambler with the matroid M. +Now we claim that every subsequent item, in other words an item in ei+1, . . . , +ek, is either selected by both gamblers or rejected by both gamblers. Suppose +the contrary and consider the first item ej, i + 1 ≤ j ≤ k that is selected by +one gambler and rejected by another gambler. Let S := {e1, e2, . . . , ej−1} and +let T be the set of items selected by the gambler with M from the set S. Thus +25 + +the gambler with N selected T ∪ {ei} from the set S. So T ∪ {ei} is a basis of +(L\x) |S and T is a basis of (L/x) |S. Thus, both T ∪{ei} and T ∪{x} are bases +of L |S. If only one of the two gamblers accepts the item sj then the matroid +L |S∪{sj} has two bases of different cardinality, attaining a contradiction and +finishing the proof. +Thus we have that the thresholds T ′ +e, e ∈ E\{x} guarantee at least EPROPHM +as the expected total value of the gambler with N. To prove that the constructed +mechanism is 1/(2α + 2)-competitive it is enough to show the following claim. +Note that in our construction we used α-competitive non-adaptive mechanism +for the matroid M and 2-competitive non-adaptive mechanism for the uniform +matroid of rank 1 on P \ {x}. +Claim 6. γ 1 +αEPROPHM + (1 − γ) 1 +2EPROPHP \{x} ⩾ +1 +2α+2EPROPHN +Proof. Let us consider the inclusion-maximal set Iopt on which the prophet +achieves PROPHN. Let Copt be a random variable corresponding to the unique +circuit of Iopt ∪ {x} in L. Recall that x is not a free element of L so such a +circuit exists and is unique and contains x. +First consider the events when |Copt| ⩾ 3. +Note that by definition of a +circuit, for every y ∈ Copt \ {x} the set (Iopt ∪ {x}) \ {y} is independent in +L. Hence, for every y ∈ Copt \ {x} the set Iopt \ {y} is independent in M. So +we have that conditioned on |Copt| ⩾ 3 we have PROPHM ≥ w(Iopt \ {y}) +for every y ∈ Copt \ {x}. +Let yopt be the random variable representing the +element in Copt \{x} of smallest value. Then conditioned on |Copt| ⩾ 3, we have +w(Copt \ {yopt, x}) ≥ w(C \ {x})/2. Thus, conditioned on |Copt| ⩾ 3 we have +PROPHM ⩾ w(Iopt \ {yopt}) = w(Iopt \ Copt) + w(Copt \ {yopt}) +≥ w(Iopt \ Copt) + 1 +2w(Copt \ {x}) ≥ 1 +2w(Iopt) = 1 +2PROPHN . +Second consider the event that |Copt| < 3. Since x is not a loop of L by +definition, we have |Copt| = 2 and so Copt = {x, xopt} for some random variable +element xopt ∈ P \{x}. For the event |Copt| ≥ 3 let us define the random variable +element xopt to be an arbitrary element in Copt \ {x}. Thus, if |Copt| < 3 we +have PROPHP \{x} ≥ w(xopt). Now let us define Jopt := Iopt \ {xopt} and note +that Jopt is independent in the matroid M. Moreover, since Iopt is the set on +which the prophet achieves PROPHN, we have that conditioned on |Copt| < 3 +the prophet achieves PROPHM on the set Jopt. +Combining everything together we have +γ 1 +αEPROPHM + (1 − γ)1 +2EPROPHP \{x} = +1 +α + 1EPROPHM + +1 +2α + 2EPROPHP \{x} ≥ +E +�w(xopt) +2α + 2 + PROPHM +α + 1 +���� |Copt| < 3 +� +P [|Copt| < 3] +26 + ++ E +�PROPHM +α + 1 +���� |Copt| ⩾ 3 +� +P [|Copt| ⩾ 3] = +E +�w(xopt) +2α + 2 + w(Iopt \ {xopt}) +α + 1 +���� |Copt| < 3 +� +P [|Copt| < 3] ++ E +�PROPHM +α + 1 +���� |Copt| ⩾ 3 +� +P [|Copt| ⩾ 3] ≥ +E +�PROPHN +2α + 2 +���� |Copt| < 3 +� +P [|Copt| < 3] ++ E +�PROPHM +α + 1 +���� |Copt| ⩾ 3 +� +P [|Copt| ⩾ 3] ≥ +1 +2α + 2EPROPHN . +Lemma 10. Let N be a matroid obtained from a matroid M by a sequence of +t projections. Let L be the set of loops in the matroid N. Let there exist an +α-competitive non-adaptive mechanism for the matroid M. Then there exists +a non-adaptive mechanism for N\L such that the expected total value of this +mechanism is at least +1 +α·3t EPROPHM\L. +In the context of Lemma 10, every set that is independent for the matroid N\ +L is also independent for the matroid M \ L. Hence, we have EPROPHM\L ≥ +EPROPHN\L. Thus in case t = 1, Lemma 10 leads us to the following corol- +lary. +Corollary 3. Let N be a projection of the matroid M. +If there is an α- +competitive non-adaptive mechanism for M then there exists a 3α-competitive +non-adaptive mechanism for N. +Proof of Lemma 10. Let us prove the statement by induction. +Of course, in +case t = 0 we have M = N and the statement is trivially true. +Let us now assume that t is at least 1. Let N ′ be a matroid such that N ′ +is obtained from the matroid M by a sequence of t − 1 projections and N is a +projection of N ′. Since N is a projection of N ′ there is a matroid P = (E, S) +and x ∈ E such that P \ x = N ′ and P/x = N. Let L′ be the set of loops in +the matroid N ′. +By induction hypothesis, there exist thresholds T ′ +e, e ∈ E \ (L′ ∪ {x}) such +that the gambler with the matroid N ′\L′ achieves at least +1 +α·3t−1 EPROPHM\L′ +as the expected total value. +Let us assume that to compute thresholds T ′ +e, +e ∈ E \(L′ ∪{x}) the values of items in L were set to be 0 while the distribution +of values for other items remain the same. Since L′ ⊆ L, analogously to Lemma 7 +we can define thresholds +T ′′ +e := +� ++∞ +if e ∈ L +T ′ +e +otherwise +27 + +such that the gambler with the matroid N ′\L achieves at least +1 +α·3t−1 EPROPHM\L +as the expected total value. Let T ′′′ +e , e ∈ E \(L∪{x}) be the thresholds guaran- +teeing 2-competitive non-adaptive mechanism as in Theorem 1 for the uniform +matroid of rank 1 on the ground set E \ (L ∪ {x}). +The constructed mechanism for the matroid N\L selects one of two threshohold +sets at random, where first set of thresholds T ′′ +e , e ∈ E \ (L ∪ {x}) is selected +with probability 1/3 and the thresholds T ′′′ +e , e ∈ E \ (L ∪ {x}) with probability +2/3. Note that the thresholds T ′′ +e , e ∈ E \ (L ∪ {x}) were designed for the +matroid N ′ \ L but are used for the matroid N \ L; hence less items might be +selected than when it is used for N ′ \ L. Also note, that the thresholds T ′′′ +e , +e ∈ E \ (L ∪ {x}) are used for N \ L but were designed for the uniform matroid +of rank 1. +For the analysis, let Ialg be the random variable indicating the items set +selected by the gambler with matroid N ′ \ L when the thresholds T ′′ +e , e ∈ +E \ (L ∪ {x}) are used. Analogously to a claim in the proof of Lemma 9, we can +assume that when the thresholds T ′′ +e , e ∈ E\(L∪{x}) are used the gambler with +N \L select all items in Ialg with an exception for possibly one item. Let xopt be +the random variable indicating the element of maximum value in E \ (L ∪ {x}). +To finish the proof it is enough to show the following inequality +1 +3E[w(Ialg) − w(xopt)] + 2 +3 +1 +2E[w(xopt)] ≥ +1 +α · 3t EPROPHM\L . +To obtain this inequality we can do estimations as follows +1 +3E[w(Ialg)−w(xopt)]+2 +3 +1 +2E[w(xopt)] = 1 +3E[w(Ialg)] ≥ 1 +3 +1 +α · 3t−1 EPROPHM\L . +Now let us combine Corollary 3 and Lemma 9. +Lemma 11. Let M and N be matroids such that dist(M, N) ≤ t. If there exists +an α-competitive non-adaptive mechanism for the matroid M with α ≥ 2 then +there exists a 3tα-competitive non-adaptive mechanism for the matroid N. +Proof. Note that for α ≥ 2 we have 3α ≥ 2α + 2. Since N can be obtained +from M by a sequence of t projection and lift steps, we can use Corollary 3 or +Lemma 9 for each of these steps to obtain the desired competitiveness ratio. +6.4.2 +Minor-closed families theorem +Lemma 12 (Lemma 6 in [HN20]). Let p and n be integers such that p ⩽ n − 2 +and p is prime. The matroid U2,n is not representable over the field Fp. +The following Structural Hypothesis is due to Geelen, Gerards and Whittle. +The proof of this Structural Hypothesis has not appeared in print. +28 + +Hypothesis 1. Let p be a prime number and M is a proper minor-closed class +of matroids representable over Fp. +Then there exist k, n, t such that every M ∈ M is a restriction of an Fp- +representable matroid M ′ having a full tree-decomposition (T, X) of thickness at +most k so that for every v ∈ V (T ) if M ′ |clM′(Xv) has a M(Kn) minor, then +there exists a 2-column sparse matroid N with dist(M ′ |clM′ (Xv), N) ⩽ t. +Proof of Theorem 8. Let k, n, t are as stated in the Structural Hypothesis 1 on +M. +Let M1 be the set of matroids on distance t or less from some 2-column +sparse matroid and are representable over Fp. +By Theorem 4 all 2-column +sparse matroids have a 32-competitive non-adaptive mechanism. By Lemma 11 +there exists a (3t · 32)-competitive mechanism for matroids in M1. +Let M2 be the set of matroids without M(Kn) minor and are representable +over Fp. By Lemma 12 all matroids in M2 do not have U2,p+2 as a minor. +Then by Corollary 2, we have that there is a pp3n-competitive non-adaptive +mechanism for every matroid in M2. +By the Structural Hypothesis 1 we have that every M ∈ M is a restriction +of some M ′ with a full tree-decomposition (T, X) of thickness at most k so that +for every v ∈ V (T ) M ′ |clM′ (Xv)∈ M1 ∪ M2. +Thus by Theorem 13, matroid M ′ has a γ := (max(3t · 32, pp3n) · pk+1)- +competitive non-adaptive mechanism. By Lemma 7 the matroid M has also a +γ-competitive non-adaptive mechanism. +29 + +References +[AKW19] +Pablo D. Azar, Robert Kleinberg, and S. Matthew Weinberg. Prior +independent mechanisms via prophet inequalities with limited in- +formation. Games and Economic Behavior, 118:511–532, 2019. +[CFPP21] +Constantine Caramanis, Matthew Faw, Orestis Papadigenopoulos, +and Emmanouil Pountourakis. Single-sample prophet inequalities +revisited. ArXiv, abs/2103.13089, 2021. +[CGKM20] Shuchi Chawla, Kira Goldner, Anna R Karlin, and J Benjamin +Miller. Non-adaptive matroid prophet inequalities. arXiv preprint +arXiv:2011.09406, 2020. +[CHMS10] Shuchi Chawla, Jason D Hartline, David L Malec, and Balasubra- +manian Sivan. Multi-parameter mechanism design and sequential +posted pricing. In Proceedings of the forty-second ACM symposium +on Theory of computing, pages 311–320, 2010. +[DK14] +Michael Dinitz and Guy Kortsarz. Matroid secretary for regular and +decomposable matroids. SIAM Journal on Computing, 43(5):1807– +1830, 2014. +[DK15] +Paul Dütting and Robert Kleinberg. Polymatroid prophet inequal- +ities. In Algorithms-ESA 2015, pages 437–449. Springer, 2015. +[Edm65] +Jack Edmonds. Minimum partition of a matroid into independent +subsets. J. Res. Nat. Bur. Standards Sect. B, 69:67–72, 1965. +[FSZ16] +Moran Feldman, Ola Svensson, and Rico Zenklusen. Online con- +tention resolution schemes. +In Proceedings of the twenty-seventh +annual ACM-SIAM symposium on Discrete algorithms, pages 1014– +1033. SIAM, 2016. +[FSZ21] +Moran Feldman, Ola Svensson, and Rico Zenklusen. Online con- +tention resolution schemes with applications to bayesian selection +problems. SIAM Journal on Computing, 50(2):255–300, 2021. +[Gee11] +Jim Geelen. +Small cocircuits in matroids. +European Journal of +Combinatorics, 32(6):795–801, 2011. +[GW19] +Nikolai Gravin and Hongao Wang. Prophet inequality for bipartite +matching: Merits of being simple and non adaptive. In Proceedings +of the 2019 ACM Conference on Economics and Computation, pages +93–109, 2019. +[HN20] +Tony Huynh and Peter Nelson. +The matroid secretary problem +for minor-closed classes and random matroids. SIAM Journal on +Discrete Mathematics, 34(1):163–176, 2020. +30 + +[KS77] +Ulrich Krengel and Louis Sucheston. +Semiamarts and finite val- +ues. Bulletin of the American Mathematical Society, 83(4):745–747, +1977. +[KW12] +Robert Kleinberg and Seth Matthew Weinberg. Matroid prophet +inequalities. In Proceedings of the forty-fourth annual ACM sympo- +sium on Theory of computing, pages 123–136, 2012. +[MTW16] +Tengyu Ma, Bo Tang, and Yajun Wang. The simulated greedy algo- +rithm for several submodular matroid secretary problems. Theory +of Computing Systems, 58(4):681–706, 2016. +[Mye81] +Roger B Myerson. Optimal auction design. Mathematics of opera- +tions research, 6(1):58–73, 1981. +[Oxl06] +James G. Oxley. Matroid Theory. Oxford graduate texts in mathe- +matics. Oxford University Press, 2006. +[SC84] +Ester Samuel-Cahn. Comparison of threshold stop rules and maxi- +mum for independent nonnegative random variables. the Annals of +Probability, pages 1213–1216, 1984. +[Sch03] +A. Schrijver. +Combinatorial Optimization: +Polyhedra and Effi- +ciency. Number Bd. 1 in Algorithms and Combinatorics. Springer, +2003. +[Sey80] +Paul D Seymour. Decomposition of regular matroids. Journal of +combinatorial theory, Series B, 28(3):305–359, 1980. +[Sot13] +José A Soto. Matroid secretary problem in the random-assignment +model. SIAM Journal on Computing, 42(1):178–211, 2013. +31 + diff --git a/5dAzT4oBgHgl3EQfu_2Z/content/tmp_files/load_file.txt b/5dAzT4oBgHgl3EQfu_2Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..92385d0271d1c7dc5d022f582cc5ac795276cccf --- /dev/null +++ b/5dAzT4oBgHgl3EQfu_2Z/content/tmp_files/load_file.txt @@ -0,0 +1,970 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf,len=969 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='01700v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='GT] 4 Jan 2023 Non-Adaptive Matroid Prophet Inequalities Kanstantsin Pashkovich, Alice Sayutina University of Waterloo Department of Combinatorics & Optimization 200 University Avenue West Waterloo, ON, Canada N2L 3G1 Abstract We consider the matroid prophet inequality problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' This problem has been extensively studied in the case of adaptive mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In par- ticular, there is a tight 2-competitive mechanism for all matroids [KW12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' However, it is not known what classes of matroids admit non-adaptive mechanisms with constant guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Recently, in [CGKM20] it was shown that there are constant-competitive non-adaptive mechanisms for graphic matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In this work, we show that various known classes of matroids admit constant-competitive non-adaptive mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 1 Introduction Let us consider the classical prophet inequality problem [KS77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A gambler observes a sequence of non-negative independent random variables X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn, which correspond to a sequence of values for n items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The gambler knows the distributions of X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The gambler is allowed to accept at most one item;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' and the gambler is interested in maximizing the value of the accepted item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' However, the gambler cannot simply select an item of the maximum value, because the values of the n items are revealed to the gambler one by one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' and each time a value of the current item is revealed the gambler has to make an irrevocable choice whether to accept the current item or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' What stopping rule the gambler should use to maximize the expected value of the item they accept?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The gambler knows only the distributions of X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn while a prophet knows the realization of X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, in contrast to the gambler the prophet can always obtain the maximum item’s value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The seminal result of Krengel and Sucheston [KS77] showed that the gambler can obtain at least a half of the expected value obtained by the prophet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The classical prophet inequality problem led to a series of works on different variants of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A natural variant of the problem is the generalization 1 of the problem where a gambler can buy more than one item, but the set of bought items should satisfy a known feasibility constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Formally, let us be given a collection S ⊆ 2[n] of item sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then both gambler and prophet can select any item set S from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' So S defines a feasibility constraint for selecting items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In most standard examples of feasibility constraints, S can be defined as a collection of all item sets with cardinality at most k for some natural number k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' More generally S can be defined as a collection of all independent sets in some matroid, in this case we speak about the matroid prophet inequality problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The result in [SC84] showed that in the single-item setting a gambler can obtain at least half of the prophet value by using the following threshold-rule: determine a constant T as a function of known distributions and accept the first item exceeding T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' This rule results in a 2-competitive mechanism, similar to the adaptive approach of [KS77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, that this approximation guarantee is known to be tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There is also another method to set a threshold presented in [KW12], which also results in a 2-competitive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' This was extended by Chawla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' in [CHMS10] and [CGKM20] to the setting of several items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The results presented in [KW12] further extend to the matroid prophet in- equalities, where accepted items need to form an independent set in a known matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' It leads to a 2-competitive mechanism for every matroid, matching the single-item setting result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' However, unlike the mechanism in the single-item setting, the mechanism for matroids is adaptive: the thresholds for items are computed based on the previously accepted items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By [KW12], there also exists a constant-competitive adaptive mechanism for feasibility constraints defined as an intersection of constant number of matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The mechanism by Kleinberg and Weinberg was further extended to a 2-competitive mechanism for polyma- troids by Dütting and Kleinberg in [DK15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Gravin and Wang [GW19] studied the bipartite matching version of this problem: in their version, the arriving items are the edges of the (known) bi- partite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Gravin and Wang obtained a 3-competitive non-adaptive mech- anism, which assigns thresholds to each vertex in the graph and an edge is accepted only if its weight is at least the sum of the thresholds associated with its endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Feldman, Svensson and Zenklusen [FSZ16] studied online item selection mechanisms called “online contention resolution schemes" (OCRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' They showed that given special properties, OCRS translate directly into a constant-competitive prophet inequality for the same problem against almighty adversary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' an ad- versary which knows in advance realizations of all the items and the random bits generated by an algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' As a result, they develop a constant-competitive mechanism for prophet inequalities of the intersection of a constant number of matroids, knapsack and matching constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Those mechanisms are “almost” non-adaptive in a sense that they fix thresholds for all items, however their mech- anisms also impose a subconstraint: an item cannot be accepted if together with previously accepted items it forms one of the fixed forbidden sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Finally, in a later version of their paper [FSZ21], they prove that pure non- adaptive mechanisms cannot achieve a constant-competitive approximation even against a “normal” adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' They construct a family of gammoid matroids 2 showing a lower bound of Ω(log n/ log log n) for a guarantee of non-adaptive mechanisms on gammoids with n elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There have been works studying similar setups with other goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Chawla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' [CHMS10] studied a Bayesian item selection process in a fixed item ar- rival order or against an adversary in control of the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' They studied it from a perspective of the revenue maximization for the auctioneer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The per- formance is constant-competitive compared to the well-known Myerson mech- anism [Mye81], which achieves the largest possible expected revenue among truthful mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The mechanism by Chawla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' [CHMS10] has an ad- vantage that it determines static thresholds together with a subconstraint so that each agent can be offered take-it-or-leave-it prices in an online fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Recently, Chawla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' [CGKM20] developed a 32-competitive non-adaptive mechanism for graphic matroids against adversary item ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='1 Our results First, we list the known results for non-adaptive mechanism that were mentioned in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 1 (Uniform Rank 1 Matroid [SC84]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists a 2-competitive non-adaptive mechanism for single-item setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 2 (Graphic Matroid [CGKM20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists a 32-competitive non-adaptive mechanism for graphic matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now let us list our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In case of a simple graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' a graph with no parallel edges or loops, we can slightly improve the above theorem by considering essentially the same mechanism as [CGKM20] but considering a different scaling of a point from the matroid polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We provide this result for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists a 16-competitive non-adaptive mechanism for graphic matroids in the case of simple graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Furthermore, the mechanism [CGKM20] can be generalized to the setting of k-column sparse matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' This result we need later to obtain Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 4 (k-Column Sparse Matroids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists a (2k+2k)-competitive non-adaptive mechanism for k-column sparse matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, that Theorem 2 of [CGKM20] follows from Theorem 4, since a graphic matroid is also a 2-column sparse matroid over F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Using analogous approach to the one in [Sot13], we also develop a mechanism for cographic matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 5 (Cographic Matroids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists a 6-competitive non-adaptive mechanism for cographic matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The approach in [Sot13] immediately leads to the following result for γ-sparse matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 3 Theorem 6 (γ-Sparse Matroids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists a γ-competitive non-adaptive mechanism for γ-sparse matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Combining the above results and using classic Seymour’s decomposition re- sults we obtain the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 7 (Regular Matroids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists a 256-competitive non-adaptive mechanism for regular matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Subject to the Structural Hypothesis 1 due to Geelen, Gerards and Whittle, which is stated later, we can also derive the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Subject to the Structural Hypothesis 1, for every prime number p there exists a constant-competitive mechanism for every proper minor-closed class of matroids representable over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We also would like to observe that some of the recent results on “single sample prophet inequalities” (SSPI) lead to non-adaptive constant-competitive mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For this, the single sample required by the gambler in SSPI can be directly sampled by our gambler from the available distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In partic- ular, the results in [AKW19] and [CFPP21] on laminar matroids and truncated partition matroids inspired by the mechanism in [MTW16] lead to non-adaptive mechanisms for prophet inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To obtain these results, it is crucial that the mechanism in [MTW16] does not involve subconstraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' each item is accepted as long as the item is not in the “observation phase”, the item passes its threshold based only on the “observation phase” and the item forms an in- dependent set with previously accepted items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In comparison, it is not clear how from the results on regular matroids in [AKW19] based on the mechanism in [DK14] one can obtain non-adaptive mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' So the following results can be directly obtained from [AKW19] and [CFPP21], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 9 (Laminar Matroid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='6-competitive non-adaptive mechanism for laminar matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 10 (Truncated Partition Matroid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists an 8-competitive non-adaptive mechanism for truncated partition matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 2 Comparison to known results Our results for cographic matroids and k-column sparse matroids are obtained through modifications of the arguments in [Sot13] and [CGKM20], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The results on regular matroids and minor-closed families of matroids follow the approach outlined in [HN20] for the secretary problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' As necessary building blocks we use our results for cographic and 2-column sparse matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that a biggest challenge for us is the compatibility of non-adaptive thresholds with contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Indeed, standard tools for deriving mechanisms for contraction 4 minors need subconstraints, while subconstraints are not permitted in non- adaptive mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To obtain our results, we resolve this issue only in the context of matroids representable over finite fields, see arguments in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' It would be interesting to see whether analogous results for contraction minors hold with no assumption about representability over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 3 Preliminaries In this paper, we consider the matroid prophet inequality problem, where items arrive online in adversarial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Here, the adversary knows the distributions of all X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn and knows the gambler’s mechanism, but the realization of X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn is not known to the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Based on the available information, the adversary can decide on the order in which items and their values are observed by the gambler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='1 Prophet inequality Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M be a matroid on the ground set [n] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , n}, where [n] corresponds to n items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let ⃗X := (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn) be non-negative independent random variables representing the values of these n items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For every subset of items S ⊆ [n] we define its weight as follows w(S) := � i∈S Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let PROPHM be the random variable corresponding to the value obtained by the prophet PROPHM := max S∈I(M) w(S) , where I(M) is a collection of independent sets for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let EPROPHM be the expectation of the value obtained by prophet EPROPHM := E[PROPHM] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us be given a number α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We call a mechanism α-competitive (alternatively, we say that the mech- anism guarantees an α-approximation) on the matroid M if the expected value obtained by the gambler via this mechanism is at least 1 αEPROPHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We call a mechanism α-competitive (alternatively, we say that the mech- anism guarantees an α-approximation) on the matroid class M if this mechanism is α-competitive for every matroid M ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2 Non-adaptive mechanism We say that a mechanism is non-adaptive if it has the following structure: Given the distributions of ⃗X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn), the mechanism determines the values of thresholds ⃗T = (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Tn), where each Ti, i ∈ [n] is a real number or +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If the value of item i ∈ [n] is observed, the gambler accepts the item i if and only both conditions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' the observed value of Xi is at least Ti 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' the item i together with all previously selected items forms an inde- pendent set with respect to the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, that a non-adaptive mechanism does not change thresholds during its course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' So, none of the thresholds depends on the realization of ⃗X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Another crucial feature of a non-adaptive mechanism is that the mechanism works only with the original matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A non-adaptive mechanism does not allow us to define a new matroid M ′, such that a set of items is independent in M ′ only if it is independent in M, and modify the condition (2) based on M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In this work, we focus on non-adaptive mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' From here and later we use the term mechanism to refer to non-adaptive mechanisms exclusively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In this work, non-adaptive mechanisms are allowed to make non- deterministic decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Hence, we allow a non-adaptive mechanism to construct the thresholds ⃗T = (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Tn) non-deterministically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To measure the performance of such a mechanism we use the expected total value, where the expectation is taken not only with respect to ⃗X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn) but also with respect to ⃗T = (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='3 Matroids We provide a review of matroids here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Experienced readers should consider skipping or skimming this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For further results about matroids, consider consulting [Oxl06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A matroid M = (E, S) is a pair of a finite ground set E and a collection S ⊆ 2E of independent sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The collection S ⊆ 2E of subsets of E satisfies the following conditions: (i) Empty set is an independent set, so ∅ ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' (ii) The collection S is closed with respect to taking subsets, so for all A ⊆ B ⊆ E if B is in S then A is in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' (iii) The collection S satisfies so called augmenation property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In other words, for all A, B ⊆ E such that A, B ∈ S and |A| > |B|, there exists c ∈ A \\ B such that B ∪ {c} ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 6 A subset of E is called dependent if it is not in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The inclusion-maximal independent sets are called bases and the inclusion-minimal dependent sets are called circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For every two bases, their cardinalities are equal: for every bases A and B of M we have |A| = |B|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A rank function for the matroid M is a function rM : 2E → N such that for every A ⊆ E the value rM(A) equals the cardinality of an inclusion-maximal independent subset of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In the cases when the choice of the matroid is clear from the context, we write r instead of rM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Given a matroid M, we can define the dual matroid M ∗ over the same ground set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A set A is independent for matroid M ∗ if and only if E \\ A contains a basis of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' An element c ∈ E is called a loop in M if rM(c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' An element c ∈ E is called a free element in M if rM∗(c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To put it another way, an element c is free, if and only if for every set A, which is independent in M, A∪{c} is also independent in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We say that elements c and d ∈ E are parallel in matroid M, denoted by c ∥ d, if rM(c) = rM(d) = rM({c, d}) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' One can show that “being parallel” defines an equivalence relation on the non-loop elements of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A matroid is called simple if it has no loops and no parallel edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M = (E, S) be a matroid and A ⊆ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The contraction of M by A, de- noted as M/A, is a matroid over ground set E\\A with the following independent sets {S ⊆ E \\ A : S ∪ A′ ∈ S} , where A′ is an inclusion-maximal independent subset of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The restriction of M to A, denoted as M |A or M \\ A, is a matroid over the ground set A where a set S ⊆ A is independent in M |A if and only if it is independent in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A matroid M ′ is called a simple version of M if M ′ is obtained from M by deleting all loops and contracting every parallel class of elements into a single element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For matroids M, N, we say that N is a minor of M = (E, S) if N is isomorphic to M/A\\B for some disjoint sets A, B ⊆ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A matroid class M is called minor-closed if for any M ∈ M every minor of M is also in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us now list some of the classical examples of matroids, which were ex- tensively studied in the context of various mathematical fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A uniform matroid M = (E, S) of rank k is matroid where S := {A ⊆ E : |A| ≤ k} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' When |E| = n, we denote the uniform matroid of rank k as Uk,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A graphic matroid over graph G = (V, E) is a matroid M = (E, S), where S := {A ⊆ E : A is acyclic} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The graphic matroid over graph G is denoted as M(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 7 A cographic matroid over graph G = (V, E) is a dual matroid M = (E, S) to the graphic matroid over the same graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In this case we have S := {A ⊆ E : (V, E\\A) has the same number of components as (V, E)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A vector matroid M = (E, S) is a matroid such that there is a vector space V and a map φ : E → V satisfying S := {A ⊆ E : multiset φ(A) is linearly independent} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Given a field F, we say that M is representable over field F if M is iso- morphic to the vector matroid where V is a vector space over field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A matroid is called regular if it is representable over every field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A matroid is called binary if it is representable over F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A k-column sparse matroid M = (E, S) is a matroid such that there is a field F and dimension m and a map φ : E → Fm such that S := {A ⊆ E : multiset φ(A) is linearly independent over F} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' and moreover φ(c) ∈ Fm has at most k nonzero coordinates for every c ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A γ-sparse matroid M = (E, S) is a matroid such that the inequality |S| ⩽ γrM(S) holds for every S ⊆ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A laminar matroid M = (E, S) is a matroid such that there exists a laminar family F over the ground set E and there are numbers cF ∈ N, F ∈ F such that S := {A ⊆ E : |A ∩ F| ≤ cF for every F ∈ F} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Moreover, if F = {E, E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Ek}, where E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Ek form a partition of the ground set E, then M is called a truncated partition matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Recall, that a family F is called laminar if for every A, B ∈ F we have A ⊆ B or B ⊆ A or A ∩ B = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Given a matroid M = (E, S) we can define the corresponding polytope PM ⊆ RE as the convex hull of points corresponding to the characteristic vectors of independent sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The polytope PM is known to admit the following outer description [Sch03].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' PM = {x ∈ RE : x ≥ 0 and x(S) ≤ rM(S) for every S ⊆ E} , where x(S) stands for � c∈S xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For a matroid M = (E, S) and a set A ⊆ E we can define the closure of A as the following set clM(A) := {c ∈ E | rM(A ∪ {c}) = rM(A)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 8 For a matroid M = (E, S), we call the following function ⊓M : E × E → Z a local connectivity function ⊓M(X, Y ) = r(X) + r(Y ) − r(X ∪ Y ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The following function λM : E → Z⩾0 is called a connectivity function λM(X) := ⊓M(X, E \\ X) = r(X) + r(E \\ X) − r(E) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Informally, connectivity functions measure dependence with respect to the matroid between parts of the ground set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To illustrate it, let us consider the connectivity function for vector matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Suppose M = (E, S) is a vector matroid defined by a vector space V and a map φ : E → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then we have λM(S) =r(S) + r(E \\ S) − r(E) = dim(span φ(S)) + dim(span φ(E \\ S)) − dim(φ(E)) = dim ((span φ(S)) ∩ (span φ(E \\ S))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='4 Ex-ante relaxation to the matroid polytope The goal of ex-ante relaxation [FSZ16] or [CGKM20] is to reduce the origi- nal problem to the problem where item values are distributed as independent Bernoulli random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, that both problems are using the same ma- troid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In the original problem item values ⃗X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn) are independent ran- dom variables with known distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For i ∈ [n] let Fi be the cumulative distribution function of Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The reduction of the original problem to a new problem is done using a point p in the matroid polytope PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us first show that there is a point p ∈ PM with properties that prove to be desirable later following the argumentation in [CGKM20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Given a matroid M over the ground set [n] and random variables ⃗X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn), there exists p ∈ PM such that EPROPHM ⩽ n � i=1 piti , where ti := E[Xi | Xi ⩾ F −1 i (1 − pi)] for every i ∈ [n]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let Iopt be a random variable indicating an optimal independent set in M with respect to ⃗X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In case when for some realization of ⃗X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn) there are several optimal independent sets, Iopt can be selected as any of these sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For i ∈ [n], let pi be the probability that element 1Here, we assume that for every i ∈ [n] the event Xi = F −1 i (1 − pi) happens with the zero probability, which is true for all continuous distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In case of discrete distributions one needs to introduce appropriate tie-breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 9 i is in Iopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that p = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , pm) is a convex combination of independent sets of M, and so lies in PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Due to EPROPHM = E[� i∈Iopt Xi], it remains to show that E[ � i∈Iopt Xi] ⩽ n � i=1 piti .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We have E[ � i∈Iopt Xi] = n � i=1 P[i ∈ Iopt]E[Xi | i ∈ Iopt] = n � i=1 piE[Xi | i ∈ Iopt] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For every i ∈ [n] we have that ti and E[Xi | i ∈ Iopt] are expectations of the same random variable Xi but conditioned on the event Xi ⩾ F −1 i (1−pi) and on the event i ∈ Iopt, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, that the probability of both these events equals pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' However, the expectation of Xi conditioned on Xi ⩾ F −1 i (1 − pi) is the “largest” conditional expectation of Xi on an event of probability pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, we have piE[Xi | i ∈ Iopt] ⩽ piti for every i ∈ [n] and so we get the desired inequality n � i=1 piE[Xi | i ∈ Iopt] ⩽ n � i=1 piti .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us show how one can use the point p = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , pn) guaranteed by Lemma 1 to reduce the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us define independent Bernoulli random variables ⃗X′ = (X′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , X′ n) as follows, for each i ∈ [n] X′ i = � ti with probability pi 0 with probability 1 − pi , where ti := E[Xi | Xi ⩾ F −1 i (1 − pi)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us assume that we have a non-adaptive mechanism for the original ma- troid M and item values ⃗X′ = (X′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , X′ n), which sets nonnegative thresholds ⃗T ′ = (T ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , T ′ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By definition of ⃗X′ = (X′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , X′ n), for every i ∈ [n] the exact value of T ′ i is not relevant per se, but it is crucial whether ti ≥ T ′ i or ti < T ′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If for some i ∈ [n] we have T ′ i > ti then this item i is “inactive” and so is never selected by the gambler working with M and ⃗X′ = (X′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , X′ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The key is to construct a non-adaptive mechanism for the original matroid M and item values ⃗X′ = (X′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , X′ n) with positive thresholds ⃗T ′ = (T ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , T ′ n) such that for each item i ∈ [n] the probability that i is selected by the gambler is at least αpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now we can use such a non-adaptive mechanism for the original matroid M and item values ⃗X′ = (X′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , X′ n) to construct a non-adaptive α-competitive mechanism for the same matroid M and random variables ⃗X = 10 (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us define the thresholds ⃗T = (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Tn) as follows, for every i ∈ [n] Ti := � +∞ if ti < T ′ i F −1 i (1 − pi) otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To see that the thresholds ⃗T = (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Tn) lead to an α-competitive mech- anism for M and ⃗X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn), let us couple random variables X′ i with random variables Xi as follows X′ i := � ti if Xi ≥ F −1 i (1 − pi) 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that ⃗X′ = (X′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , X′ n) are independent Bernoulli random variables, where for each i ∈ [n] the variable X′ i equals ti with probability pi and equals 0 with probability 1 − pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' When ⃗X′ are coupled with ⃗X this way, Xi and X′ i have the same expected value when conditioned on X′ i being ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The mechanism with thresholds ⃗T selects an item i ∈ [n] when run for ⃗X only if the mechanism with thresholds ⃗T ′ selects the item i when run for ⃗X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Moreover, for both of these algorithms, conditionally on the event that the item i is selected the expected value of i equals ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now, α-competitiveness guarantee of the thresholds ⃗T for M and ⃗X follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 4 Graphic and k-column sparse matroids First, we construct a 16-competitive non-adaptive mechanism for graphic ma- troids without parallel edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Our construction is done through the ex-ante re- laxation to the matroid polytope, following the works in [FSZ16] or [CGKM20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Later, we present a constant-competitive non-adaptive mechanism for k-column sparse matroids whenever k is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='1 Graphic matroids Now we are ready to provide a 16-competitive non-adaptive mechanism for graphic matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The provided mechanism is essentially the one constructed in [CGKM20] but with saving a factor of 2 in the guarantee, which is achieved by rescaling the point from the matroid polytope by 2 and not by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us be given a simple graph G = (V, E) and let us consider the corre- sponding graphic matroid M over the ground set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Recall that a subset of E is independent with respect to M if and only if it is acyclic in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us also assume that the graph G has n edges and so E = {e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , en}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let p = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , pn) be a point in the polytope PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus we assume that for every i ∈ [n] the coordinate pi of p corresponds to the edge ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then there exists an orientation of edges E = {e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , en} in the graph G = (V, E) such that for every vertex v ∈ V we have � i∈[n]:ei∈δ−(v) pi ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Observe that the average degree of a vertex in a forest on |V | vertices is at most (2|V | − 2)/|V | = 2 − 1/|V | ⩽ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us use this fact to prove the desired statement by induction on the number of vertices in the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If the graph G has at most two vertices then the orientation is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Other- wise, since p is a convex combination of points corresponding to forests in G, we have that the average of the value � i∈[n]:ei∈δ(v) pi over all vertices v ∈ V is at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus there exists a vertex v ∈ V such that we have � i∈[n]:ei∈δ(v) pi ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We orient all edges incident to v as edges in δ−(v), so these edges are incoming with respect to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then we remove the vertex v and all edges incident to it and orient the remaining edges according to the orientation guaranteed by the inductive hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now we present an algorithm for graphic matroids of simple graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Algorithm 1 A non-adaptive 16-competitive mechanisms for graphic matroids of a simple graph 1: Let p be a point in the polytope PM so that the statement of Lemma 1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 2: Let the edges of the original graph G = (V, E) be oriented so that the statement of Lemma 2 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 3: For every edge ei ∈ E, i ∈ [n], mark the edge ei as “discarded" independently at random with probability 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 4: Select a cut S ⊆ V uniformly at random, mark all edges not in [S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' S] as “discarded".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Here, [S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' S] stands for the set of edges which are oriented such that their tail is in S and their head is in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 5: Set thresholds ⃗T = (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Tn) as follows, for each i ∈ [n] Ti := � +∞ if ei is “discarded” F −1 i (1 − pi) otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For every i ∈ [n], we have P[ei is selected | Xi ≥ Ti and ei is not “discarded”] ≥ 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us assume that the vertex v is the head of the oriented edge ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us also assume that ei is not marked as “discarded” and Xi ≥ Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since the edge ei is not “discarded”, the edge ei is in the selected set [S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' S].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Hence, every not “discarded” edge incident to v has the vertex v as its head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, as long as no other edge with the head at the vertex v is selected by the gambler, the gambler has to select ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We claim, that with probability at least 1/2 no other edge with the head at v was selected by the gambler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let I be the event indicating that "the gambler selected an edge ej, j ̸= i such that v is the head of ej", in other words “there is j ∈ [n], j ̸= i such that 12 v is the head of ej and Xj ≥ Tj and ej is not “discarded”".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let J indicate the event that "ei is not marked as “discarded” after the selection of the cut", in other words, "the head of ei is in S and the tail of ei is in S".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us show P[I | J] ≤ 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By the union bound, we have P[I | J] ⩽ � j∈[n]\\{i}:ej∈δ−(v) P[Xj ≥ Tj and ej is not “discarded” | J] Note that for each edge ej ∈ δ−(v) we have P[Xj ≥ Tj|J] = pj and we also have P[ej is not “discarded”|J] = 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that any edge is not “discarded” in Step 3 of Algorithm 1 with probability 1/2, and not “discarded” in Step 4 of Algorithm 1 with probability 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' However, since the probabilities are with respect to the edge ej ∈ δ−(v) and are counted conditioned on the event J, the conditioned probability of not being “discarded” in Step 4 of Algorithm 1 is 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Moreover, even conditioned on J the events "Xj ≥ Tj" and "ej is not “discarded”" are independent events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus we have � j∈[n]\\{i}:ej∈δ−(v) P[Xj ≥ Tj and ej is not “discarded” | J] ≤ � j∈[n]\\{i}:ej∈δ−(v) pj/4 ⩽ 1/2 , where the last inequality follows from the orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We are ready to prove Theorem 3 by showing that Algorithm 1 is a 16- competitive for graphic matroids without parallel edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Lemma 3 for every i ∈ [n] the probability of edge ei being accepted conditional on Xi ≥ Ti and being not “discarded” is at least 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Overall, the probability of edge ei being accepted is at least 1 16pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus mechanism guarantees at least �n i=1 1 16piti of the expected total value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Lemma 1, we have � i∈[n] 1 16piti ⩾ 1 16EPROPHM, finishing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2 k-column sparse matroids There are known constant-competitive mechanisms for k-column sparse ma- troids in the context of the secretary problem [Sot13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' However they do not immediately lead to a non-adaptive mechanism of constant competitiveness guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The reason for that are not the updated thresholds but implicit changes to the considered matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Here, we present a constant competitive mechanism for k-column sparse matroid class for each constant k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, graphic matroids form a subclass of 2-column sparse matroids .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Because of their significance, 2-column sparse matroids are also known in literature as represented frame matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Later, we use 2-column sparse matroids to prove results in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 13 Suppose M is a k-column sparse matroid over field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In this section, we prove that there exists a (2k+2k)-competitive mechanism for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Suppose a k-sparse representation of M = (E, S) is defined by a map φ : E → Fd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, if for some element t ∈ E the vector φ(t) is a zero vector then c is a loop and therefore can be removed from consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now we consider an undirected hyper-multigraph G with vertex set [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Each matroid element t ∈ E induces a hyperedge et in this graph between non-zero coordinates of φ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Formally, the hyperedge et is defined as follows et := {i ∈ [d] : φ(t)i ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We say that a vertex i ∈ [d] of the hyper-multigraph G is incident to every edge e of G such that i ∈ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For a vertex i ∈ [d] we denote the collection of incident hyperedges by δ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The degree of a vertex i in the hyper-multigraph G equals |δ(i)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Suppose I is an independent set of the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then the average degree of a vertex is at most k when one considers the hyper-multigraph with vertices [d] and hyperedges {et : t ∈ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Observe that |I| ⩽ d because having more than d vectors in d-dimensional vector space Fd leads to a a linear dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since M is k-column sparse, we have that every edge in {et : t ∈ I} is incident to at most k vertices in [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Hence, the total degree is at most kd and thus the average degree of a vertex is at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now we consider orientations of the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' An orientation of the graph G is a function ϕ which maps every edge et into one vertex of G incident to et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We call ϕ(et) to be the head of the edge et, and all other vertices, if any, to be tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For every vertex i ∈ [d] we denote the set of incoming edges by δ−(i), formally δ−(i) = {et : ϕ(et) = i, t ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let p be a point in the polytope PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We assume that for every t ∈ E, the coordinate pt of p corresponds to the element t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then there exists an orientation ϕ of hyperedges in the hyper-mulrigraph G such that for every vertex i ∈ [d] we have � t∈E:et∈δ−(i) pt ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The proof of Lemma 4 is analogous to the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now let us describe an algorithm for k-column sparse matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For every t ∈ E we have P[t is selected | Xt ≥ Tt and t is not “discarded”] ≥ 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that item t ∈ E is accepted whenever Xt ≥ Tt and no other item was selected from non-discarded edges in δ−(ϕ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By the union bound, for every event J we can upper bound the probability that P[there j ∈ E \\ {t} such that j is selected and ej ∈ δ−(ϕ(t)) | J] ⩽ � j∈E\\{t}:ej∈δ−(ϕ(t)) P[ej is not “discarded” and Xj ≥ Tj | J] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 14 Algorithm 2 A non-adaptive 2k+2k-competitive mechanisms for k-column sparse matroids 1: Let p be a point in the polytope PM so that the statement of Lemma 1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 2: Let the edges of the hyper-multigraph G be oriented so that the statement of Lemma 4 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 3: For every edge ei ∈ E, i ∈ [n], mark the edge ei as “discarded" independently at random with probability 1 − 1 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 4: Select a cut S ⊆ [d] uniformly at random, mark all edges not in [S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' S] as “discarded”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Here, [S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' S] stands for the set of edges which are oriented such that all their tails are in S and their head is in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In particular, for t ∈ E we say that et lies in a cut [S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' S] with respect to the orientation ϕ if ϕ(et) ∈ S and for every i ∈ et \\ {ϕ(et)} we have i ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 5: Set thresholds {Tt : t ∈ E} as follows, for each t ∈ E Tt := � +∞ if t is “discarded” F −1 t (1 − pt) otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let J indicate the event that "et is not marked as “discarded” after the selection of the cut".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then for each j ∈ E \\{t} we have P[ej is not “discarded” and Xj ≥ Tj | J] ≤ 1 2kpj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Lemma 4, we have � j∈E:ej∈δ−(ϕ(t)) pj ⩽ k, leading to the desired inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that the proof of Lemma 5 is analogous to the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We are ready to prove Theorem by showing that the Algorithm 2 is a 2k+2k-competitive for k-column sparse matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For every item t ∈ E we have P[Xt ≥ Tt] = pt and P[t is not “discarded”] ≥ 1 2k+1k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Lemma 5, we have that with probability at least 1/2 the item t is selected when it is not “discarded” and Xt ≥ Tt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus the expected total value of Algorithm 2 is at least � j∈E 1 2k+2kpjtj which is at least 1 2k+2kEPROPHM by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 5 Cographic and gamma-sparse matroids 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='1 Cographic matroids Let us revisit a mechanism of Soto [Sot13] for the cographic matroid secretary problem which is based on the following corollary of Edmond’s matroid parti- tioning theorem [Edm65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' This mechanism leads to a non-adaptive mechanism for cographic matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 15 Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let G = (V, E) be a three edge-connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then there exist spanning trees H1, H2, H3 in G such that the union of their complements contains all the edges E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' E = (E \\ H1) ∪ (E \\ H2) ∪ (E \\ H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Algorithm 3 A non-adaptive 3-competitive mechanisms for cographic matroids in the case of three edge-connectivity 1: Let H1, H2 and H3 be the spanning trees as in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 2: Uniformly at random select a spanning tree H∗ from H1, H2 and H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Set thresholds {Te : e ∈ E} as follows, for each e ∈ E Te := � +∞ if e is not in H∗ 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let G = (V, E) be a three edge-connected graph and let M be the cographic matroid over G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then Algorithm 3 is a 3-competitive non-adaptive mechanism for the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The expected total value of the mechanism provided by Algorithm 3 equals E[� e∈E\\H∗ Xe] which can be estimated as follows E[ � e∈E\\H∗ Xe] = 1 3E[ � i∈[3] � e∈E\\Hi Xe] ≥ 1 3E[ � e∈E Xe] ≥ 1 3EPROPHM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The next theorem provides a proof for Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let G = (V, E) be a graph and let M be the cographic matroid over G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then Algorithm 4 is a 6-competitive non-adaptive mechanism for the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We can assume that G does not have bridges, because every such bridge is a loop in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus these edges can be selected neither by the gambler nor by the prophet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' So we can assume G = G′ and M = M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In the case when each connected component of G is three edge-connected, then Algorithm 4 runs Algorithm 3 for each component to obtain a 3-competitive non-adaptive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Otherwise, there is one or more pairs of edges e,e′ such that {e, e′} corre- sponds to a cut in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In this case, the edges e,e′ correspond to parallel elements of the cographic matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Algorithm 4 considers the partition of E into classes of parallel elements C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us construct the matroid M ′′ from M by contracting all but one edge in each class C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, that the ground set of M ′′ has k elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Abusing the notation we refer to these elements of the ground set as 16 Algorithm 4 A non-adaptive 6-competitive mechanisms for cographic matroids 1: Delete all loops of M to obtain a matroid M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Remove all bridges from G = (V, E) and obtain a graph G′ = (V ′, E′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 2: Let C1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Ck be equivalence classes of M ′ with respect to the relation of being parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Construct the matroid M ′′ from M ′ by contracting all but one edge in each class C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, that the ground set of M ′′ has k elements and matroid M ′′ is the cographic matroid over a graph G′′, where each connected component of G′′ is three edge-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Abusing the notation we refer to the elements of the ground set of M ′′ as C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 3: Let H1, H2 and H3 be forests in G′′ such that the restriction of H1, H2 and H3 to each connected component of G′′ satisfies Proposition 6 for the respective connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 4: Uniformly at random select a forest H∗ from H1, H2 and H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 5: For each i ∈ [k] select thresholds T e, e ∈ Ci according to Theorem 1 when the gambler is allowed to accept only one item of Ci and the distributions of Xe, e ∈ Ci are the same as original distributions of values for e ∈ Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 6: Set thresholds {Te : e ∈ E} as follows, for each e ∈ E Te := � T e if e ∈ Ci and Ci ∈ H∗ for some i ∈ [k] +∞ otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The matroid M ′′ is isomorphic to the cographic matroid over a graph G′′, where each connected component of G′′ is three edge-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Following Lemma 6, Algorithm 4 constructs forests H1, H2, H3 for the graph G′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' So Algorithm 4 leads us to a 6-competitive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Indeed, the prophet with M and with the original distributions of Xe, e ∈ E performs exactly as the prophet with M ′′ and with the corresponding distributions of X′′ i := maxe∈Ci Xe, i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By selecting forests in Algorithm 4 the gambler acheives in expectation E[� i∈[k] X′′ i ]/3 when all classes C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Ck are singletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' However, for classes that are not singletons we need to take into account an- other 2 approximation factor with respect to the prophet, who can achieve the expected value E[X′′ i ] for each i ∈ [k], while the gambler is guaranteed in ex- pectation to achieve only E[X′′ i ]/2 for each i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2 Gamma-sparse matroids Let us also revisit a mechanism of Soto [Sot13] for γ-sparse matroids to verify that it directly leads to a non-adaptive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M = (E, S) be a γ-sparse matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists a γ- competitive non-adaptive mechanism for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' First observe that the point x := 1/γ lies in the matroid polytope PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 17 Indeed, it is non-negative and for every set S ⊆ E(M) we have x(S) = |S|/γ ⩽ rM(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then x can be expressed as a convex combination of indicator variables corresponding to the independent sets of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In other words, we have x = � S∈S αS1S for some α ⩾ 0, � S∈S αS = 1, where 1S refers to the characteristic vector of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now sample an independent set S in matroid M randomly with probabil- ity αS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let the gambler select all items in S and let the gambler leave all the items not in S unselected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If Xe is the random variable corresponding to the weight of element e ∈ E(M), then this mechanism results in a total expected value as follows � S∈S αS � e∈S E[Xe] = � e∈E (1/γ)E[Xe] = E[ � e∈E Xe]/γ ⩾ EPROPH/γ , finishing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Observe that Proposition 1 implies that for a three edge-connected graph G, the cographic matroid of G is 3-sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus Lemma 6 is a corollary of Theo- rem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Similarly, for a planar graph G the graphic matroid is 3-sparse, leading us to the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let G is a planar graph and let M be the corresponding graphic matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There is a 3-competitive non-adaptive mechanism for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 6 Representable matroids Many results in the theory of matroids make use of minors coming from re- strictions and contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To get access to the toolbox provided by matroid theory, we need to understand how prophet inequality guarantees change when we consider minors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='1 Preliminaries Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M be a matroid and let matroid N be a restriction of the ma- troid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If there exists an α-competitive non-adaptive mechanism on M, then there is an α-competitive non-adaptive mechanism for N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To obtain a mechanism for the matroid N, we can impose thresholds +∞ for the items that were removed from the ground set to obtain the restriction N from the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The remaining items are assigned the same thresholds in both mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A similar result for contractions is harder to obtain in the case of non- adaptive mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Indeed, a straightforward approach would require us to impose the thresholds +∞ for the contracted items, while using the given 18 mechanism on the remaining items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Unfortunately, this would also require us to “change" the underlying matroid, in other words a gambler might be forced to reject an item even though its value is over the assigned threshold and its addition to the currently selected items keeps the selected set independent with respect to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Because of this difficulty, in this work we provide a matching result for contractions only for matroids representable over a finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' This result is sufficient for the purpose of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M = (E, S) be a matroid representable over the field Fp for some p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let T ⊆ E be a subset of the ground set such that λM(T ) ⩽ k for some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then there exists S ⊆ T so that every set that is independent in M |S is also independent in M/T and EPROPHM|S ⩾ 1 pk+1 EPROPHM/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Recall that T stands for the complement of T with respect to the ground set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Consider the representation of the matroid M over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let φ : E → Fm p be the map describing the representation of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, for every S ⊆ E we have that the set φ(S) = {φ(e) ∈ Fm p : e ∈ S} is independent over the field Fp if and only if S is an independent set for the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since λM(T ) ⩽ k holds, by definition of λM we have rM(T ) + rM(T ) − rM(E) ⩽ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We have rM(R) = dim span(φ(R)) for every R ⊆ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, we have dim span φ(E) = dim span φ(T )+dim span φ(T )−dim � (span φ(T )) ∩ (span φ(T)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' and so dim � (span φ(T )) ∩ (span φ(T )) � ⩽ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since we are working over the field Fp, the linear space L := (span φ(T )) ∩ (span φ(T )) has at most pk vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let C be the orthogonal complement of the linear space L in the space span φ(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, we can represent span φ(T ) as L ⊕ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For every vector v ∈ span φ(T ) we denote v orthogonal projection to L and C by v |L and v |C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For each vector a ∈ L, define the set Ta := {t ∈ T : φ(t) |L= a, φ(t) ̸= a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that by definition for every a ∈ L we have Ta ∩ L = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now let us select a uniformly at random from L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Ea[EPROPHM|Ta ] ≥ 1 pk EPROPHM/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To prove the desired inequality, we prove the corresponding inequality for any realization of item values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' From now on we consider the realization of item values fixed and thus we prove the following inequality Ea[PROPHM|Ta] ≥ 1 pk PROPHM/T 19 Let us consider the set Iopt on which the prophet achieves PROPHM/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that the set Iopt does not contain any item e such that φ(e) is in L, because every such an item e is a loop in M/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, the set Iopt can be partitioned into sets Iopt,a, a ∈ L where Iopt,a is a subset of Ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The set Iopt is independent in M/T and so Iopt is also independent in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Hence the sets Iopt,a, a ∈ L are also independent in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus for every a ∈ L, PROPHM|Ta ⩾ w(Iopt,a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then we have Ea[PROPHM|Ta] ⩾ � a∈L w(Iopt,a) |L| = 1 |L|w(Iopt) ⩾ 1 pk PROPHM/T , finishing the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us now select a∗ ∈ L such that EPROPHM|Ta is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By the previous claim, we have PROPHM|Ta∗ ⩾ 1 pk PROPHM/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now for every c ∈ C define set Hc := {t ∈ Ta∗ : (φ(t) |C) · c = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now let us select c uniformly at random from C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Ec[EPROPHM|Hc ] ≥ 1 pEPROPHM|T a∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To prove the desired inequality, we prove the corresponding inequality for any realization of item values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' From now on we consider the realization of item values fixed and thus we prove the following inequality Ec[PROPHM|Hc ] ≥ 1 pPROPHM|T a∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let Iopt be the set corresponding to PROPHM|Ta∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, we have that for every e ∈ Iopt, φ(e) is not in L and hence φ(e) |C is not the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Due to Pc[c · t = 1] = 1/p, for every t ∈ Ta∗, we have Ec[w(Iopt∩Hc)] = � t∈Iopt Pc[c·t = 1]w(t) = 1 p � t∈Iopt w(t) = 1 pw(Iopt) = PROPHM|Ta∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Finally, since Iopt is independent in M so is Iopt ∩ Hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, we have Ec[PROPHM|Hc ] ≥ 1 pPROPHM|T a∗ , finishing the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now let us select c∗ so that EPROPHM|Hc is maximized and let S∗ := Hc∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then we have EPROPH(M |S∗) ⩾ 1 pk+1 EPROPH(M/T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Finally, we need to show that every set independent in M |S∗ is an indepen- dent set in M/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Suppose the contrary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' there exists a set that is independent 20 in M |S∗ but is not an independent set in M/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then span S∗ has a non-trivial intersection with span T, suppose x ∈ (span φ(S∗)) ∪ (span φ(T )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us show that x is a zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since x ∈ span S∗, we have x = � s∈S∗ αsφ(s) for some αs ∈ Fp, s ∈ S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us consider the projections of x on C and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since x ∈ span φ(T ) we have that x lies in L and so x |C is the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus x |C= � s∈S∗ αs(φ(s) |C) is the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that by definition, φ(s) |L= a∗ and c∗ · (φ(s) |C) = 1 hold for every s ∈ S∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus over the field Fp we have � s∈S∗ αs = � s∈S∗ αs(c∗ · (φ(s) |C)) = c∗ · � � s∈S∗ αs(φ(s) |C) � = c∗ · (x |C) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now let us consider x |L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We have x |L= � s∈S∗ αs(φ(s) |L) = � � s∈S∗ αs � a∗ , where the last expression equals the zero vector since � s∈S∗ αs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus we have a vector x ∈ L ⊕ C such that both projections x |L and x |C are the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Hence, the vector x is the zero vector, finishing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2 Tree Decompositions Similarly to the approach [HN20] for the matroid secretary problem, we exten- sively use the tree decomposition of matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A tree decomposition of bounded thickness allows us to construct non-adaptive mechanisms with good approxi- mation ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Before proceeding with these constructions, let us introduce tree decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A tree decomposition of a matroid M = (E, S) is a pair (T, X) where T is a tree and X = {Xv ⊆ E : v ∈ V (T )}, where sets in X form a partition of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Here, we refer to the vertex and edge sets of the tree T as V (T ) and E(T ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Given an edge e = {v1, v2} ∈ E(T ) of the tree T , let T1 and T2 be two connected components of T − e, in other word the removal of the edge e from T leads to two connected components T1 and T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The thickness of the edge e = (v1, v2) is denoted as λ(e) and is defined as follows λ(e) := λM(∪v∈V (T1)Xv) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The thickness of the tree decomposition is the maximum thickness of the edge e in E(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M be a family of matroids, M be a matroid and (T, X) be a tree decom- position of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We say that tree decomposition (T, X) is M-tree decomposition 21 if M |clM(Xv)∈ M holds for every v ∈ V (T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let tk(M) be a set of matroids which have M-tree decomposition of thickness at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let Mα,p be the family of matroids which admit α-competitive non-adaptive mechanisms and are representable over the finite field Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then for every natural number k and every matroid M in tk(Mα,p), the matroid M has an (αpk+1)-competitive non-adaptive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For a natural number m, let tk,m(Mα,p) be the set of matroids which have an Mα,p-tree decomposition (T, X) of thickness at most k satisfying |V (T )| = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us prove the statement of the lemma by induction on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The base case follows from the definition of the family Mα,p and the fact that Mα,p = tk,1(Mα,p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us now show how to do the inductive step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us assume m ≥ 2 and consider a matroid M = (E, S) in tk,m(Mα,p) with its Mα,p-tree decomposition (T, X) of thickness at most k and with |V (T )| = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let ℓ be a leaf of the tree T and let u be the neighbour of the vertex ℓ in the tree T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Observe that the tree (V (T ) \\ {ℓ}, E(T )\\ {ℓu}) together with the subfamily {Xw : w ∈ V (T ) \\ {ℓ}} defines an Mα,p-tree decomposition of the matroid M \\ Xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus the matroid M \\ Xℓ is in M ∈ tk,m−1(Mα,p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Hence, by the inductive hypothesis there are thresholds T ′ e, e ∈ E \\ Xℓ guaranteeing αpk+1- competitiveness of the gambler in comparison to the prophet on the matroid M \\ Xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There are thresholds T ′′ e , e ∈ Xℓ leading to an (α · pk+1)-competitive non-adaptive mechanism for matroid M |Xℓ, such that the gambler always selects a set that is independent in M/Xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Lemma 8 there exists a set S ⊆ Xl such that every set independent in M |S is also independent in the matroid M/Xℓ and EPROPHM|S ⩾ 1 pk+1 EPROPHM/Xℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By definition of Mα,p and the appearance of Xℓ in the tree decomposition, we have that M |Xℓ is in the family Mα,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Lemma 7, since S is a subset of Xℓ the matroid M |S is also in the family Mα,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, there are thresholds T ′′ e , e ∈ S that lead to an α-competitive non-adaptive mechanism on M |S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The thresholds T ′′ e , e ∈ Xℓ \\ S can be defined as +∞, finishing the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now we can define thresholds Te, e ∈ E for all elements of the matroid M as follows Te := � T ′ e if e ̸∈ Xℓ T ′′ e otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us now demonstrate that such thresholds Te, e ∈ E lead to an (αpk+1)- competitive non-adaptive mechanism for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 22 First, by the above claim the selected items from Xℓ always form an inde- pendent set in M/Xℓ when used with the thresholds Te, e ∈ Xℓ on the matroid M |Xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus the definition of the thresholds guarantees that in expectation the value of selected items from Xℓ is at least EPROPHM|Xℓ/(αpk+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' and in ex- pectation the value of selected items from E\\Xℓ is at least EPROPHM\\Xℓ/(αpk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To finish the proof, note that we have PROPHM|Xℓ + PROPHM\\Xℓ ≥ PROPHM and so EPROPHM|Xℓ + EPROPHM\\Xℓ ≥ EPROPHM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='3 Regular matroids In this section, we prove Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Before we proceed to the proof, let us define key notions related to regular matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A subset of the matroid’s ground set is called a circuit, if it is an inclusion- minimal dependent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A cycle is a subset of the ground set which can be partitioned into a disjoint union of circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M1 = (E1, S1), M2 = (E2, S2) be two binary matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then the matroid sum M1△M2 has the ground set E1△E2 and the cycles of M1△M2 are all sets of the form C1△C2, where C1 is a cycle for M1 and C2 is a cycle for M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Consider two binary matroids M1 = (E1, S1), M2 = (E2, S2) and M = M1△M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If |E1 ∩ E2| = 0, and E1 ̸= ∅, E2 ̸= ∅, M is called a 1-sum of M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If |E1 ∩ E2| = 1, |E1| ≥ 3, |E2| ≥ 3 and E1 ∩ E2 is not a loop of M1 or M2 or their dual matroids, M is called a 2-sum of M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If |E1 ∩ E2| = 3, |E1| ≥ 7, |E2| ≥ 7 and E1 ∩ E2 is a circuit in both M1 and M2, and E1 ∩ E2 does not contain a circuit in their dual matroids, then M is called a 3-sum of M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Seymour’s regular matroid decomposition theorem [Sey80], every regular matroid M can be obtained from graphic, cographic or a special matroid R10 through a sequence of 1-sums, 2-sums or 3-sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' This gives a tree decomposition (T, X) of thickness at most 2 so that each M |Xv, v ∈ V (T ) is either a graphic, cographic or a special matroid R10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By performing parallel extensions of the elements to be deleted before each 2-sum and 3-sum, we construct a matroid M ′, so that M is a restriction of M ′ and M ′ has a tree decomposition (T, X ′) so that each M ′ |clM′ (X′v), v ∈ V (T ) is either graphic, cographic or a parallel extension of R10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 23 By Theorem 2, every graphic matroid has a 32-competitive non-adaptive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Theorem 5, every cographic matroid has a 6-competitive non- adaptive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since matroid R10 has ground set of size 10, by Theorem 1 every parallel extension of R10 has a 20-competitive non-adaptive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that by definition every regular matroid is representable over finite field F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, by Theorem 13 with p = 2, k = 2 and α = 32 there is a 256- competitive non-adaptive mechanism for matroid M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since M is a restriction of M ′, by Lemma 7, there is a 256-competitive non-adaptive mechanism for M, finishing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='4 Minor-closed representable matroid families In this section we show that every minor-clossed subclass of matroids repre- sentable over Fp has a constant-competitive non-adaptive mechanism, where the constant is a function only of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The proof of this fact is analogous to the proof in [HN20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 14 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='3 in [Gee11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Given natural numbers q ⩾ 2 and n ⩾ 1, let M = (E, S) be a matroid with no U2,q+2 or M(Kn) minors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then we have |E| ≤ qq3nrM(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Given natural numbers q ⩾ 2 and n ⩾ 1, let M = (E, S) be a matroid with no U2,q+2 or M(Kn) minors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then there exists a qq3n-competitive non-adaptive mechanism for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If M has no U2,q+2 or M(Kn) minors, then every restriction of M also has no U2,q+2 or M(Kn) minors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus for every X ⊆ E we have |X| ⩽ qq3nrM(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' So, M is a qq3n-sparse matroid and by Theorem 12 there exists a qq3n-competitive non-adaptive mechanism for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='1 Projections and lifts Let M be a matroid and x be an element of the ground set, which is a not a loop and not a free element of the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then M/x is called a projection of M \\ x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' M \\ x is called a lift of M/x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that here and later we write M/x and M \\ x instead of M/{x} and M \\ {x}, repsectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M and N be two matroids with the same ground set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We say that the distance between M and N is t, denoted by dist(M, N) = t if t is the smallest integer such that there exists a sequence of matroids P0, P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Pt where P0 = M and Pt = N and for every i ∈ [t] the matroid Pi is either a lift or a projection of Pi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let N be a lift of the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If there is an α-competitive non- adaptive mechanism for M then there exists a (2α+2)-competitive non-adaptive mechanism for N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 24 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since N is a lift of M, there exists a matroid L = (E, S) and an element x of its ground set, such that M = L/x, N = L\\x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Here, x is not a loop and not a free element of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let P be the set of elements in L that are parallel to x, in other words P := {x′ ∈ E : x′ ∥ x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that N |P \\{x} is a uniform matroid of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note also that elements in P \\{x} are loops in M and so EPROPHM = EPROPHM\\P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let T ′ e, e ∈ E \\ {x} be the thresholds imposed by an α-competitive non- adaptive mechanism for the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let T ′′ e , e ∈ P be the thresholds guaranteeing 2-competitive non-adaptive mechanism as in Theorem 1 for the uniform matroid of rank 1 on the ground set P \\{x};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' and let T ′′ e , e ∈ E\\(P ∪{x}) be +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We select one of these two sets of thresholds for the matroid N as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The constructed mechanism for the matroid N selects one of those two sets at random, where first set of thresholds T ′ e, e ∈ E \\ {x} is selected with probability γ := α/(α + 1) and the second set T ′′ e , e ∈ E \\ {x} with probability 1 − γ = 1/(α + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Next part is dedicated to the analysis of how thresholds T ′ e, e ∈ E \\ {x} perform on the matroid N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, that these thresholds are coming from a mechanism for the matroid M, while they are used for the matroid N with probability γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We show that the total expected value achieved by thresholds T ′ e, e ∈ E \\ {x} on N is at least the total expected value achieved by these thresholds on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For this we can assume that for every realization of item values, the orders of items in matroid N and M are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To see that this assumption is valid, we can assume that the order for N is chosen in an adversarial way and is used also as the items order for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us assume that the items order for M and N is the same for a given realization of item values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us also assume that for every item e ∈ E \\ {x} the threshold T ′ e is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then the gambler with matroid N selects all items that the gambler with matroid M selects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We fix the item values realization and items order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let e1, e2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , ek be the items with their values being at least their threshold and with the corre- sponding order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now we need to show that if the gambler with matroid N selects items greedily from e1, e2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , ek starting from e1, then the set of selected items is a superset of the items greedily selected by the gambler with matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If both gamblers end up selecting exactly the same set of items, then proof of the claim is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Otherwise consider the first index i ∈ [k] such that the item ei is selected by exactly one of the two gamblers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since N = L\\x and M = L/x we have that it is only possible if ei is selected by the gambler with the matroid N and rejected by the gambler with the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now we claim that every subsequent item, in other words an item in ei+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , ek, is either selected by both gamblers or rejected by both gamblers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Suppose the contrary and consider the first item ej, i + 1 ≤ j ≤ k that is selected by one gambler and rejected by another gambler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let S := {e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , ej−1} and let T be the set of items selected by the gambler with M from the set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus 25 the gambler with N selected T ∪ {ei} from the set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' So T ∪ {ei} is a basis of (L\\x) |S and T is a basis of (L/x) |S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, both T ∪{ei} and T ∪{x} are bases of L |S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If only one of the two gamblers accepts the item sj then the matroid L |S∪{sj} has two bases of different cardinality, attaining a contradiction and finishing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus we have that the thresholds T ′ e, e ∈ E\\{x} guarantee at least EPROPHM as the expected total value of the gambler with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To prove that the constructed mechanism is 1/(2α + 2)-competitive it is enough to show the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that in our construction we used α-competitive non-adaptive mechanism for the matroid M and 2-competitive non-adaptive mechanism for the uniform matroid of rank 1 on P \\ {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' γ 1 αEPROPHM + (1 − γ) 1 2EPROPHP \\{x} ⩾ 1 2α+2EPROPHN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us consider the inclusion-maximal set Iopt on which the prophet achieves PROPHN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let Copt be a random variable corresponding to the unique circuit of Iopt ∪ {x} in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Recall that x is not a free element of L so such a circuit exists and is unique and contains x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' First consider the events when |Copt| ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that by definition of a circuit, for every y ∈ Copt \\ {x} the set (Iopt ∪ {x}) \\ {y} is independent in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Hence, for every y ∈ Copt \\ {x} the set Iopt \\ {y} is independent in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' So we have that conditioned on |Copt| ⩾ 3 we have PROPHM ≥ w(Iopt \\ {y}) for every y ∈ Copt \\ {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let yopt be the random variable representing the element in Copt \\{x} of smallest value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then conditioned on |Copt| ⩾ 3, we have w(Copt \\ {yopt, x}) ≥ w(C \\ {x})/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, conditioned on |Copt| ⩾ 3 we have PROPHM ⩾ w(Iopt \\ {yopt}) = w(Iopt \\ Copt) + w(Copt \\ {yopt}) ≥ w(Iopt \\ Copt) + 1 2w(Copt \\ {x}) ≥ 1 2w(Iopt) = 1 2PROPHN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Second consider the event that |Copt| < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since x is not a loop of L by definition, we have |Copt| = 2 and so Copt = {x, xopt} for some random variable element xopt ∈ P \\{x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For the event |Copt| ≥ 3 let us define the random variable element xopt to be an arbitrary element in Copt \\ {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, if |Copt| < 3 we have PROPHP \\{x} ≥ w(xopt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now let us define Jopt := Iopt \\ {xopt} and note that Jopt is independent in the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Moreover, since Iopt is the set on which the prophet achieves PROPHN, we have that conditioned on |Copt| < 3 the prophet achieves PROPHM on the set Jopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='Combining everything together we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='γ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='αEPROPHM + (1 − γ)1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2EPROPHP \\{x} = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='α + 1EPROPHM + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2α + 2EPROPHP \\{x} ≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='�w(xopt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2α + 2 + PROPHM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='α + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='���� |Copt| < 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='P [|Copt| < 3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='+ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='�PROPHM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='α + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='���� |Copt| ⩾ 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='P [|Copt| ⩾ 3] = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='�w(xopt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2α + 2 + w(Iopt \\ {xopt}) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='α + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='���� |Copt| < 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='P [|Copt| < 3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='+ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='�PROPHM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='α + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='���� |Copt| ⩾ 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='P [|Copt| ⩾ 3] ≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='�PROPHN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2α + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='���� |Copt| < 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='P [|Copt| < 3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='+ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='�PROPHM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='α + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='���� |Copt| ⩾ 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='P [|Copt| ⩾ 3] ≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2α + 2EPROPHN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let N be a matroid obtained from a matroid M by a sequence of t projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let L be the set of loops in the matroid N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let there exist an α-competitive non-adaptive mechanism for the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then there exists a non-adaptive mechanism for N\\L such that the expected total value of this mechanism is at least 1 α·3t EPROPHM\\L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In the context of Lemma 10, every set that is independent for the matroid N\\ L is also independent for the matroid M \\ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Hence, we have EPROPHM\\L ≥ EPROPHN\\L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus in case t = 1, Lemma 10 leads us to the following corol- lary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let N be a projection of the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If there is an α- competitive non-adaptive mechanism for M then there exists a 3α-competitive non-adaptive mechanism for N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us prove the statement by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Of course, in case t = 0 we have M = N and the statement is trivially true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us now assume that t is at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let N ′ be a matroid such that N ′ is obtained from the matroid M by a sequence of t − 1 projections and N is a projection of N ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since N is a projection of N ′ there is a matroid P = (E, S) and x ∈ E such that P \\ x = N ′ and P/x = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let L′ be the set of loops in the matroid N ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By induction hypothesis, there exist thresholds T ′ e, e ∈ E \\ (L′ ∪ {x}) such that the gambler with the matroid N ′\\L′ achieves at least 1 α·3t−1 EPROPHM\\L′ as the expected total value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us assume that to compute thresholds T ′ e, e ∈ E \\(L′ ∪{x}) the values of items in L were set to be 0 while the distribution of values for other items remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since L′ ⊆ L, analogously to Lemma 7 we can define thresholds T ′′ e := � +∞ if e ∈ L T ′ e otherwise 27 such that the gambler with the matroid N ′\\L achieves at least 1 α·3t−1 EPROPHM\\L as the expected total value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let T ′′′ e , e ∈ E \\(L∪{x}) be the thresholds guaran- teeing 2-competitive non-adaptive mechanism as in Theorem 1 for the uniform matroid of rank 1 on the ground set E \\ (L ∪ {x}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The constructed mechanism for the matroid N\\L selects one of two threshohold sets at random, where first set of thresholds T ′′ e , e ∈ E \\ (L ∪ {x}) is selected with probability 1/3 and the thresholds T ′′′ e , e ∈ E \\ (L ∪ {x}) with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that the thresholds T ′′ e , e ∈ E \\ (L ∪ {x}) were designed for the matroid N ′ \\ L but are used for the matroid N \\ L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' hence less items might be selected than when it is used for N ′ \\ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Also note, that the thresholds T ′′′ e , e ∈ E \\ (L ∪ {x}) are used for N \\ L but were designed for the uniform matroid of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For the analysis, let Ialg be the random variable indicating the items set selected by the gambler with matroid N ′ \\ L when the thresholds T ′′ e , e ∈ E \\ (L ∪ {x}) are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Analogously to a claim in the proof of Lemma 9, we can assume that when the thresholds T ′′ e , e ∈ E\\(L∪{x}) are used the gambler with N \\L select all items in Ialg with an exception for possibly one item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let xopt be the random variable indicating the element of maximum value in E \\ (L ∪ {x}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To finish the proof it is enough to show the following inequality 1 3E[w(Ialg) − w(xopt)] + 2 3 1 2E[w(xopt)] ≥ 1 α · 3t EPROPHM\\L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To obtain this inequality we can do estimations as follows 1 3E[w(Ialg)−w(xopt)]+2 3 1 2E[w(xopt)] = 1 3E[w(Ialg)] ≥ 1 3 1 α · 3t−1 EPROPHM\\L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now let us combine Corollary 3 and Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M and N be matroids such that dist(M, N) ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If there exists an α-competitive non-adaptive mechanism for the matroid M with α ≥ 2 then there exists a 3tα-competitive non-adaptive mechanism for the matroid N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that for α ≥ 2 we have 3α ≥ 2α + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since N can be obtained from M by a sequence of t projection and lift steps, we can use Corollary 3 or Lemma 9 for each of these steps to obtain the desired competitiveness ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2 Minor-closed families theorem Lemma 12 (Lemma 6 in [HN20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let p and n be integers such that p ⩽ n − 2 and p is prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The matroid U2,n is not representable over the field Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The following Structural Hypothesis is due to Geelen, Gerards and Whittle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The proof of this Structural Hypothesis has not appeared in print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 28 Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let p be a prime number and M is a proper minor-closed class of matroids representable over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then there exist k, n, t such that every M ∈ M is a restriction of an Fp- representable matroid M ′ having a full tree-decomposition (T, X) of thickness at most k so that for every v ∈ V (T ) if M ′ |clM′(Xv) has a M(Kn) minor, then there exists a 2-column sparse matroid N with dist(M ′ |clM′ (Xv), N) ⩽ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let k, n, t are as stated in the Structural Hypothesis 1 on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M1 be the set of matroids on distance t or less from some 2-column sparse matroid and are representable over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Theorem 4 all 2-column sparse matroids have a 32-competitive non-adaptive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Lemma 11 there exists a (3t · 32)-competitive mechanism for matroids in M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M2 be the set of matroids without M(Kn) minor and are representable over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Lemma 12 all matroids in M2 do not have U2,p+2 as a minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then by Corollary 2, we have that there is a pp3n-competitive non-adaptive mechanism for every matroid in M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By the Structural Hypothesis 1 we have that every M ∈ M is a restriction of some M ′ with a full tree-decomposition (T, X) of thickness at most k so that for every v ∈ V (T ) M ′ |clM′ (Xv)∈ M1 ∪ M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus by Theorem 13, matroid M ′ has a γ := (max(3t · 32, pp3n) · pk+1)- competitive non-adaptive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Lemma 7 the matroid M has also a γ-competitive non-adaptive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 29 References [AKW19] Pablo D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Azar, Robert Kleinberg, and S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' [Sey80] Paul D Seymour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Decomposition of regular matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Journal of combinatorial theory, Series B, 28(3):305–359, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' [Sot13] José A Soto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Matroid secretary problem in the random-assignment model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' SIAM Journal on Computing, 42(1):178–211, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 31' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} diff --git a/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf b/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0adaf0058a1e1bbdbe6c59f6de6a0ebb286e6f53 --- /dev/null +++ b/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1b578983b5d04843cefafc656ac55156b252ba7b6e555eedc94dc7f2c7a8061d +size 4197471 diff --git a/6dFKT4oBgHgl3EQf_S4j/vector_store/index.faiss b/6dFKT4oBgHgl3EQf_S4j/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..b02b752c257beb2b92e4c19d6ee4314f052cf58e --- /dev/null +++ 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100644 index 0000000000000000000000000000000000000000..f3b527cb6fefa5c16c4f20cbcb86d44e82c01ccb --- /dev/null +++ b/8NE1T4oBgHgl3EQf7gWJ/content/tmp_files/2301.03535v1.pdf.txt @@ -0,0 +1,788 @@ +1 +RISs and Sidelink Communications in Smart Cities: +The Key to Seamless Localization and Sensing +Hui Chen, Member, IEEE, Hyowon Kim, Member, IEEE, Mustafa Ammous, Student Member, IEEE, +Gonzalo Seco-Granados, Senior Member, IEEE, George C. Alexandropoulos, Senior Member, IEEE, +Shahrokh Valaee, Fellow, IEEE, and Henk Wymeersch, Senior Member, IEEE +Abstract—A smart city involves, among other elements, intelli- +gent transportation, crowd monitoring, and digital twins, each of +which requires information exchange via wireless communication +links and localization of connected devices and passive objects +(including people). Although localization and sensing (L&S) are +envisioned as core functions of future communication systems, +they have inherently different demands in terms of infrastructure +compared to communications. Wireless communications gener- +ally requires a connection to only a single access point (AP), +while L&S demand simultaneous line-of-sight propagation paths +to several APs, which serve as location and orientation anchors. +Hence, a smart city deployment optimized for communication +will be insufficient to meet stringent L&S requirements. In this +article, we argue that the emerging technologies of reconfigurable +intelligent surfaces (RISs) and sidelink communications constitute +the key to providing ubiquitous coverage for L&S in smart cities +with low-cost and energy-efficient technical solutions. To this end, +we propose and evaluate AP-coordinated and self-coordinated +RIS-enabled L&S architectures and detail three groups of appli- +cation scenarios, relying on low-complexity beacons, cooperative +localization, and full-duplex transceivers. A list of practical issues +and consequent open research challenges of the proposed L&S +systems is also provided. +Index Terms—Smart cities, reconfigurable intelligent surfaces, +localization, sensing, sidelink communication. +I. INTRODUCTION +The emerging concept of smart cities aims to improve ac- +cessibility to public services, advance digitization of the urban +environment, and monitor various human-oriented processes +as well as assets, by harmonizing diverse digital technologies +at a city level [1]. This broad concept integrates various +independent applications to bring improvements both at the +societal level (such as smart homes, smart transportation, sup- +ply chains, and environment monitoring), and at the individual +level (such as indoor navigation and extended reality (XR)). To +realize the concept of smart cities, reliable, low-latency, and +high-speed communication systems (to support information +exchange and management among interconnected devices), as +well as accurate localization and sensing (L&S) (to support +communication and provide situation-awareness services), are +of great importance. In this article, we use the term localization +to indicate the position (and possibly orientation) estimation of +a target user equipment (UE), and the term sensing to specify +the position estimation of passive objects (i.e., objects without +networking infrastructure or non-cooperating ones). +By exploiting the large antenna array sizes and wide +bandwidth of millimeter-wave/THz systems, recent research +activities in industry and academia on integrated sensing, +localization, and communication (ISLAC) are growing. ISLAC +is able to utilize communication infrastructures and signals +to enable synergies with L&S for diverse applications. To +this end, several standardization efforts and 3GPP activities +have been recently studied, such as the definition of new +radio positioning requirements, evaluation methodologies, and +techniques (dependent on radio access technology and not), in +TR 38.855 [2] as well as the development of Wi-Fi sensing +technology (in both sub-7 GHz and mmWave spectrum), in +the IEEE 802.11bf standard [3]. +While high angular and delay resolution (due to large +arrays and wide signal bandwidths) facilitate L&S tasks, +signal coverage is one of the major challenges, especially +for high-frequency systems which suffer from high path loss +and high blockage probability. In contrast to communications, +which is possible to work with only one point-to-point link, +L&S functions necessitate access to multiple access points +(APs). Reflective reconfigurable intelligent surfaces (RISs) +constitute a promising emerging technology for extending +coverage and dynamically programming signal propagation +with almost zero-energy consumption [4], thus, supporting, +or even enabling in certain cases, wireless communications, +as well as L&S tasks in various scenarios [5]. However, since +they are incapable of generating their own signals and only +modify the analog waveforms impinging on them, separate +signal generation sources are needed. +The sources generating signals for RIS-assisted L&S can +be: i) APs with full communication capabilities, ii) low- +complexity beacons that broadcast or receive L&S reference +signals; or iii) the UEs themselves that are involved in the +L&S tasks. The latter two cases are particularly relevant when +power-hungry APs, whose deployment is usually costly, needs +to be avoided. To fulfill the ubiquitous L&S requirements in +out-of-coverage areas or partially-covered ones (e.g., indoor +UEs and vehicles in tunnels), sidelink communications via +the PC5 interface can be particularly useful [6], as mentioned +in the 3GPP TR 38.845 report [7]. In addition, cooperative +localization between several UEs can be employed to enhance +or enable L&S. Furthermore, when a UE is equipped with +a full-duplex transceiver [8] —a technology that is widely +discussed in ISLAC for sensing purposes—, it can both +simultaneously transmit and receive the signal to localize itself +with a single RIS anchor. +In this article, we argue that RIS technology in conjunction +with sidelink communications can provide seamless L&S so- +lutions, speeding up the intelligent transformation of cities into +arXiv:2301.03535v1 [eess.SP] 9 Jan 2023 + +2 +Beacon-Assisted Localization +Beacon +Self-L&S with +a Full-Duplex +Transceiver +Beacon-UE Channel +RIS Anchor Channel +RIS Localization Channel +Sidelink Channel +Sensing Channel +Scenarios +Cooperative Localization +RIS +A2 +A1 +B1 +B2 +C1 +C2 +C3 +RIS-attached +Vehicle +AP +A3 +B3 +Figure elements by macrovector on Freepik +Fig. 1. RIS-enabled seamless L&S scenarios in smart cities: a) beacon-assisted localization, b) cooperative localization, and c) self-L&S with a full-duplex +transceiver. +smart entities. We particularly elaborate on the representative +RIS-enabled L&S scenarios illustrated in Figure 1: beacon- +assisted localization, cooperative localization, and self-L&S +with a full-duplex transceiver, describing both AP-coordinated +and AP-free architectures for different coverage scenarios. We +discuss the open challenges with the proposed green (low- +cost and energy-efficient) L&S systems in smart cities and list +potential directions for future research. +II. L&S ARCHITECTURES AND PROTOCOLS +In this section, we describe the different entity types of the +proposed L&S system for smart cities, relying on RISs and +sidelink communications, as well as its enabling architectures, +depending on whether an AP is present for L&S coordination. +A. Entity Types and Overall Architecture +In the proposed RIS-enabled L&S system, there are several +types of entities: APs, beacons, RISs, and UEs, as shown in +Figure 2. The APs (e.g., a gNB macro base station) provide +cellular services to the devices in coverage. A beacon could be +a roadside unit (RSU) that is capable of sending and receiving +L&S reference signals via sidelink communications. In this +way, the explicit involvement of expensive and power-hungry +APs with full communication protocols is not needed for L&S +purposes. RISs serve as reference anchors to assist ISLAC +tasks, and are controlled by a dedicated entity responsible +for realizing communication functions. Finally, UEs may have +different hardware capabilities (e.g., single/multiple antennas, +half-/full-duplex) that play different roles in L&S (e.g., a target +UE to be localized, an assistant UE with known or measured +location to assist L&S, or a server/coordinator in performing +L&S tasks). Although all the devices involved in the targeted +tasks will have sidelink communication capabilities, beacons +usually have fewer power constraints than UEs, while the +controllers of reflective RISs are expected to work in low- +power mode without sending or processing L&S reference +signals [4]. We further consider that RISs coordinate their +reflective beamforming, either using time division or phase +profile codes in the time domain. +For the scenarios considered in this article, we propose two +different architectures: one based on AP coordination and the +other on self-coordination. The former architecture works for +UEs (and other L&S-related devices) inside a coverage area or +in partial-covered areas (where sidelink is available), requiring +a specific AP to serve as a L&S coordinator, e.g., interact +with the location management function (LMF) via the NR +positioning protocol A (NRPPa) [9]. The self-coordinated ar- +chitecture relies explicitly on sidelink communications, being +particularly suitable for UEs in the out-of-coverage of APs, or +for UEs connected to APs that cannot meet the latency and +spatial resolution requirements (e.g., legacy 3G/4G APs). In +both architectures, L&S signals can be generated by a beacon, +an assistant UE, or a target UE, depending on the network +topologies and the specific application scenario. + +3 +Beacon-UE Channel +Sidelink Channel +RIS Channel +AP-Coordinated Control Link +Self-Coordinated Control Link +L&S Coordinator +Controller +RIS-1 +In-coverage +Partial coverage +Out-of-coverage +AP +Beacon +UE-1 +UE-2 +UE-3 +UE-4 +UE-5 +RIS-2 +RIS-3 +Fig. 2. UEs performing L&S in different coverage areas. The AP-coordinated +architecture can be used for UEs located in in-coverage or partial-coverage +(sidelink communications required) areas. When the UEs and all the L&S +devices cannot access any surrounding APs (i.e., lying in out-of-coverage +areas), the self-coordinated architecture is the only option for L&S services. +B. AP-Coordinated Architecture +This architecture relies on one or several APs to allocate +the available radio resources and ensure timing among the +connected devices, which are the UEs, the RIS controllers, and +dedicated beacon nodes. This network management scheme is +similar to the mode-1 in sidelink communications [6], and the +L&S protocols can be summarized into the following 6 steps: +1) The target UE triggers an L&S request to the AP (which +is selected as the task coordinator). +2) The coordinator exchanges related location information +with the core network (e.g., via the NRPPa), determines +L&S configurations (e.g., broadcasting, sidelink, and +RIS phase profiles), as well as selects and notifies nearby +beacons, UEs, and RISs that are involved in the L&S +task. +3) The coordinator allocates time-frequency resources for +L&S to all involved devices and triggers the RIS con- +troller(s) to configure their phase profiles for the whole +duration of the estimation process. +4) The beacons and/or UEs transmit L&S reference signals, +which are reflected by the involved RIS(s) and received +by the target UE. +5) The collected measurements are used by the target UE +for the localization, and/or sensing tasks. Alternatively, +this computation can be offloaded at the localization +server (e.g., the coordinating AP or another UE with +high computational power). +6) The target UE updates the task coordinator with the +estimated L&S results; this optional step can serve as +prior information for future use. +For the partial-coverage scenarios, the out-of-coverage UEs +need to establish sidelink communications with devices that +are covered by APs. Then, the L&S tasks can be performed +similarly to the aforementioned steps. +C. Self-Coordinated Architecture +An AP-free architecture is required for cases where the +devices involved in L&S tasks are located in out-of-coverage +areas. Similar to the mode-2 sidelink communications [6], +L&S tasks can be autonomously realized by selecting a +specific device as the localization coordinator, as follows: +1) The target UE discovers nearby devices (e.g., beacons, +RISs, and other UEs) and obtains their location infor- +mation (if available). +2) Based on the discovered neighbors, the target UE de- +termines a L&S task coordinator (could be itself) and +notifies it of the L&S configurations. +3) The target UE triggers a L&S request to the coordi- +nator and performs the same actions as with the AP- +coordinated architecture (i.e., steps 2)–6)). +III. RIS-ENABLED L&S SCENARIOS +In this section, we will present three representative RIS- +enabled L&S scenarios for smart city applications relying on +the aforementioned architectures and protocols. We mainly +focus on localization applications and list potential position- +ing scenarios in systems with minimum infrastructure and +resources. For example, if positioning can be done with a +single antenna, it can also be achieved using a multi-antenna +array. The same applies to narrowband (NB)/wideband (WB) +signals, single/multiple anchors, and availability/unavailability +of a line-of-sight (LOS) path between the UE and the active +signal source. +A. Beacon-Assisted Localization +With low-complexity beacons, high flexibility in the in- +stallation and deployment of L&S systems is feasible. A +typical use case could be a train station with multiple low- +cost beacons broadcasting L&S reference signals to UEs to +navigate indoors, via the support of RISs. In this category, we +consider UE localization (with the aid of one or more RISs) +and RIS localization (with the aid of several beacons). +A1) Single-RIS-Enabled UE Localization: In a WB system +where the LOS channel is available, the delays of the LOS +and RIS paths can be estimated based on the L&S reference +signals sent from a beacon. Assuming the RIS and beacon +states are known, the angle-of-departure (AOD) at the RIS +can also be estimated. The target UE can then be localized +by the intersection of a hyperbola (i.e., time-difference-of- +arrival (TDOA) of the LOS and RIS paths) and the line in +the direction of the AOD at the RIS. When the target UE +is equipped with an antenna array, 3D orientation can also +be estimated based on the estimated angle-of-arrivals (AOAs). +This is the basic RIS-enabled localization scenario that only +requires a single low-complexity beacon [10]. +A2) Multi-RIS-Enabled UE Localization: If multiple RISs +are simultaneously available, the requirements for LOS and +WB are unnecessary. The AODs from different RISs can be +estimated and used to localize the UE by intersecting the AOD +lines. In this way, localization tasks can be completed using +NB signals, which saves bandwidth resources for communi- +cations. Figure 3 shows the position error bound (PEB) of the +target UE with different positions inside a 5 × 10 m2 area. As +shown, with multiple RISs, the UE is localizable even under +blockage of the LOS path between the beacon and the UE. + +4 +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +x axis [m] +0 +1 +2 +3 +4 +5 +y axis [m] +Beacon +RIS +Wall +0.001 +0.01 +0.1 +1 [m] +Fig. 3. Scenarios A1 and A2: PEB (in meters) with different UE locations +in a multi-RIS-aided localization scenario. The target UE can be localized +with one RIS and a beacon-UE LOS path, or with at least 2 RISs under LOS +blockage conditions. +However, the localization tasks cannot be performed when +only one anchor (beacon or RIS) is visible to the UE (see +the yellow triangular area around the point [3, 3] m). +A3) RIS Localization via Multi-Static Sensing: In a +scenario where passive UEs or objects are coated with RISs, +the localization (or sensing, depending on scenarios) can be +performed semi-passively with only a small amount of energy +needed for localization coordination and RIS phase profile +control. Such localization tasks can estimate the positions (and +orientations) of RIS-coated objects by using several beacons +with known positions. Note that the geometrical constraints +can largely reduce the difficulties in these scenarios. For +example, the orientation of an RIS can be assumed as 1D +(e.g., placed on the top of a car facing up). In addition, the +adoption of antenna arrays at the beacons can further simplify +the RIS localization problem. +B. Cooperative Localization +Sidelink communications (e.g., device-to-device (D2D) +communications, or the vehicular version known as vehicle- +to-everything (V2X) communications) has been introduced in +the millimeter-wave band for information exchange between +vehicles, opening the road for numerous use cases, such +as platooning, collision avoidance and autonomous driving +[6]. The combination of RISs and sidelink is expected to +provide low-latency and high-reliability communications [11]. +Interestingly, it can also be exploited to assist L&S. Sidelink +communications enables UEs to participate in a cooperative +manner in the sharing of position and surrounding information +within a local network, and in performing relative location +estimation using sidelink signals [12]. We will focus on the +latter case, where the absolute location can be estimated +using RIS anchors (i.e., with known position and orientation +information), without any APs or beacons. A typical scenario +could be cooperative vehicular networks in urban areas with +severe AP and GPS signal blockages. +We next discuss single- and multi-RIS-involved localization +scenarios, where single-antenna UEs cooperate to estimate +their positions via WB sidelink signals. We also consider a +more general scenario where RIS-coated objects are involved, +resulting in cooperative RIS localization. +B1) Single-RIS-Enabled Cooperative Localization: Con- +sider a scenario with several UEs and one RIS anchor, where +the UEs wish to estimate their positions. We assume that each +UE can send sidelink signals (i.e., being the transmitter (TX)) +to other UEs, which arrive at the receiver (RX) UEs via two +paths (i.e., the UE-UE and UE-RIS-UE paths). By proper +control of the RIS elements, those two paths can be separated, +and thus, two delay measurements can be obtained. Due to the +unknown states of the UEs, both the AOD and AOA at the +RIS for every TX-RX pair of UEs are unknown and cannot +be directly estimated. However, we can estimate the spatial +frequency information at the RIS for every TX-RX pair. This +scenario requires at least three UEs to cooperate and render +their locations feasibly without ambiguities. Figure 4 compares +the PEBs for three UEs in an RIS-enabled versus beacon-aided +(equipped with an antenna array) 3D cooperative localization +scenario as a function of the number of RIS elements. +B2) Multi-RIS-Enabled One-Way Sidelink Localization: +In the scenario with at least two RISs, one-way sidelink +communications is sufficient to localize both the TX and RX +UEs. Assume that one UE takes the role of the TX and the +other UE is the RX. With two RISs, three delay measurements +can be obtained between the TX and RX via the LOS and +the two RIS paths. However, that would require an optimal +joint design of the reflection elements at both RISs to be able +to separate the paths at the RX. In addition, once the RIS +paths are separated, we can also estimate the spatial frequency +information at each RIS. Thus, those collected measurements +can be utilized to estimate the locations of the TX and RX. +B3) Cooperative RIS Localization: Let us consider a more +general scenario where one RIS (or several) with a known state +is used to localize multiple UEs (with sidelink capabilities) and +objects (coated with an RIS). This scenario is challenging due +to the high complexity of the network and a large number of +unknowns. However, with a proper design of all the involved +RIS profiles and the transmission protocol, this problem can +be decomposed into a cooperative localization (see B1) and +an RIS localization (see A3) problems. Similar to B1, at least +several UEs (depending on scenarios) need to take the role +of the TX, and transmit sidelink signals to the other UEs via +the direct and indirect paths. Once the UEs are localized, the +RIS localization task can be solved similarly to A3, and the +estimation results can be refined by processing all the available +information. +C. Self-L&S with a Full-Duplex Transceiver +When a UE is equipped with a full-duplex transceiver (like +radar) [8], the multi-RIS setup and cooperation between UEs +are unnecessary. Instead, this UE can perform self-positioning +with a single RIS and use the multipath components to map the +environment over time; this process is known as monostatic +simultaneous localization and mapping (SLAM). It is noted +that SLAM is not limited to full-duplex UEs, and bistatic +SLAM can also be performed in use cases A1-A3 and B1- +B3. +We next present three beacon-free L&S scenarios with a +full-duplex UE. + +5 +0 +20 +40 +60 +80 +100 +120 +140 +160 +0 +0.5 +1 +1.5 +UE-1 (RIS) +UE-1 (beacon) +UE-2 (RIS) +UE-2 (beacon) +UE-3 (RIS) +UE-3 (beacon) +Number of RIS Elements +PEB [m] +Fig. 4. +PEBs of RIS-enabled vs. beacon-aided cooperative localization for +different RIS sizes and a fixed random phase profile setup. It is shown that, +with a sufficient number of RIS elements, an active anchor can be replaced +with a passive one (RIS) without performance degradation. +C1) RIS-Enabled Self-Localization: Consider a system +with a single-antenna UE and an RIS, where the UE transmits +L&S reference signals and receives their back-scattered ver- +sions, i.e., the UE-RIS-UE (controlled path) and UE-landmark- +UE (uncontrolled path) signals. One option for the RIS phase +profiles is to consider directional reflective beams, which can +be efficiently designed when the UE position uncertainty (even +under mobility cases) is available [13]. The delay and angle +information at the RIS of the UE-RIS-UE channel can be +estimated for this scenario, and then used to localize the UE. In +Figure 5, beampatterns at the RIS with two different phase pro- +files are illustrated, focusing on the UE uncertainty region. As +demonstrated, the optimized phase profiles of [13] can provide +sufficient beamforming gain, compared with directional phase +profiles. This gain can offer improved L&S performance. +C2) RIS-Enabled SLAM: If a UE is equipped with a +full-duplex multiple-input multiple-output (MIMO) antenna +array [8], SLAM can be enabled. Similar to scenario C1, the +signals from different paths can be resolvable with optimized +RIS phase profiles and precoders/combiners. The following +channel parameters can be estimated: i) the signal propagation +delay, the AOD at the RIS, and the AOA at the UE for the +UE-RIS-UE channel; as well as ii) the delay and AOA at +the UE for each UE-landmark-UE channel. In addition to the +position information obtained from the controlled path (as in +scenario C1), the angular resolution offered by the UE array +can be leveraged to map/sense the environment. With multiple +estimations, the localization and radio mapping performance +can be improved using state-of-the-art filters (e.g., Poisson +multi-Bernoulli filter). +C3) RIS Localization with a Full-Duplex Array: Consider +the more general scenario from C1 including one anchor +RIS mounted on a wall, a UE equipped with a full-duplex +MIMO transceiver, and several objects coated with RISs (e.g., +mounted on the front and rear side of a vehicle). In addition to +the signals reflected from the anchor RIS (as also in scenario +C1), the UE also receives single-bounce reflected signals from +the RISs mounted on the objects. The time delay, AOA, and +the amplitude of the channel gain for each signal path can be +estimated, which can be used for the localization of both itself +and the RISs-coated objects. When multiple UEs are present +and cooperate in the estimation process, the orientation of the +(a) Directional Phase Profiles +(b) Optimized Phase Profiles +Fig. 5. The reflective beamforming gain (in dB) with a 50 × 50 RIS using +(a) a directional phase profile, and (b) an optimized phase profile via [13]. +The red squares represent the UE angular uncertainty region, which needs to +be fully covered by an effective beampattern design. +RISs-coated objects can also be obtained. In a scenario without +any anchors, this RIS localization can also help in estimating +the relative locations of the active UE and passive UEs. +IV. OPEN RESEARCH CHALLENGES +With the assistance of low-complexity beacons, cooperative +localization, and full-duplex radios, the L&S coverage for +smart city applications can be significantly extended. However, +there exist several practical issues that need to be thoroughly +investigated. In this section, we discuss the most critical +challenges with the proposed RIS-enabled L&S system and +list possible directions for future research. +A. Anchor Deployment Optimization +The placement of the anchors (e.g., beacons and RISs) is +critical to meet the L&S key performance indicator (KPI) re- +quirements within a service area (e.g., error bounds lower than +a certain threshold, as shown in Figure 3). The deployment +involves both the position and orientation optimization of the +anchors, taking into account the blockage in the surrounding +environment. RIS-aided SLAM can help in creating such an +environment map, which can be supported by cooperative +sidelink UEs. Heuristic optimization solutions can then be +applied for finding optimal anchor sites, extending approaches +from the literature [14]. + +e.100100150evation20 +Gro40 +.M06080J60 +azllBeaiot0 +muthele100100150evation20 +G[o]40 +.=060 +B80ncertain60 +azilReaiol +Vmuth6 +B. Resource Allocation and Coordination +RIS-aided L&S systems involve APs, beacons, RISs, and +UEs, making them inherently heterogeneous. Resource al- +location for L&S tasks, including power allocation, time- +frequency allocation, beamforming design, and scheduling +must be carefully designed to ensure a favorable trade-off +with conventional communication services. Depending on the +KPI requirements of the applications that send L&S ser- +vice requests, new objectives that consider integrated L&S +and communications should be formulated and satisfied. An +important part of resource allocation is RIS phase profile +optimization and multiplexing [10]. Broad RIS beams lead to +coverage reduction, while narrow pencil beams are sensitive to +misalignment. Hence, highly adaptive RIS profile designs are +needed, relying, when possible, on prior UE and object state +information. When RISs are large, the near-field effects need +to be taken into consideration and the beamforming designs +should account for the curvature of arrival, resulting in beam- +focusing designs. RIS multiplexing can be addressed by time +multiplexing, temporal coding, and making use of high path +loss for spatial reuse. The afore-described resource allocation +problems can be tackled by a combination of traditional +optimization-based methods (e.g., convex optimization) and +learning-based methods (e.g., reinforcement learning). +C. Estimation Algorithms +From an algorithmic perspective, there are challenges re- +lated to channel parameter estimation, tracking in dynamic +environments, and calibration. Channel parameter estimation +in the presence of severe multipath is difficult since almost +passive reflective RISs have no local signal processing capa- +bilities. Moreover, in cooperative localization (scenarios B1 +and B2) and RIS localization (e.g., scenarios A3, B3, and +C3) tasks, the AOAs/AODs at the RISs are coupled, meaning +that we can no longer estimate them independently. Instead, +only spatial frequencies (containing coupled AOAs and AODs +information) can be obtained, requiring novel algorithms for +further processing. More refined channel parameter estimation +also requires accurate channel models and the RISs’ impact +on them, such as the near-field effect, beam squint effect, RIS +element failures, and anchor calibration errors. +Due to mobility, difficult conditions such as signal blockage, +unresolvable signal paths, and severe path loss will affect L&S +performance. Multiple RISs can be involved to handle such +blockages, offering coverage extension. As the carrier fre- +quency increases, the signal resolutions become higher, and the +effect of the severe path loss can be mitigated by adopting RISs +with larger sizes or active RISs with reflection amplification +capabilities [15]. Sensing also suffers from inherent complica- +tions, such as an unknown number of objects, unknown types +of objects, unknown detection probabilities for signal paths, +extended objects, and multi-bounce observations. Dedicated +filters should be developed to address these complications and +get integrated into the L&S framework. +Finally, in terms of calibration, anchor geometry error and +hardware impairments (HWIs) are two important aspects. We +note that the geometrical calibration of an RIS is similar to RIS +localization (as described in scenarios A3, B3, and C3), which +requires a calibration agent that incorporates other sources of +localization estimations (i.e., sensor fusion). While for HWIs, +the channel model could be too complicated when considering +each specific impairment (e.g., mutual coupling and phase +noise), learning-based methods can be considered to unveil +their impact and drive practical algorithmic designs. +D. Understanding Anchor Hardware Alternatives +There are also opportunities to improve L&S coverage via +variations of the hardware deployed at the beacons, RISs, and +UEs. On the beacon and UE sides, multi-panel arrays (i.e., 3D +arrays) could be implemented for further coverage extension. +On the RIS side, new types of RISs are emerging beyond +almost passive reflective RISs [15]. As previously mentioned, +an active RIS can be used to boost the signal energy (i.e., +change both the amplitude and phase of the incident signal) for +improved coverage. A receiving RIS (also known as a hybrid +RIS or a simultaneously reflecting and sensing RIS) can enable +parameter estimation at the RIS side, offering extra degrees +of freedom for the design of L&S estimation approaches. +Omni-directional RISs, intended to realize simultaneous reflec- +tion and refraction (i.e., 360◦ coverage), enable simultaneous +indoor and outdoor 3D localization. A non-reciprocal RIS +that integrates nonreciprocal phase shifters allows full-duplex +communications, and a delay-adjustable RIS is capable of +adjusting the delays of signals reflected by different RIS +elements, which contributes to the alleviation of the beam +squint effect. All of these alternatives have implications on +L&S services and merit further study. +E. Privacy, Security, and Social Acceptance Issues +Cooperative L&S require extensive information exchange of +local measurements between devices, which may cause privacy +issues. For example, the RX in scenario B2 can estimate the +position of the TX with a one-way pilot signal transmission. In +addition, different types of cyber attacks can reduce the L&S +service availability, or even provide an undetected erroneous +location estimation, which is unacceptable for safety-critical +applications. Currently, several security management systems +have been standardized (e.g., IEEE 1609.2), and security +threats have been identified for sidelink communications. +However, the discussions on L&S task-related security issues +are still at the initial stage, and potential threats need to +be explored and eliminated. A final aspect related to the +widespread adoption of RISs lies in their social acceptance. +RISs should be integrated in a way that they blend into +the environment (ideally, be transparent). To this end, the +benefits of RISs to improve safety and reduce electromagnetic +emissions should be demonstrated and disseminated. +V. CONCLUSION AND OUTLOOK +The smart city paradigm constitutes the epitome of the +widespread adoption of digital services for societal needs. +It is envisioned to profit people and city-level businesses, +offering efficient, safe, and comfortable living spaces as well + +7 +as everyday-life smart-living applications. To achieve this +overarching goal, seamless wireless communications among +diverse devices and L&S are of paramount importance, en- +abling information exchange, device localization, and mapping +of the environment. In this article, we discussed the key +to achieving low-cost and energy-efficient seamless L&S, +namely, reflective RISs in conjunction with sidelink commu- +nications. We presented AP-coordinated and AP-free system +architectures and detailed three RIS-enabled L&S scenarios, +each including several use cases and most relying on sidelink +communications. As became apparent, instead of using APs +with full communication capabilities, low-complexity beacons +and RISs can be widely-deployed to enable green L&S smart +city applications. In addition, when multiple UEs with sidelink +communication capabilities can be connected in the same +network, cooperative localization can relieve the requirement +for multiple anchors. Furthermore, when devices are equipped +with full-duplex transceivers, they can localize themself and +map their surrounding environment with only a single RIS +anchor. Finally, an extended list of open research challenges +relevant to the proposed RIS-enabled seamless L&S concept +was presented, including the necessity for anchor deployment +optimization and optimized resource allocation schemes, al- +gorithmic and privacy issues, as well as the role of multi- +functional RISs. +ACKNOWLEDGMENTS +This work was supported, in part, by the European Com- +mission through the EU H2020 RISE-6G project under grant +101017011, and by the 6G-Cities project from Chalmers +Transport Area of Advance. +REFERENCES +[1] S. Kisseleff et al., “Reconfigurable intelligent surfaces for smart cities: +Research challenges and opportunities,” IEEE Open J. Commun. Soc., +vol. 1, pp. 1781–1797, Nov. 2020. +[2] “3GPP +TR +38.855 +V16.0.0: +Study +on +NR +positioning +support +(Release +16) +(accessed +on +28-Dec-2022),” +Mar. +2019. +[On- +line]. Available: https://portal.3gpp.org/desktopmodules/Specifications/ +SpecificationDetails.aspx?specificationId=3501 +[3] C. 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Ko et al., “V2X-based vehicular positioning: Opportunities, chal- +lenges, and future directions,” IEEE Wireless Commun., vol. 28, no. 2, +pp. 144–151, Mar. 2021. +[13] H. Kim et al., “RIS-enabled and access-point-free simultaneous radio +localization and mapping,” arXiv preprint arXiv:2212.07141, 2022. +[14] A. Albanese et al., “LOKO: localization-aware roll-out planning for +future mobile networks,” IEEE Trans. Mobile Comput., (Early Access), +2022. +[15] M. Jian et al., “Reconfigurable intelligent surfaces for wireless commu- +nications: Overview of hardware designs, channel models, and estima- +tion techniques,” Intell. Converged Netw., vol. 3, no. 1, pp. 1–32, Mar. +2022. +Hui Chen (hui.chen@chalmers.se) is a postdoctoral researcher at Chalmers +University of Technology, Sweden. +Hyowon Kim (hyowon@chalmers.se) is a postdoctoral researcher at Chalmers +University of Technology, Sweden. +Mustafa Ammous (mustafa.ammous@mail.utoronto.ca) is a Ph.D. student at +University of Toronto, Canada. +Gonzalo Seco-Granados (gonzalo.seco@uab.cat) is a professor at Universitat +Autonoma of Barcelona, Spain. +George C. Alexandropoulos (alexandg@di.uoa.gr) is an assistant professor +at the Department of Informatics and Telecommunications, National and +Kapodistrian University of Athens, Greece. +Shahrokh Valaee (valaee@ece.utoronto.ca) is a professor at University of +Toronto, Canada. +Henk Wymeersch (henkw@chalmers.se) is a professor at Chalmers Univer- +sity of Technology, Sweden. + diff --git a/8NE1T4oBgHgl3EQf7gWJ/content/tmp_files/load_file.txt b/8NE1T4oBgHgl3EQf7gWJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3384b3a028c7f80cf7d48b34ebf36397ca03ebc2 --- /dev/null +++ b/8NE1T4oBgHgl3EQf7gWJ/content/tmp_files/load_file.txt @@ -0,0 +1,456 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf,len=455 +page_content='1 RISs and Sidelink Communications in Smart Cities: The Key to Seamless Localization and Sensing Hui Chen, Member, IEEE, Hyowon Kim, Member, IEEE, Mustafa Ammous, Student Member, IEEE, Gonzalo Seco-Granados, Senior Member, IEEE, George C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Alexandropoulos, Senior Member, IEEE, Shahrokh Valaee, Fellow, IEEE, and Henk Wymeersch, Senior Member, IEEE Abstract—A smart city involves, among other elements, intelli- gent transportation, crowd monitoring, and digital twins, each of which requires information exchange via wireless communication links and localization of connected devices and passive objects (including people).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Although localization and sensing (L&S) are envisioned as core functions of future communication systems, they have inherently different demands in terms of infrastructure compared to communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Wireless communications gener- ally requires a connection to only a single access point (AP), while L&S demand simultaneous line-of-sight propagation paths to several APs, which serve as location and orientation anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Hence, a smart city deployment optimized for communication will be insufficient to meet stringent L&S requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In this article, we argue that the emerging technologies of reconfigurable intelligent surfaces (RISs) and sidelink communications constitute the key to providing ubiquitous coverage for L&S in smart cities with low-cost and energy-efficient technical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' To this end, we propose and evaluate AP-coordinated and self-coordinated RIS-enabled L&S architectures and detail three groups of appli- cation scenarios, relying on low-complexity beacons, cooperative localization, and full-duplex transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' A list of practical issues and consequent open research challenges of the proposed L&S systems is also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Index Terms—Smart cities, reconfigurable intelligent surfaces, localization, sensing, sidelink communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' INTRODUCTION The emerging concept of smart cities aims to improve ac- cessibility to public services, advance digitization of the urban environment, and monitor various human-oriented processes as well as assets, by harmonizing diverse digital technologies at a city level [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' This broad concept integrates various independent applications to bring improvements both at the societal level (such as smart homes, smart transportation, sup- ply chains, and environment monitoring), and at the individual level (such as indoor navigation and extended reality (XR)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' To realize the concept of smart cities, reliable, low-latency, and high-speed communication systems (to support information exchange and management among interconnected devices), as well as accurate localization and sensing (L&S) (to support communication and provide situation-awareness services), are of great importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In this article, we use the term localization to indicate the position (and possibly orientation) estimation of a target user equipment (UE), and the term sensing to specify the position estimation of passive objects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', objects without networking infrastructure or non-cooperating ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' By exploiting the large antenna array sizes and wide bandwidth of millimeter-wave/THz systems, recent research activities in industry and academia on integrated sensing, localization, and communication (ISLAC) are growing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' ISLAC is able to utilize communication infrastructures and signals to enable synergies with L&S for diverse applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' To this end, several standardization efforts and 3GPP activities have been recently studied, such as the definition of new radio positioning requirements, evaluation methodologies, and techniques (dependent on radio access technology and not), in TR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='855 [2] as well as the development of Wi-Fi sensing technology (in both sub-7 GHz and mmWave spectrum), in the IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='11bf standard [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' While high angular and delay resolution (due to large arrays and wide signal bandwidths) facilitate L&S tasks, signal coverage is one of the major challenges, especially for high-frequency systems which suffer from high path loss and high blockage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In contrast to communications, which is possible to work with only one point-to-point link, L&S functions necessitate access to multiple access points (APs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Reflective reconfigurable intelligent surfaces (RISs) constitute a promising emerging technology for extending coverage and dynamically programming signal propagation with almost zero-energy consumption [4], thus, supporting, or even enabling in certain cases, wireless communications, as well as L&S tasks in various scenarios [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' However, since they are incapable of generating their own signals and only modify the analog waveforms impinging on them, separate signal generation sources are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' The sources generating signals for RIS-assisted L&S can be: i) APs with full communication capabilities, ii) low- complexity beacons that broadcast or receive L&S reference signals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' or iii) the UEs themselves that are involved in the L&S tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' The latter two cases are particularly relevant when power-hungry APs, whose deployment is usually costly, needs to be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' To fulfill the ubiquitous L&S requirements in out-of-coverage areas or partially-covered ones (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', indoor UEs and vehicles in tunnels), sidelink communications via the PC5 interface can be particularly useful [6], as mentioned in the 3GPP TR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='845 report [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In addition, cooperative localization between several UEs can be employed to enhance or enable L&S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Furthermore, when a UE is equipped with a full-duplex transceiver [8] —a technology that is widely discussed in ISLAC for sensing purposes—, it can both simultaneously transmit and receive the signal to localize itself with a single RIS anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In this article, we argue that RIS technology in conjunction with sidelink communications can provide seamless L&S so- lutions, speeding up the intelligent transformation of cities into arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='03535v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='SP] 9 Jan 2023 2 Beacon-Assisted Localization Beacon Self-L&S with a Full-Duplex Transceiver Beacon-UE Channel RIS Anchor Channel RIS Localization Channel Sidelink Channel Sensing Channel Scenarios Cooperative Localization RIS A2 A1 B1 B2 C1 C2 C3 RIS-attached Vehicle AP A3 B3 Figure elements by macrovector on Freepik Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' RIS-enabled seamless L&S scenarios in smart cities: a) beacon-assisted localization, b) cooperative localization, and c) self-L&S with a full-duplex transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' smart entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' We particularly elaborate on the representative RIS-enabled L&S scenarios illustrated in Figure 1: beacon- assisted localization, cooperative localization, and self-L&S with a full-duplex transceiver, describing both AP-coordinated and AP-free architectures for different coverage scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' We discuss the open challenges with the proposed green (low- cost and energy-efficient) L&S systems in smart cities and list potential directions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' L&S ARCHITECTURES AND PROTOCOLS In this section, we describe the different entity types of the proposed L&S system for smart cities, relying on RISs and sidelink communications, as well as its enabling architectures, depending on whether an AP is present for L&S coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Entity Types and Overall Architecture In the proposed RIS-enabled L&S system, there are several types of entities: APs, beacons, RISs, and UEs, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' The APs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', a gNB macro base station) provide cellular services to the devices in coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' A beacon could be a roadside unit (RSU) that is capable of sending and receiving L&S reference signals via sidelink communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In this way, the explicit involvement of expensive and power-hungry APs with full communication protocols is not needed for L&S purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' RISs serve as reference anchors to assist ISLAC tasks, and are controlled by a dedicated entity responsible for realizing communication functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Finally, UEs may have different hardware capabilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', single/multiple antennas, half-/full-duplex) that play different roles in L&S (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', a target UE to be localized, an assistant UE with known or measured location to assist L&S, or a server/coordinator in performing L&S tasks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Although all the devices involved in the targeted tasks will have sidelink communication capabilities, beacons usually have fewer power constraints than UEs, while the controllers of reflective RISs are expected to work in low- power mode without sending or processing L&S reference signals [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' We further consider that RISs coordinate their reflective beamforming, either using time division or phase profile codes in the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' For the scenarios considered in this article, we propose two different architectures: one based on AP coordination and the other on self-coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' The former architecture works for UEs (and other L&S-related devices) inside a coverage area or in partial-covered areas (where sidelink is available), requiring a specific AP to serve as a L&S coordinator, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', interact with the location management function (LMF) via the NR positioning protocol A (NRPPa) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' The self-coordinated ar- chitecture relies explicitly on sidelink communications, being particularly suitable for UEs in the out-of-coverage of APs, or for UEs connected to APs that cannot meet the latency and spatial resolution requirements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', legacy 3G/4G APs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In both architectures, L&S signals can be generated by a beacon, an assistant UE, or a target UE, depending on the network topologies and the specific application scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' 3 Beacon-UE Channel Sidelink Channel RIS Channel AP-Coordinated Control Link Self-Coordinated Control Link L&S Coordinator Controller RIS-1 In-coverage Partial coverage Out-of-coverage AP Beacon UE-1 UE-2 UE-3 UE-4 UE-5 RIS-2 RIS-3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' UEs performing L&S in different coverage areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' The AP-coordinated architecture can be used for UEs located in in-coverage or partial-coverage (sidelink communications required) areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' When the UEs and all the L&S devices cannot access any surrounding APs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', lying in out-of-coverage areas), the self-coordinated architecture is the only option for L&S services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' AP-Coordinated Architecture This architecture relies on one or several APs to allocate the available radio resources and ensure timing among the connected devices, which are the UEs, the RIS controllers, and dedicated beacon nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' This network management scheme is similar to the mode-1 in sidelink communications [6], and the L&S protocols can be summarized into the following 6 steps: 1) The target UE triggers an L&S request to the AP (which is selected as the task coordinator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' 2) The coordinator exchanges related location information with the core network (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', via the NRPPa), determines L&S configurations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', broadcasting, sidelink, and RIS phase profiles), as well as selects and notifies nearby beacons, UEs, and RISs that are involved in the L&S task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' 3) The coordinator allocates time-frequency resources for L&S to all involved devices and triggers the RIS con- troller(s) to configure their phase profiles for the whole duration of the estimation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' 4) The beacons and/or UEs transmit L&S reference signals, which are reflected by the involved RIS(s) and received by the target UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' 5) The collected measurements are used by the target UE for the localization, and/or sensing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Alternatively, this computation can be offloaded at the localization server (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', the coordinating AP or another UE with high computational power).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' 6) The target UE updates the task coordinator with the estimated L&S results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' this optional step can serve as prior information for future use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' For the partial-coverage scenarios, the out-of-coverage UEs need to establish sidelink communications with devices that are covered by APs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Then, the L&S tasks can be performed similarly to the aforementioned steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Self-Coordinated Architecture An AP-free architecture is required for cases where the devices involved in L&S tasks are located in out-of-coverage areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Similar to the mode-2 sidelink communications [6], L&S tasks can be autonomously realized by selecting a specific device as the localization coordinator, as follows: 1) The target UE discovers nearby devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', beacons, RISs, and other UEs) and obtains their location infor- mation (if available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' 2) Based on the discovered neighbors, the target UE de- termines a L&S task coordinator (could be itself) and notifies it of the L&S configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' 3) The target UE triggers a L&S request to the coordi- nator and performs the same actions as with the AP- coordinated architecture (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', steps 2)–6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' RIS-ENABLED L&S SCENARIOS In this section, we will present three representative RIS- enabled L&S scenarios for smart city applications relying on the aforementioned architectures and protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' We mainly focus on localization applications and list potential position- ing scenarios in systems with minimum infrastructure and resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' For example, if positioning can be done with a single antenna, it can also be achieved using a multi-antenna array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' The same applies to narrowband (NB)/wideband (WB) signals, single/multiple anchors, and availability/unavailability of a line-of-sight (LOS) path between the UE and the active signal source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Beacon-Assisted Localization With low-complexity beacons, high flexibility in the in- stallation and deployment of L&S systems is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' A typical use case could be a train station with multiple low- cost beacons broadcasting L&S reference signals to UEs to navigate indoors, via the support of RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In this category, we consider UE localization (with the aid of one or more RISs) and RIS localization (with the aid of several beacons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' A1) Single-RIS-Enabled UE Localization: In a WB system where the LOS channel is available, the delays of the LOS and RIS paths can be estimated based on the L&S reference signals sent from a beacon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Assuming the RIS and beacon states are known, the angle-of-departure (AOD) at the RIS can also be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' The target UE can then be localized by the intersection of a hyperbola (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', time-difference-of- arrival (TDOA) of the LOS and RIS paths) and the line in the direction of the AOD at the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' When the target UE is equipped with an antenna array, 3D orientation can also be estimated based on the estimated angle-of-arrivals (AOAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' This is the basic RIS-enabled localization scenario that only requires a single low-complexity beacon [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' A2) Multi-RIS-Enabled UE Localization: If multiple RISs are simultaneously available, the requirements for LOS and WB are unnecessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' The AODs from different RISs can be estimated and used to localize the UE by intersecting the AOD lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In this way, localization tasks can be completed using NB signals, which saves bandwidth resources for communi- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Figure 3 shows the position error bound (PEB) of the target UE with different positions inside a 5 × 10 m2 area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' As shown, with multiple RISs, the UE is localizable even under blockage of the LOS path between the beacon and the UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' 4 5 4 3 2 1 0 1 2 3 4 5 x axis [m] 0 1 2 3 4 5 y axis [m] Beacon RIS Wall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='1 1 [m] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Scenarios A1 and A2: PEB (in meters) with different UE locations in a multi-RIS-aided localization scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' The target UE can be localized with one RIS and a beacon-UE LOS path, or with at least 2 RISs under LOS blockage conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' However, the localization tasks cannot be performed when only one anchor (beacon or RIS) is visible to the UE (see the yellow triangular area around the point [3, 3] m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' A3) RIS Localization via Multi-Static Sensing: In a scenario where passive UEs or objects are coated with RISs, the localization (or sensing, depending on scenarios) can be performed semi-passively with only a small amount of energy needed for localization coordination and RIS phase profile control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Such localization tasks can estimate the positions (and orientations) of RIS-coated objects by using several beacons with known positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Note that the geometrical constraints can largely reduce the difficulties in these scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' For example, the orientation of an RIS can be assumed as 1D (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', placed on the top of a car facing up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In addition, the adoption of antenna arrays at the beacons can further simplify the RIS localization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Cooperative Localization Sidelink communications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', device-to-device (D2D) communications, or the vehicular version known as vehicle- to-everything (V2X) communications) has been introduced in the millimeter-wave band for information exchange between vehicles, opening the road for numerous use cases, such as platooning, collision avoidance and autonomous driving [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' The combination of RISs and sidelink is expected to provide low-latency and high-reliability communications [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Interestingly, it can also be exploited to assist L&S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Sidelink communications enables UEs to participate in a cooperative manner in the sharing of position and surrounding information within a local network, and in performing relative location estimation using sidelink signals [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' We will focus on the latter case, where the absolute location can be estimated using RIS anchors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', with known position and orientation information), without any APs or beacons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' A typical scenario could be cooperative vehicular networks in urban areas with severe AP and GPS signal blockages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' We next discuss single- and multi-RIS-involved localization scenarios, where single-antenna UEs cooperate to estimate their positions via WB sidelink signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' We also consider a more general scenario where RIS-coated objects are involved, resulting in cooperative RIS localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' B1) Single-RIS-Enabled Cooperative Localization: Con- sider a scenario with several UEs and one RIS anchor, where the UEs wish to estimate their positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' We assume that each UE can send sidelink signals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', being the transmitter (TX)) to other UEs, which arrive at the receiver (RX) UEs via two paths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', the UE-UE and UE-RIS-UE paths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' By proper control of the RIS elements, those two paths can be separated, and thus, two delay measurements can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Due to the unknown states of the UEs, both the AOD and AOA at the RIS for every TX-RX pair of UEs are unknown and cannot be directly estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' However, we can estimate the spatial frequency information at the RIS for every TX-RX pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' This scenario requires at least three UEs to cooperate and render their locations feasibly without ambiguities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Figure 4 compares the PEBs for three UEs in an RIS-enabled versus beacon-aided (equipped with an antenna array) 3D cooperative localization scenario as a function of the number of RIS elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' B2) Multi-RIS-Enabled One-Way Sidelink Localization: In the scenario with at least two RISs, one-way sidelink communications is sufficient to localize both the TX and RX UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Assume that one UE takes the role of the TX and the other UE is the RX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' With two RISs, three delay measurements can be obtained between the TX and RX via the LOS and the two RIS paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' However, that would require an optimal joint design of the reflection elements at both RISs to be able to separate the paths at the RX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In addition, once the RIS paths are separated, we can also estimate the spatial frequency information at each RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Thus, those collected measurements can be utilized to estimate the locations of the TX and RX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' B3) Cooperative RIS Localization: Let us consider a more general scenario where one RIS (or several) with a known state is used to localize multiple UEs (with sidelink capabilities) and objects (coated with an RIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' This scenario is challenging due to the high complexity of the network and a large number of unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' However, with a proper design of all the involved RIS profiles and the transmission protocol, this problem can be decomposed into a cooperative localization (see B1) and an RIS localization (see A3) problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Similar to B1, at least several UEs (depending on scenarios) need to take the role of the TX, and transmit sidelink signals to the other UEs via the direct and indirect paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Once the UEs are localized, the RIS localization task can be solved similarly to A3, and the estimation results can be refined by processing all the available information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Self-L&S with a Full-Duplex Transceiver When a UE is equipped with a full-duplex transceiver (like radar) [8], the multi-RIS setup and cooperation between UEs are unnecessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Instead, this UE can perform self-positioning with a single RIS and use the multipath components to map the environment over time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' this process is known as monostatic simultaneous localization and mapping (SLAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' It is noted that SLAM is not limited to full-duplex UEs, and bistatic SLAM can also be performed in use cases A1-A3 and B1- B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' We next present three beacon-free L&S scenarios with a full-duplex UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' 5 0 20 40 60 80 100 120 140 160 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='5 UE-1 (RIS) UE-1 (beacon) UE-2 (RIS) UE-2 (beacon) UE-3 (RIS) UE-3 (beacon) Number of RIS Elements PEB [m] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' PEBs of RIS-enabled vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' beacon-aided cooperative localization for different RIS sizes and a fixed random phase profile setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' It is shown that, with a sufficient number of RIS elements, an active anchor can be replaced with a passive one (RIS) without performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' C1) RIS-Enabled Self-Localization: Consider a system with a single-antenna UE and an RIS, where the UE transmits L&S reference signals and receives their back-scattered ver- sions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', the UE-RIS-UE (controlled path) and UE-landmark- UE (uncontrolled path) signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' One option for the RIS phase profiles is to consider directional reflective beams, which can be efficiently designed when the UE position uncertainty (even under mobility cases) is available [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' The delay and angle information at the RIS of the UE-RIS-UE channel can be estimated for this scenario, and then used to localize the UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In Figure 5, beampatterns at the RIS with two different phase pro- files are illustrated, focusing on the UE uncertainty region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' As demonstrated, the optimized phase profiles of [13] can provide sufficient beamforming gain, compared with directional phase profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' This gain can offer improved L&S performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' C2) RIS-Enabled SLAM: If a UE is equipped with a full-duplex multiple-input multiple-output (MIMO) antenna array [8], SLAM can be enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Similar to scenario C1, the signals from different paths can be resolvable with optimized RIS phase profiles and precoders/combiners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' The following channel parameters can be estimated: i) the signal propagation delay, the AOD at the RIS, and the AOA at the UE for the UE-RIS-UE channel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' as well as ii) the delay and AOA at the UE for each UE-landmark-UE channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In addition to the position information obtained from the controlled path (as in scenario C1), the angular resolution offered by the UE array can be leveraged to map/sense the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' With multiple estimations, the localization and radio mapping performance can be improved using state-of-the-art filters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', Poisson multi-Bernoulli filter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' C3) RIS Localization with a Full-Duplex Array: Consider the more general scenario from C1 including one anchor RIS mounted on a wall, a UE equipped with a full-duplex MIMO transceiver, and several objects coated with RISs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', mounted on the front and rear side of a vehicle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In addition to the signals reflected from the anchor RIS (as also in scenario C1), the UE also receives single-bounce reflected signals from the RISs mounted on the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' The time delay, AOA, and the amplitude of the channel gain for each signal path can be estimated, which can be used for the localization of both itself and the RISs-coated objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' When multiple UEs are present and cooperate in the estimation process, the orientation of the (a) Directional Phase Profiles (b) Optimized Phase Profiles Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' The reflective beamforming gain (in dB) with a 50 × 50 RIS using (a) a directional phase profile, and (b) an optimized phase profile via [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' The red squares represent the UE angular uncertainty region, which needs to be fully covered by an effective beampattern design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' RISs-coated objects can also be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In a scenario without any anchors, this RIS localization can also help in estimating the relative locations of the active UE and passive UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' OPEN RESEARCH CHALLENGES With the assistance of low-complexity beacons, cooperative localization, and full-duplex radios, the L&S coverage for smart city applications can be significantly extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' However, there exist several practical issues that need to be thoroughly investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In this section, we discuss the most critical challenges with the proposed RIS-enabled L&S system and list possible directions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Anchor Deployment Optimization The placement of the anchors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', beacons and RISs) is critical to meet the L&S key performance indicator (KPI) re- quirements within a service area (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', error bounds lower than a certain threshold, as shown in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' The deployment involves both the position and orientation optimization of the anchors, taking into account the blockage in the surrounding environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' RIS-aided SLAM can help in creating such an environment map, which can be supported by cooperative sidelink UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Heuristic optimization solutions can then be applied for finding optimal anchor sites, extending approaches from the literature [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='100100150evation20 Gro40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='M06080J60 azllBeaiot0 muthele100100150evation20 G[o]40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='=060 B80ncertain60 azilReaiol Vmuth6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Resource Allocation and Coordination RIS-aided L&S systems involve APs, beacons, RISs, and UEs, making them inherently heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Resource al- location for L&S tasks, including power allocation, time- frequency allocation, beamforming design, and scheduling must be carefully designed to ensure a favorable trade-off with conventional communication services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Depending on the KPI requirements of the applications that send L&S ser- vice requests, new objectives that consider integrated L&S and communications should be formulated and satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' An important part of resource allocation is RIS phase profile optimization and multiplexing [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Broad RIS beams lead to coverage reduction, while narrow pencil beams are sensitive to misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Hence, highly adaptive RIS profile designs are needed, relying, when possible, on prior UE and object state information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' When RISs are large, the near-field effects need to be taken into consideration and the beamforming designs should account for the curvature of arrival, resulting in beam- focusing designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' RIS multiplexing can be addressed by time multiplexing, temporal coding, and making use of high path loss for spatial reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' The afore-described resource allocation problems can be tackled by a combination of traditional optimization-based methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', convex optimization) and learning-based methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', reinforcement learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Estimation Algorithms From an algorithmic perspective, there are challenges re- lated to channel parameter estimation, tracking in dynamic environments, and calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Channel parameter estimation in the presence of severe multipath is difficult since almost passive reflective RISs have no local signal processing capa- bilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Moreover, in cooperative localization (scenarios B1 and B2) and RIS localization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', scenarios A3, B3, and C3) tasks, the AOAs/AODs at the RISs are coupled, meaning that we can no longer estimate them independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Instead, only spatial frequencies (containing coupled AOAs and AODs information) can be obtained, requiring novel algorithms for further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' More refined channel parameter estimation also requires accurate channel models and the RISs’ impact on them, such as the near-field effect, beam squint effect, RIS element failures, and anchor calibration errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Due to mobility, difficult conditions such as signal blockage, unresolvable signal paths, and severe path loss will affect L&S performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Multiple RISs can be involved to handle such blockages, offering coverage extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' As the carrier fre- quency increases, the signal resolutions become higher, and the effect of the severe path loss can be mitigated by adopting RISs with larger sizes or active RISs with reflection amplification capabilities [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Sensing also suffers from inherent complica- tions, such as an unknown number of objects, unknown types of objects, unknown detection probabilities for signal paths, extended objects, and multi-bounce observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Dedicated filters should be developed to address these complications and get integrated into the L&S framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Finally, in terms of calibration, anchor geometry error and hardware impairments (HWIs) are two important aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' We note that the geometrical calibration of an RIS is similar to RIS localization (as described in scenarios A3, B3, and C3), which requires a calibration agent that incorporates other sources of localization estimations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', sensor fusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' While for HWIs, the channel model could be too complicated when considering each specific impairment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', mutual coupling and phase noise), learning-based methods can be considered to unveil their impact and drive practical algorithmic designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Understanding Anchor Hardware Alternatives There are also opportunities to improve L&S coverage via variations of the hardware deployed at the beacons, RISs, and UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' On the beacon and UE sides, multi-panel arrays (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', 3D arrays) could be implemented for further coverage extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' On the RIS side, new types of RISs are emerging beyond almost passive reflective RISs [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' As previously mentioned, an active RIS can be used to boost the signal energy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', change both the amplitude and phase of the incident signal) for improved coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' A receiving RIS (also known as a hybrid RIS or a simultaneously reflecting and sensing RIS) can enable parameter estimation at the RIS side, offering extra degrees of freedom for the design of L&S estimation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Omni-directional RISs, intended to realize simultaneous reflec- tion and refraction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', 360◦ coverage), enable simultaneous indoor and outdoor 3D localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' A non-reciprocal RIS that integrates nonreciprocal phase shifters allows full-duplex communications, and a delay-adjustable RIS is capable of adjusting the delays of signals reflected by different RIS elements, which contributes to the alleviation of the beam squint effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' All of these alternatives have implications on L&S services and merit further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Privacy, Security, and Social Acceptance Issues Cooperative L&S require extensive information exchange of local measurements between devices, which may cause privacy issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' For example, the RX in scenario B2 can estimate the position of the TX with a one-way pilot signal transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In addition, different types of cyber attacks can reduce the L&S service availability, or even provide an undetected erroneous location estimation, which is unacceptable for safety-critical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Currently, several security management systems have been standardized (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=', IEEE 1609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='2), and security threats have been identified for sidelink communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' However, the discussions on L&S task-related security issues are still at the initial stage, and potential threats need to be explored and eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' A final aspect related to the widespread adoption of RISs lies in their social acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' RISs should be integrated in a way that they blend into the environment (ideally, be transparent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' To this end, the benefits of RISs to improve safety and reduce electromagnetic emissions should be demonstrated and disseminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' CONCLUSION AND OUTLOOK The smart city paradigm constitutes the epitome of the widespread adoption of digital services for societal needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' It is envisioned to profit people and city-level businesses, offering efficient, safe, and comfortable living spaces as well 7 as everyday-life smart-living applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' To achieve this overarching goal, seamless wireless communications among diverse devices and L&S are of paramount importance, en- abling information exchange, device localization, and mapping of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In this article, we discussed the key to achieving low-cost and energy-efficient seamless L&S, namely, reflective RISs in conjunction with sidelink commu- nications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' We presented AP-coordinated and AP-free system architectures and detailed three RIS-enabled L&S scenarios, each including several use cases and most relying on sidelink communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' As became apparent, instead of using APs with full communication capabilities, low-complexity beacons and RISs can be widely-deployed to enable green L&S smart city applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' In addition, when multiple UEs with sidelink communication capabilities can be connected in the same network, cooperative localization can relieve the requirement for multiple anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Furthermore, when devices are equipped with full-duplex transceivers, they can localize themself and map their surrounding environment with only a single RIS anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Finally, an extended list of open research challenges relevant to the proposed RIS-enabled seamless L&S concept was presented, including the necessity for anchor deployment optimization and optimized resource allocation schemes, al- gorithmic and privacy issues, as well as the role of multi- functional RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported, in part, by the European Com- mission through the EU H2020 RISE-6G project under grant 101017011, and 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' 1–32, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Hui Chen (hui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='chen@chalmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='se) is a postdoctoral researcher at Chalmers University of Technology, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Hyowon Kim (hyowon@chalmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='se) is a postdoctoral researcher at Chalmers University of Technology, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Mustafa Ammous (mustafa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='ammous@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='utoronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='ca) is a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' student at University of Toronto, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Gonzalo Seco-Granados (gonzalo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='seco@uab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='cat) is a professor at Universitat Autonoma of Barcelona, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' George C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Alexandropoulos (alexandg@di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='uoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='gr) is an assistant professor at the Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, Greece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Shahrokh Valaee (valaee@ece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='utoronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='ca) is a professor at University of Toronto, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content=' Henk Wymeersch (henkw@chalmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} +page_content='se) is a professor at Chalmers Univer- sity of Technology, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'} diff --git a/8tE1T4oBgHgl3EQfCAKJ/content/tmp_files/2301.02859v1.pdf.txt b/8tE1T4oBgHgl3EQfCAKJ/content/tmp_files/2301.02859v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..52e15ca2f1eae3d33a0a96c42ca4050e541d122c --- /dev/null +++ b/8tE1T4oBgHgl3EQfCAKJ/content/tmp_files/2301.02859v1.pdf.txt @@ -0,0 +1,1175 @@ +D-Optimal and Nearly D-Optimal Exact Designs for +Binary Response on the Ball +Martin Radloff† and Rainer Schwabe‡ +Abstract: In this paper the results of Radloff and Schwabe (2019a) will be +extended for a special class of symmetrical intensity functions. This includes +binary response models with logit and probit link. To evaluate the position +and the weights of the two non-degenerated orbits on the k-dimensional ball +usually a system of three equations has to be solved. The symmetry allows +to reduce this system to a single equation. As a further result, the number of +support points can be reduced to the minimal number. These minimally sup- +ported designs are highly efficient. The results can be generalized to arbitrary +ellipsoidal design regions. +Key words and phrases: Binary response models, D-optimality, k-dimensional +ball, logit and probit model, multiple regression models, simplex. +1. Introduction +Spherical design spaces can occur in engineering or physics problems where the validity of +a model may be assumed on a spherical region around a target value. So (linear) models +on spherical design spaces were investigated early in publications like Kiefer (1961) and +Farrell et al (1967) which discuss polynomial regression on the ball. These ideas were +followed up by papers in which also only linear problems were focused. So Lau (1988) +fitted polynomials on the k-dimensional unit ball by using canonical moments. In Dette +et al (2005, 2007) and Hirao et al (2015) harmonic polynomials and Zernike polynomials +were used to be fit on the unit disc (2-dimensional unit ball), the 3- and k-dimensional +unit ball. On the other hand generalized linear models are also well-examined and used +in practical application. Logit and probit models, for example, in one dimension on an +interval have already been investigated by Ford et al (1992) and Biedermann et al (2006). +But there seems to be no available literature which combines both topics. +In our publication Radloff and Schwabe (2019b) we took the first step to bring non- +linearity or generalized linear models, respectively, and spherical design regions together. +These results were extended to a wider class of non-linear models in our follow-up paper +Radloff and Schwabe (2019a). +†corresponding author: Martin Radloff, Institute for Mathematical Stochastics, Otto-von-Guericke- +University, PF 4120, 39016 Magdeburg, Germany, martin.radloff@ovgu.de +‡Rainer Schwabe, Institute for Mathematical Stochastics, Otto-von-Guericke-University, PF 4120, +39016 Magdeburg, Germany, rainer.schwabe@ovgu.de +arXiv:2301.02859v1 [stat.ME] 7 Jan 2023 + +Martin Radloff, Rainer Schwabe +Exact Designs on the Ball +For better comprehensibility, we will start with the model description and give a brief +overview of the findings so far. Then we will consider a special class of intensity func- +tions which allows to reduce the the complexity of finding (locally) D-optimal designs. +Afterwards we will tackle the problem, that the optimal designs are not exact designs in +general, by establishing highly efficient designs on the ball. +2. General Model Description +As in Radloff and Schwabe (2019b) and Radloff and Schwabe (2019a), where we described +(locally) D-optimal designs for two special classes of linear and non-linear models on a +k-dimensional unit ball Bk = {x ∈ Rk : x2 +1 + . . . + x2 +k ≤ 1} with k ∈ N, we solely focus +(non-linear) multiple regression models, which means the linear predictor is +f(x)⊤β = β0 + β1x1 + . . . + βkxk +with regression function f : Bk → Rk+1, x �→ (1, x1, . . . , xk)⊤, and parameter vector +β = (β0, β1, . . . , βk)⊤ ∈ Rk+1. The one-support-point (or elemental) information matrix +should be representable in the form +M(x, β) = λ +� +f(x)⊤β +� +f(x)f(x)⊤ +with an intensity (or efficiency) function λ which only depends on the value of the linear +predictor f(x)⊤β. These one-support-point (or elemental) information matrices are the +base for the information matrix of a (generalized) design ξ with independent observations +M(ξ, β) = +� +M(x, β) ξ(dx) = +� +λ +� +f(x)⊤β +� +f(x)f(x)⊤ξ(dx) . +Here generalized design means an arbitrary probability measure on the design region Bk. +These information matrices allow to define the (local) D-optimality, which is one of the +most popular criteria in experimental design theory. A design ξ∗ +β0 with regular infor- +mation matrix M(ξ∗ +β0, β0) is called (locally) D-optimal (at β0) if det(M(ξ∗ +β0, β0)) ≥ +det(M(ξ, β0)) holds for all suitable probability measures ξ on the design space — here +Bk. This optimality criterion can be interpreted as the minimization of the volume of the +(asymptotic) confidence ellipsoid. +3. Prior Results +In Radloff and Schwabe (2016) we stated results on equivariance and invariance. +By +rotating the design space Bk — the k-dimensional unit ball — and the parameter space +Rk+1 in an analogous way the linear predictor of the multiple regression problem reduces +to +f(x)⊤β = β0 + β1x1 +and +β1 ≥ 0 . +(3.1) +Using the rotation invariance with fixed x1, this means the invariance to all orthogonal +transformations in O(k) which let the x1-component unchanged, the (locally) D-optimal +(generalized) design ξ∗ can be decomposed (ξ∗ = ξ∗ +1 ⊗ η) in a marginal probability measure +ξ∗ +1 on [−1, 1] for x1 and a probability kernel η given x1. For fixed x1 the kernel η(x1, ·) is +2 + +Martin Radloff, Rainer Schwabe +Exact Designs on the Ball +the uniform distribution on the surface of a (k − 1)-dimensional ball with radius +� +1 − x2 +1 +— the orbit at position x1. +As a consequence the multidimensional problem collapses to a one-dimensional marginal +problem. Only the positions of the orbits and their weights have to be determined. To get +an exact design the uniform orbits have to be discretized, for example, by using regular +simplices. +In our first paper — Radloff and Schwabe (2019b) — we started with models where the +intensity function belongs to the class of monotonous functions. Such models have already +been investigated in one dimension, for example, by Konstantinou et al (2014) and on +multidimensional cuboids or orthants by Schmidt and Schwabe (2017). These authors +gave the following four conditions on the intensity function λ: +(A1) λ is positive on R and twice continuously differentiable. +(A2) The first derivative λ′ is positive on R. +(A3) The second derivative u′′ of u = 1 +λ is injective on R. +(A4) The function λ′ +λ is non-increasing. +Condition (A2) is the motivation for the name class of monotonous intensity functions. +The intensity functions of this class have to satisfy always (A1) to (A3). (A4) is an extra +condition to guarantee uniqueness. For a concise notation +q(x1) = λ(β0 + β1x1) +is used and the properties (A1), (A2), (A3) and (A4) transfer to q for β1 > 0, respectively, +and vice versa. Poisson regression with intensity function qP(x1) = exp(β0 + β1x1) and +negative binomial regression as well as special proportional hazard models with censoring, +see Schmidt and Schwabe (2017), satisfy all four conditions. +If β1 = 0 then the intensity function q is always a constant. This yields to a (locally) +D-optimal design as it can be found in linear models. In Pukelsheim (1993, section 15.12) +such a design consists of the equally weighted vertices of a regular simplex inscribed in the +unit sphere, the boundary of the design space. The orientation of the simplex is arbitrary. +The main result for β1 > 0 in Radloff and Schwabe (2019b) is recited for the readers’ +convenience. +Theorem 1. There is a (locally) D-optimal design for the multiple regression prob- +lem (3.1) with β1 > 0 and intensity function satisfying (A1)-(A3) which has one support +point equal to (1, 0, . . . , 0)⊤ and the other k support points are the vertices of an arbi- +trarily rotated (k − 1)-dimensional regular simplex which is maximally inscribed in the +intersection of the k-dimensional unit ball and a hyperplane with x1 = x∗ +12. +• For k ≥ 2 the position x∗ +12 ∈ (−1, 1) is solution of +q′(x∗ +12) +q(x∗ +12) = 2 (1 + kx∗ +12) +k (1 − x∗ 2 +12 ) . +If additionally (A4) is satisfied, the solution x∗ +12 is unique. +3 + +Martin Radloff, Rainer Schwabe +Exact Designs on the Ball +• For k = 1 the position x∗ +12 ∈ [−1, 1) is either solution of +q′(x∗ +12) +q(x∗ +12) = +2 +1 − x∗ +12 +, +if such a solution exists in [−1, 1), or otherwise x∗ +12 = −1. +If additionally (A4) is satisfied, the solution x∗ +12 is unique. +The design is equally weighted with +1 +k+1. +It should be noted, that for fixed β this theorem does not need (A1) to (A4) on the +entire real line R. It is enough to have it in the ball and so on x1 ∈ [−1, 1] for q and on +[β0 − β1, β0 + β1] for λ, respectively. But the model has to satisfy the conditions always +on the whole real line. +In our second paper — Radloff and Schwabe (2019a) — the conditions (A2) and (A3) +were replaced by (A2′) and (A3′) and a fifth property (A5) was added. +(A2′) λ is unimodal with mode c(A2′) +λ +∈ R. +(A3′) There exists a threshold c(A3′) +λ +∈ R so that the second derivative u′′ of u = 1 +λ is +both injective on (−∞, c(A3′) +λ +] and injective on [c(A3′) +λ +, ∞). +(A5) u = 1 +λ dominates z2 asymptotically for z → ∞. +In this context condition (A2′) means that there exists a c(A2′) +λ +∈ R so that λ′ is positive +on (−∞, c(A2′) +λ +) and negative on (c(A2′) +λ +, ∞). +Hence, there is only one local maximum +which is simultaneously the global maximum. So the class of intensity functions, which +satisfy (A1), (A2′) and (A3′), is called class of unimodal intensity functions. +Indeed (A2) or (A3) do not imply (A2′) or (A3′), respectively. As mentioned before, we +only focus on the unit ball and the interval x1 ∈ [−1, 1] for q or [β0 − β1, β0 + β1] for λ. +So in our special case (A2) and (A3) can be transferred to (A2′) and (A3′) by using an +arbitrary cλ > β0 + β1, which means that cq lies outside the interval [−1, 1] and only one +branch of the function is considered. +Property (A5) means +lim +z→∞ +���� +u(z) +z2 +���� = ∞ . +This means that u(z) = +1 +λ(z) goes faster to (±) infinity than z2 for z → ∞. +As (A1) to (A4) the conditions (A2′), (A3′) and (A5) transfer from the intensity function +λ to the abbreviated form q for β1 > 0 and vice versa — analogously c(·) +q += +c(·) +λ −β0 +β1 +with +(·) is (A2′), (A3′) or empty. +The logit model has the intensity function +qlogit(x1) = +exp(β0 + β1x1) +(1 + exp(β0 + β1x1))2 +and probit model has +qprobit(x1) = +φ2(β0 + β1x1) +Φ(β0 + β1x1) · (1 − Φ(β0 + β1x1)) +4 + +Martin Radloff, Rainer Schwabe +Exact Designs on the Ball +with the density function φ and cumulative distribution function Φ of the standard normal +distribution. Both models satisfy all five conditions (A1), (A2′), (A3′), (A4), (A5) and +share a common c(A2′) +λ += c(A3′) +λ += 0, say cλ = 0. Analogously cq = − β0 +β1 for q. +Beside these two models other models like the complementary log-log model, see Ford +et al (1992), with intensity function λcomp log log(z) = +exp(2z) +exp(exp(z))−1 satisfy all five conditions +with c(A2′) +λ +≈ 0.466011 and c(A3′) +λ +≈ 0.049084, but here mode c(A2′) +λ +and threshold c(A3′) +λ +do +not coincide. +We showed that if the (concise) intensity function q satisfies (A1), (A2′), (A3′) and (A5) +the (locally) D-optimal design ξ∗ = ξ∗ +1 ⊗η is concentrated on exactly two orbits, which are +the support points of the marginal design ξ∗ +1. The idea of the proof is based on Biedermann +et al (2006) and Konstantinou et al (2014). +The next theorem is the main result of our second paper — Radloff and Schwabe (2019a) +— and is reproduced for the readers’ convenience. It characterizes the positions of the +two support points of the optimal marginal design ξ∗ +1. +Theorem 2. For k ≥ 2 the simplified problem (3.1) with β1 > 0 and intensity function q +satisfying (A1), (A2′), (A3′) and (A5) has a (locally) D-optimal marginal design ξ∗ +1 with +exactly 2 support points x∗ +11 and x∗ +12 with x∗ +11 > x∗ +12 and weights w1 = ξ∗ +1(x∗ +11) and w2 = +ξ∗ +1(x∗ +12). +There are 3 cases: +(a) If c(A2′) +q +> 1 and c(A3′) +q +/∈ [−1, 1], then x∗ +11 = 1, w1 = +1 +k+1, w2 = +k +k+1 and x∗ +12 ∈ (−1, 1) +is solution of +q′(x∗ +12) +q(x∗ +12) = 2 (1 + kx∗ +12) +k (1 − x∗ 2 +12 ) . +(3.2) +If additionally (A4) is satisfied, the solution x∗ +12 is unique. +(b) If c(A2′) +q +< −1 and c(A3′) +q +/∈ [−1, 1], then x∗ +12 = −1, w1 = +k +k+1, w2 = +1 +k+1 and +x∗ +11 ∈ (−1, 1) is solution of +q′(x∗ +11) +q(x∗ +11) = 2 (−1 + kx∗ +11) +k (1 − x∗ 2 +11 ) +. +(3.3) +If additionally (A4) is satisfied, the solution x∗ +11 is unique. +(c) Otherwise c(A2′) +q +∈ [−1, 1] or c(A3′) +q +∈ [−1, 1]. +Let x, y ∈ R with x > y and α ∈ +� +− 1 +2, 1 +2 +� +be solution of the equation system: +q′(x) +q(x) + +2 +x−y + (k−1) q′(x) (1−x2) ( 1 +2 −α) + q(x) (−2 x) ( 1 +2 −α) +q(x) (1−x2) ( 1 +2 −α) + q(y) (1−y2) ( 1 +2 +α) = 0 +(3.4) +q′(y) +q(y) − +2 +x−y + (k−1) q′(y) (1−y2) ( 1 +2 +α) + q(y) (−2 y) ( 1 +2 +α) +q(x) (1−x2) ( 1 +2 −α) + q(y) (1−y2) ( 1 +2 +α) = 0 +(3.5) +1 +1 +2 −α − +1 +1 +2 +α + (k−1) +q(x) (1−x2) − q(y) (1−y2) +q(x) (1−x2) ( 1 +2 −α) + q(y) (1−y2) ( 1 +2 +α) = 0 +(3.6) +5 + +Martin Radloff, Rainer Schwabe +Exact Designs on the Ball +Figure 1: Logit model for k = 3 and β1 = 1: Dependence of x∗ +11 and x∗ +12 (solid lines) and +the corresponding weights w1 and w2 = 1 − w1 (dashed lines) on −β0 = − β0 +β1 = +cq ∈ [−1.2, 1.2]. +(c0) If x, y ∈ (−1, 1) with x > y and α ∈ (− 1 +2, 1 +2) is a solution of the equation +system, the orbit positions are x∗ +11 = x, x∗ +12 = y with weights w1 = +1 +2 − α +and w2 = 1 +2 + α. +(c1) If x +≥ +1 and y +∈ +(−1, 1), then x∗ +11 += +1, w1 += +1 +k+1, w2 += +k +k+1 +and x∗ +12 ∈ (−1, 1) is the solution of the equation (3.2). +(c2) If y ≤ −1 and x ∈ (−1, 1), then x∗ +12 = −1, w1 = +k +k+1, w2 = +1 +k+1 +and x∗ +11 ∈ (−1, 1) is the solution of the equation (3.3). +Remark 1. Instead of reproducing the whole theorem for k = 1, only the two main +changes in case (c) should be mentioned. So the weights are always w1 = w2 = 1 +2 and the +equation system (3.4)–(3.6) is replaced by +q′(x) +q(x) + +2 +x − y = 0 +and +q′(y) +q(y) − +2 +x − y = 0 . +(3.7) +To illustrate this complex issue we revisit the logit model in dimension k = 3 with β1 = 1. +We (numerically) plot the orbit positions x∗ +11 and x∗ +12 and corresponding weights w1 and +w2 = 1 − w1 depending on −β0 = − β0 +β1 = cq, see Figure 1. The cases (a) and (b) go along +with Theorem 1 and the results from Radloff and Schwabe (2019b). The cases (c1) and +(c2) yield marginal extremum solutions which are identical to (a) and (b). So for these +four cases there is always an exact minimally supported (locally) D-optimal design. As +described in Theorem 1, it consists of a pole point in x1 = −1 or else x1 = 1 and the k +vertices of a (regular) simplex which is maximally inscribed in the non-degenerated orbit. +But the problematic case is (c0) because the (locally) D-optimal (generalized) design +consists of two non-degenerated orbits and additionally the weights are rarely appropriate +for a discretization. In Radloff and Schwabe (2019a) we showed two examples for the logit +model (k = 3, β1 = 1) from which we derived (nearly) exact designs. +For −β0 = 0 the two orbit positions are symmetrical around 0, that is x∗ +11 = −x∗ +12 ≈ 0.52, +and the weights are ξ∗ +1(x∗ +11) = ξ∗ +1(x∗ +12) = +1 +2. These two orbits were discretized by two +6 + +Martin Radloff, Rainer Schwabe +Exact Designs on the Ball +2-dimensional simplices — overall 6 equally weighted support points; see Figure 2 (left +image). +For −β0 = −0.1 it is x∗ +11 ≈ 0.42, x∗ +12 ≈ −0.62 and ξ∗ +1(x∗ +11) ≈ 0.4297, while 0.4297 ≈ 3 +7. +We took the rounded design ξ≈ with the same support points x∗ +11 and x∗ +12 but with the +marginal design ξ≈ +1 (x∗ +11) = 3 +7 and ξ≈ +1 (x∗ +12) = 4 +7. So it was possible to substitute one orbit +by the vertices of a 2-dimensional simplex (3 points — an equilateral triangle) and one +by the vertices of a 2-dimensional cube or cross polytope (4 points — a square). Because +of rounding the design ξ≈ is not optimal but exact and has a high D-efficiency, which +compares the rounded design ξ≈ and the optimal design ξ∗ +β0 with respect to β0 — here +p = k + 1 = 4 and β0 = (0.1, 1, 0, 0)⊤: +EffD(ξ≈, β0) = +� +det(M(ξ≈, β0)) +det(M(ξ∗ +β0, β0)) +�1 +p +≈ 0.999757 . +These designs are not very satisfactory. On the one hand the number of support points +is not minimal. On the other hand only special cases have appropriate rational weights +which allow a discretization or otherwise the optimality is lost by rounding. Therefore we +want to establish minimal supported exact designs for the case (c0) in this paper. Mostly +these designs wont be optimal but (highly) efficient. +But we start with the reduction of the system of three equations in Theorem 2 to only one +single equation for special unimodal intensity functions — symmetrical unimodal intensity +functions — which can be found, for example, in binary response models with logit and +probit link. +4. Optimal Design for Symmetrical Unimodal +Intensity Functions +An interesting observation was made in the discussion section in Radloff and Schwabe +(2019a). For models with unimodal intensity function in which the mode and threshold +coincide (c(A2′) +λ += c(A3′) +λ += cλ) and which are symmetrical, also the two orbit positions are +symmetrical in a certain way, which we want to investigate here. For one dimension this +has been considered and shown in Ford et al (1992, Section 6.5 and 6.6), but this proof +cannot be extended to higher dimensions directly. +Definition 1. An unimodal intensity function in which the mode and threshold coincide +(c(A2′) +λ += c(A3′) +λ += cλ) will be called symmetrical to cλ if +λ(cλ + z) = λ(cλ − z) +for all z ∈ R. +The intensity functions of the logit and probit models are symmetrical with cλ = 0. But +the unimodal intensity function of the complementary log-log model has c(A2′) +λ +̸= c(A3′) +λ +and cannot be symmetrical for this reason. +Lemma 1. Let the intensity function λ be symmetrical to cλ in the situation of Theo- +rem 2 (c0). +7 + +Martin Radloff, Rainer Schwabe +Exact Designs on the Ball +• For given β0 ̸= cλ let r solve +λ′(cλ+r) +λ(cλ+r) = − +−2 k r2 (β2 +1 +c2−r2)+(β2 +1 −c2−r2)2−4 c2 r2 ++(β2 +1 −c2+r2) +� +(β2 +1 −c2−r2)2+4 (k2−1) c2 r2 +(k+1) r (r+c−β1)(r+c+β1)(r−c+β1)(r−c−β1) +(4.8) +with c := cλ − β0. Then +x = c +β1 ++ r +β1 +, +(4.9) +y = c +β1 +− r +β1 +, +(4.10) +α = +−(β2 +1 −c2−r2)+ +� +(β2 +1 −c2−r2)2+4 (k2−1) c2 r2 +4 (k+1) c r +(4.11) +is a solution of the equation system (3.4)–(3.6). +• For given β0 = cλ it is x = +r +β1, y = − r +β1 and α = 0. Here r is the solution of +λ′(cλ + r) +λ(cλ + r) = − +2 (β2 +1 − k r2) +(k + 1) r (β2 +1 − r2) . +(4.12) +Remark 2. For k = 1, see Remark 1, let λ be symmetrical to cλ. Then x = cλ−β0 +β1 ++ +r +β1 +and y = cλ−β0 +β1 +− r +β1 with r is solution of +λ′(cλ + r) +λ(cλ + r) = −1 +r +(4.13) +solve the equation system (3.7). +Lemma 1, whose proof sketch can be found in Appendix B, and Remark 2 in combination +with Theorem 2 give (locally) D-optimal designs for models with symmetrical unimodal +intensity functions. As a result we reduced the system of equations (3.4)–(3.6) to only +one single equation (4.8). +But now there is the question if condition (A4) can guarantee a unique solution as in +Theorem 1 or in Theorem 2 (a) and (b) because Theorem 2 (c), especially (c0), tells +nothing about uniqueness. But we want to add a remark about the values of r before. +Remark 3. Since the system of equations (3.4)–(3.6) in Theorem 2 (c0) should have a +solution with two inner support points for the marginal design, x, y ∈ (−1, 1) is required. +So +−1 < cλ − β0 +β1 +± r +β1 +< 1 +must be valid. This leads with β1 > 0 to r ∈ (−(cλ − β0) − β1, −(cλ − β0) + β1) and r ∈ +((cλ − β0) − β1, (cλ − β0) + β1). Consequently, both intervals must overlap. This happens +for cλ − β0 > 0 at 0 < cλ − β0 < β1 and for cλ − β0 < 0 at −β1 < cλ − β0 < 0. +Thus cλ − β0 ∈ (−β1, β1) and in particular β2 +1 > (cλ − β0)2 must hold. Then r is in the +interval (|cλ − β0| − β1, −|cλ − β0| + β1). But Theorem 2 (c) need x > y and consequently +r > 0. Hence, r ∈ (0, −|cλ − β0| + β1). +This remains valid in particular for β0 = cλ, i. e. cλ − β0 = 0. So r ∈ (−β1, β1). With +r > 0 it is r ∈ (0, β1). +8 + +Martin Radloff, Rainer Schwabe +Exact Designs on the Ball +Lemma 2. In situation of Lemma 1 let the intensity function λ additionally satisfy +condition (A4), then equation (4.8), whose right hand side is continuously continued +in −|cλ − β0| + β1, has a unique solution in r ∈ (0, |cλ − β0| + β1). +This also holds for β0 = cλ and equation (4.12), which has exactly one solution in r ∈ +(0, β1). +Remark 4. For k = 1, see Remark 2, and for an intensity function satisfying (A4) there +is only one solution of (4.13). +The proof sketch of Lemma 2 can be found in Appendix B. Lemma 2 guarantees a unique +solution in r ∈ (0, |cλ − β0| + β1). But Remark 3 points out that for Theorem 2 (c0) +we need r ∈ (0, −|cλ − β0| + β1). This means that the unique solution can result in the +two-orbit case or in the one-orbit one-pole case of Theorem 2 (c). +5. Minimally Supported Designs +In the situation of Theorem 1 and Theorem 2 (a), (b), (c1) and (c2) the designs have +always the minimal number of support points to estimate the parameter vector β. These +are k + 1 support points. +In Radloff and Schwabe (2019a) revisited here in the introductory section we indicated +exemplarily a (locally) D-optimal design for the logit model on the 3-dimensional ball +with −β0 = 0 and β1 = 1. +This design consists of six support points which are the +vertices of two regular 2-dimensional simplices — equilateral triangles; see Figure 2 (left +image). But this is not the minimum of support points to estimate the four parameters. +So the question arises whether it is possible to reduce the number of support points as it +can be found in the concept of fractional factorial designs, see, for example, Pukelsheim +(1993, section 15.11). Instead of using all vertices of the hypercube [−1, 1]k as in the +full factorial design the fractional factorial design picks only a special percentage of these +points. For k = 3 +(−1, −1, 1)⊤, (−1, 1, −1)⊤, (1, −1, −1)⊤, (1, 1, 1)⊤ +represent a 23−1-fractional factorial design. +In our issue we do not want to pick four of the six points, but we want to use the +orthogonality of the spaces spanned by the points (without the x1-component) in the +two orbits (x1 = −1 and x1 = 1) of the given 23−1-fractional factorial design. +Here +span{(−1, 1)⊤, (1, −1)⊤} ⊥ span{(−1, −1)⊤, (1, 1)⊤}. +The idea for our problem is il- +lustrated in Figure 2 (right image). +The spanned spaces by points (without the x1- +component) in the orbits are orthogonal to each other. And all points span a simplex. +As stated above a (generalized) design ξ which is rotation invariant with fixed x1 — +invariant with respect to all orthogonal transformations in O(k) which do not change +the x1-component — and which has all mass on the unit sphere can be decomposed +into a marginal design ξ1 on [−1, 1] and a probability kernel η (conditional design), i. e. +ξ = ξ1 ⊗ η. For fixed x1 the kernel η(x1, ·) is the uniform distribution on the surface of a +(k − 1)-dimensional ball with radius +� +1 − x2 +1 — the orbit at position x1. If x1 ∈ {−1, 1}, +the (k − 1)-dimensional ball with the uniform distribution reduces to a single point and +represents only a one-point-measure. +Remembering q(x1) = λ(β0 + β1x1) the related +9 + +Martin Radloff, Rainer Schwabe +Exact Designs on the Ball +information matrix, see Radloff and Schwabe (2019b), is +M(ξ1 ⊗ η, β0) = +� +� +� +� +q dξ1 +� +q id dξ1 +� +q id dξ1 +� +q id2 dξ1 +O2×(k−1) +O(k−1)×2 +1 +k−1 +� +q (1 − id2) dξ1 Ik−1 +� +� +� +(5.14) +with β0 = (β0, β1, 0, . . . , 0)⊤. +The information matrix for a design on the k-dimensional unit sphere Sk−1, which is +based on exactly two orbits, can be determined analogously to this result. Additionally +the uniform distribution does not cover the the full orbits but only sub-spheres. +Lemma 3. Let ξ1 be the two-point-measure in x11 and x12 with ξ1(x11) = +1 +2 − α and +ξ1(x12) = +1 +2 + α with α ∈ +� +− 1 +2, 1 +2 +� +. Further let η(x11, ·) be a uniform distribution on +Sm−2 +�� +1 − x2 +11 +� +× {0}k−m and likewise η(x12, ·) be a uniform distribution on {0}m−1 × +Sk−m−1 +�� +1 − x2 +12 +� +. Then the information matrix is +M(ξ1 ⊗ η, β0) = +� +� +� +� +� +� +q dξ1 +� +q id dξ1 +� +q id dξ1 +� +q id2 dξ1 +O2×(k−1) +O(k−1)×2 +c1 Im−1 +O(m−1)×(k−m) +O(k−m)×(m−1) +c2 Ik−m +� +� +� +� +� +(5.15) +with c1 = +1 +m−1 q(x11) (1−x2 +11) ( 1 +2 −α) and c2 = +1 +k−m q(x12) (1−x2 +12) ( 1 +2 +α). +Now the optimality case in Theorem 2 (c0) on two orbits should be used to investigate +when both information matrices (5.14) und (5.15) are identical. With that both related +(generalized) designs would be (locally) D-optimal. +Lemma 4. Both information matrices (5.14) and (5.15) are identical in the situation of +Theorem 2 (c0) if and only if α = 1 +2 − +m +k+1. +The proof can be found in Appendix B. +Consequently both orbits need the weights ξ1(x11) = +m +k+1 and ξ1(x12) = k−m+1 +k+1 +to coincide +both information matrices. This allows an experimental design, which has the same value +for the D-optimality criterion, consisting of two orbits with m and with k −m+1 support +Figure 2: Logit model for k = 3 and β1 = 1 and −β0 = 0: discretized (locally) D-optimal +designs with 6 or 4 support points. +10 + +1Martin Radloff, Rainer Schwabe +Exact Designs on the Ball +Figure 3: D-efficiency for the logit model with k = 3 and β1 = 1: comparison of designs +with exactly k+1 = 4 equally weighted support points in −β0 ∈ (−0.403, 0.403) +(rounded). +points. This can be done by two regular simplices — one simplex in dimension m − 1 +and one in dimension k − m. +So the simplices are the discretizations of the uniform +distributions on Sm−2 +�� +1 − x2 +11 +� +× {0}k−m and on {0}m−1 × Sk−m−1 +�� +1 − x2 +12 +� +. +Let Sm ∈ Rm×(m+1) be a matrix, where the columns represent the m + 1 vertices of an +m-dimensional regular simplex (in Rm). Then the columns of the matrix +� +� +� +x111⊤ +m +x121⊤ +k−m+1 +R1 Sm−1 +O(m−1)×(k−m+1) +O(k−m)×m +R2 Sk−m +� +� +� +with arbitrary orthogonal transformations R1 ∈ O(m − 1) and R2 ∈ O(k − m) represent +the support points of such a minimal supported design. +�� +m + 1 +m +Im + 1 − √m + 1 +m√m +1m1⊤ +m +����� − +1 +√m 1m +� +∈ Rm×(m+1) +is an example for Sm. In this notation Im stands for the standard simplex which needs +to be scaled and shifted appropriately so that it is in combination with the last vertex +− 1 +√m 1m (last column) a regular simplex on the unit sphere Sm−1. +Finally, we want to look at the D-efficiency, here with β0 = (β0, β1, 0, . . . , 0)⊤, +EffD(ξ, β0) = +� +det(M(ξ, β0)) +det(M(ξ∗ +β0, β0)) +�1 +p +∈ [0, 1] +for designs ξ with exactly p = k + 1 equally weighted support points in the region where +two non-degenerated orbits occur. +As an example, the logit model with β1 = 1 is used to determine the D-efficiency in +dimensions k = 3 and k = 6. In Figure 3 and Figure 4 only the regions for −β0 with +11 + +Martin Radloff, Rainer Schwabe +Exact Designs on the Ball +two non-degenerated orbits in the optimal design (case (c0) in Theorem 2), i. e. −β0 ∈ +(−0.403, 0.403) (rounded) for k = 3 and −β0 ∈ (−0.480, 0.480) (rounded) for k = 6, are +plotted. +For this purpose, three different types of exact designs are compared with the (locally) +D-optimal design ξ∗ +β0. +The optimal design is a generalized design with real weights. +Therefore it cannot be discretized as an exact design in general. +First, the two optimal exact designs with one pole and one orbit, which are discretized as +a regular (k−1)-dimensional simplex, are used for comparison. The orbit position remains +unchanged and is determined at the boundary values −β0 ≈ ±0.403 or −β0 ≈ ±0.480. +See the solid lines in both figures. +Second, the designs with the same orbit position as the associated design which is (locally) +optimal for −β0 are the next alternative. +Only the weights were rounded/shifted to +integral multiples of +1 +k+1. See the dotted lines. +Third, the designs with fixed design weights which are integral multiples of +1 +k+1 represent +the last model category. So only the positions of the orbits have to be optimized with +these fixed design weights. This can be done by solving only the equations (3.4) and (3.5) +with the selected weights in Theorem 2 (c). Equation (3.6) is omitted. See the dashed +lines in both plots. +The Figure 3 reveals for dimension k = 3 that there are only three positions in the +range −β0 ∈ [−0.403, 0.403] (rounded) where (locally) D-optimal designs with the min- +imal number of support points — four points — exists. For −β0 ≈ −0.403 this is the +design consisting of the pole x∗ +12 = −1 and one orbit at x∗ +11 with three points as vertices +of an equilateral triangle. Then for −β0 = 0 there are two orbits with two points each. +And, at −β0 ≈ 0.403 the design consists of one orbit at x∗ +12 with three equally weighted +support points and the pole x∗ +11 = 1. In the span between these optimality positions the +considered discretizations provide a fairly high efficiency. Using the transition directly +from pole and orbit to orbit and pole, the efficiency is always greater than 0.988 (intersec- +tion of the solid lines). If the two orbits are also discretized in between, the efficiency is +greater than 0.993 (intersection of dotted line and solid lines) or even greater than 0.997 +(intersection of dashed line and solid lines). +For dimension k = 6, see figure 4, an efficiency of more than 0.986 is possible by stepping +directly from pole and orbit with six support points to orbit with six design points and +pole. If the intermediate steps — two orbits with 2 and 5 points, 3 and 4 points, 4 and +3 points as well as 5 and 2 points — are used, then by simple rounding of the weights to +integral multiples of +1 +k+1 an efficiency greater than 0.995 (dotted lines) and with additional +optimization of the orbit positions even greater than 0.999 (dashed lines) can be achieved. +6. Conclusion +In summary it can be postulated that very efficient designs are generated based on only +k + 1 design points which is the minimal number of support points to estimate the pa- +rameter vector. It seems that higher dimensions enable designs with higher D-efficiency, +in particular using the third option of discretization. Here we only considered designs +with exactly two orbits. Thus it cannot be excluded that there are designs with a better +efficiency or even (locally) optimal designs which are supported by exactly k + 1 points. +Maybe these designs have support points which lie not on the orbit but are jittered a little +bit. This as well as a potential lower efficiency bound needs further investigations. +12 + +Martin Radloff, Rainer Schwabe +Exact Designs on the Ball +Figure 4: D-efficiency for the logit model with k = 6 and β1 = 1: comparison of designs +with exactly k+1 = 7 equally weighted support points in −β0 ∈ (−0.480, 0.480) +(rounded). +On the other side the reduction of the equation system to one single equation for deter- +mining (locally) D-optimal design for symmetrical unimodal intensity functions is a nice +feature and can help to decrease computing costs. +Also the question of optimal designs on the ball with respect to other optimality criteria +should be considered in future. +Finally, we want to emphasize that the established designs do not only work for the +unit ball. By using the concept of equivariance for linear transformations, say scaling, +reflecting and rotating, the class of design spaces can be extended to k-dimensional balls +with arbitrary radius or any k-dimensional ellipsoid. +Appendix A +Notation +Bk +k-dimensional unit ball +Bk(r) +k-dimensional ball with radius r +Sk−1 +unit sphere, which is the surface of Bk +Sk−1(r) +sphere with radius r, which is the surface of Bk(r) +Ok +k-dimensional zero-vector +Ok1×k2 +(k1 × k2)-dimensional zero-matrix +1k +k-dimensional one-vector +Ik +(k × k)-dimensional identity matrix +id +identity function +13 + +Martin Radloff, Rainer Schwabe +Exact Designs on the Ball +Appendix B +Proofs +Proof sketch of Lemma 1. By plugging (4.9) and (4.10) into (3.6) and using the sym- +metry to simplify, we get +−2 α (4 c r α+(β2 +1 −c2−r2))+4 (k−1) c r +� 1 +2 −α +� � 1 +2 +α +� +� 1 +2 −α +� � 1 +2 +α +� +(4 c r α+(β2 +1 −c2−r2)) += 0 . +In the numerator there is a polynomial of degree two in α with the two roots α∓(r) +depending on r: +α∓(r) := +− (β2 +1 − c2 − r2) ∓ +� +(β2 +1 − c2 − r2)2 + 4 (k + 1) (k − 1) c2 r2 +4 (k + 1) c r +. +Now we examine the values of α∓(r) depending on r. Only −|c| − β1, |c| − β1, −|c| + β1 +or |c| + β1 can solve the expression α∓(r) = ± 1 +2. But −|c| − β1 and |c| + β1 are not in the +interesting region for r. We have +α− (±(|c| − β1)) = ±1 +2 sign(c) +and +α+ (±(|c| − β1)) = ∓1 +2 sign(c) k − 1 +k + 1 . +Because of limr↗0 α− (r) = sign(c)∞ and limr↘0 α− (r) = − sign(c)∞ the root α−(r) has +in the interval r ∈ [|c| − β1, −|c| + β1] only values outside (− 1 +2, 1 +2). Hence, α−(r) is not a +relevant root. +Since limr→0 α+ (r) = 0 the discontinuity of the root α+(r) in r = 0 can be removed. +So α+(r) has only values in (− 1 +2, 1 +2) on the interval r ∈ [|c| − β1, −|c| + β1] and α+(r), +which is (4.11), is the only relevant root. +After inserting (4.9) and (4.10) into (3.4) as well as (4.9) and (4.10) into (3.5) and sub- +tracting both obtained equations and simplifying by using the symmetry, we get +(k + 1) λ′(cλ + r) +λ(cλ + r) += −(k − 1) +−2 r + α · 4 c +(β2 +1 − c2 − r2) + α · 4 c r − 2 +r . +Equation (4.8) follows by plugging α+(r) as α into it and by some simplifications. +For β0 = cλ, i. e. c = cλ − β0 = 0, we get directly α = 0 by inserting x = +r +β1 and y = − r +β1 +in (3.6) and exploiting the symmetry. This is inserted in (3.4) and in (3.5). The difference +between these two equations results in (4.12). +Proof sketch of Lemma 2. This proof is a lot of curve sketching. We start with β0 ̸= +cλ. The denominator of the right hand side of (4.8) has five roots in r. −|cλ −β0|−β1 < 0 +and |cλ − β0| − β1 < 0 are not in the considered interval (0, |cλ − β0| + β1). In r = −|cλ − +β0| + β1 there is a discontinuity which can be removed. In r = 0 and in r = |cλ − β0| + β1 +there are two poles. Analyzing these poles for the considered interval we see that the +values start from −∞ (r ↘ 0) and go up to +∞ (r ↗ |cλ − β0| + β1). Sophisticated +curve sketching shows that the right hand side of (4.8) is strictly monotonically increasing +on (0, |cλ − β0| + β1). So it is strictly monotonically increasing and covers (−∞, ∞). In +combination with (A4) for the left hand side of (4.8) (monotonically decreasing) there is +exactly one solution. +For β0 = cλ we can mention that the right hand side of (4.12) is also strictly monotonically +increasing on (0, β1). Hence, there is only one solution. +An analogue result holds for the situation in Remark 4. +14 + +Martin Radloff, Rainer Schwabe +Exact Designs on the Ball +Proof of Lemma 4. Rearranging equation (3.6) equivalently in two ways gives +q(x12) (1−x2 +12) ( 1 +2 +α) = q(x11) (1−x2 +11) ( 1 +2 −α) k ( 1 +2 +α)−( 1 +2 −α) +k ( 1 +2 −α)−( 1 +2 +α) +and +q(x11) (1−x2 +11) ( 1 +2 −α) = q(x12) (1−x2 +12) ( 1 +2 +α) k ( 1 +2 −α)−( 1 +2 +α) +k ( 1 +2 +α)−( 1 +2 −α) . +The two denominators are zero if and only if α = 1 +2 − +1 +k+1 and α = 1 +2 − +k +k+1, respectively. +But this cannot happen to non-degenerated orbits because 1 +2 − +k +k+1 < α < 1 +2 − +1 +k+1. +Putting both equations into the diagonal entry of the information matrix (5.14) yield +1 +k − 1 +� +q (1 − id2) dξ1 += q(x11) (1−x2 +11) ( 1 +2 −α) +� +1 +k − 1 + +1 +k − 1 · k ( 1 +2 +α)−( 1 +2 −α) +k ( 1 +2 −α)−( 1 +2 +α) +� +and +1 +k − 1 +� +q (1 − id2) dξ1 += q(x12) (1−x2 +12) ( 1 +2 −α) +� +1 +k − 1 · k ( 1 +2 −α)−( 1 +2 +α) +k ( 1 +2 +α)−( 1 +2 −α) + +1 +k − 1 +� +They are identical to the diagonal entries of the information matrix (5.15) in Lemma 3 if +and only if +1 +k−1 + +1 +k−1 · k ( 1 +2 +α)−( 1 +2 −α) +k ( 1 +2 −α)−( 1 +2 +α) = +1 +m−1 and +1 +k−1 · k ( 1 +2 −α)−( 1 +2 +α) +k ( 1 +2 +α)−( 1 +2 −α) + +1 +k−1 = +1 +k−m +which are both equivalent to α = 1 +2 − +m +k+1. +References +Biedermann S, Dette H, Zhu W (2006) Optimal designs for dose-response models with +restricted design spaces. Journal of the American Statistical Association 101:747–759 +Dette H, Melas VB, Pepelyshev A, et al (2005) Optimal designs for three-dimensional +shape analysis with spherical harmonic descriptors. The Annals of Statistics 33:2758– +2788 +Dette H, Melas VB, Pepelyshev A (2007) Optimal designs for statistical analysis with +zernike polynomials. Statistics 41:453–470 +Farrell RH, Kiefer J, Walbran A (1967) Optimum multivariate designs. In: Proceedings +of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Volume +1: Statistics. University of California Press, Berkeley, Calif., pp 113–138 +Ford I, Torsney B, Wu C (1992) The use of a canonical form in the construction of locally +optimal designs for non-linear problems. Journal of the Royal Statistical Society: Series +B (Statistical Methodology) 54:569–583 +15 + +Martin Radloff, Rainer Schwabe +Exact Designs on the Ball +Hirao M, Sawa M, Jimbo M (2015) Constructions of φp-optimal rotatable designs on the +ball. Sankhya A : The Indian Journal of Statistics 77:211–236 +Kiefer JC (1961) Optimum experimental designs v, with applications to systematic and +rotatable designs. In: Proceedings of the Fourth Berkeley Symposium on Mathematical +Statistics and Probability, Univ of California Press, pp 381–405 +Konstantinou M, Biedermann S, Kimber A (2014) Optimal designs for two-parameter +nonlinear models with application to survival models. Statistica Sinica 24:415–428 +Lau TS (1988) d-optimal designs on the unit q-ball. Journal of statistical planning and +inference 19:299–315 +Pukelsheim F (1993) Optimal Design of Experiments. Wiley Series in Probability and +Statistics +Radloff M, Schwabe R (2016) Invariance and equivariance in experimental design for +nonlinear models. In: Kunert J, Müller CH, Atkinson AC (eds) mODa 11-Advances in +Model-Oriented Design and Analysis. Springer, p 217–224 +Radloff M, Schwabe R (2019a) Locally d-optimal designs for a wider class of non-linear +models on the k-dimensional ball with applications to logit and probit models. Statis- +tical Papers 60:165–177 +Radloff M, Schwabe R (2019b) Locally d-optimal designs for non-linear models on the +k-dimensional ball. Journal of Statistical Planning and Inference 203:106–116 +Schmidt D, Schwabe R (2017) Optimal design for multiple regression with information +driven by the linear predictor. Statistica Sinica 27:1371–1384 +16 + diff --git a/8tE1T4oBgHgl3EQfCAKJ/content/tmp_files/load_file.txt b/8tE1T4oBgHgl3EQfCAKJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c5512fc2865c948515376123782cc2842726791b --- /dev/null +++ b/8tE1T4oBgHgl3EQfCAKJ/content/tmp_files/load_file.txt @@ -0,0 +1,427 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf,len=426 +page_content='D-Optimal and Nearly D-Optimal Exact Designs for Binary Response on the Ball Martin Radloff† and Rainer Schwabe‡ Abstract: In this paper the results of Radloff and Schwabe (2019a) will be extended for a special class of symmetrical intensity functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' This includes binary response models with logit and probit link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' To evaluate the position and the weights of the two non-degenerated orbits on the k-dimensional ball usually a system of three equations has to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The symmetry allows to reduce this system to a single equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' As a further result, the number of support points can be reduced to the minimal number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' These minimally sup- ported designs are highly efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The results can be generalized to arbitrary ellipsoidal design regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Key words and phrases: Binary response models, D-optimality, k-dimensional ball, logit and probit model, multiple regression models, simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Introduction Spherical design spaces can occur in engineering or physics problems where the validity of a model may be assumed on a spherical region around a target value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' So (linear) models on spherical design spaces were investigated early in publications like Kiefer (1961) and Farrell et al (1967) which discuss polynomial regression on the ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' These ideas were followed up by papers in which also only linear problems were focused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' So Lau (1988) fitted polynomials on the k-dimensional unit ball by using canonical moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In Dette et al (2005, 2007) and Hirao et al (2015) harmonic polynomials and Zernike polynomials were used to be fit on the unit disc (2-dimensional unit ball), the 3- and k-dimensional unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' On the other hand generalized linear models are also well-examined and used in practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Logit and probit models, for example, in one dimension on an interval have already been investigated by Ford et al (1992) and Biedermann et al (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' But there seems to be no available literature which combines both topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In our publication Radloff and Schwabe (2019b) we took the first step to bring non- linearity or generalized linear models, respectively, and spherical design regions together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' These results were extended to a wider class of non-linear models in our follow-up paper Radloff and Schwabe (2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' †corresponding author: Martin Radloff, Institute for Mathematical Stochastics, Otto-von-Guericke- University, PF 4120, 39016 Magdeburg, Germany, martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='radloff@ovgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='de ‡Rainer Schwabe, Institute for Mathematical Stochastics, Otto-von-Guericke-University, PF 4120, 39016 Magdeburg, Germany, rainer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='schwabe@ovgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='de arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='02859v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='ME] 7 Jan 2023 Martin Radloff, Rainer Schwabe Exact Designs on the Ball For better comprehensibility, we will start with the model description and give a brief overview of the findings so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Then we will consider a special class of intensity func- tions which allows to reduce the the complexity of finding (locally) D-optimal designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Afterwards we will tackle the problem, that the optimal designs are not exact designs in general, by establishing highly efficient designs on the ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' General Model Description As in Radloff and Schwabe (2019b) and Radloff and Schwabe (2019a), where we described (locally) D-optimal designs for two special classes of linear and non-linear models on a k-dimensional unit ball Bk = {x ∈ Rk : x2 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' + x2 k ≤ 1} with k ∈ N, we solely focus (non-linear) multiple regression models, which means the linear predictor is f(x)⊤β = β0 + β1x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' + βkxk with regression function f : Bk → Rk+1, x �→ (1, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' , xk)⊤, and parameter vector β = (β0, β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' , βk)⊤ ∈ Rk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The one-support-point (or elemental) information matrix should be representable in the form M(x, β) = λ � f(x)⊤β � f(x)f(x)⊤ with an intensity (or efficiency) function λ which only depends on the value of the linear predictor f(x)⊤β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' These one-support-point (or elemental) information matrices are the base for the information matrix of a (generalized) design ξ with independent observations M(ξ, β) = � M(x, β) ξ(dx) = � λ � f(x)⊤β � f(x)f(x)⊤ξ(dx) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Here generalized design means an arbitrary probability measure on the design region Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' These information matrices allow to define the (local) D-optimality, which is one of the most popular criteria in experimental design theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' A design ξ∗ β0 with regular infor- mation matrix M(ξ∗ β0, β0) is called (locally) D-optimal (at β0) if det(M(ξ∗ β0, β0)) ≥ det(M(ξ, β0)) holds for all suitable probability measures ξ on the design space — here Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' This optimality criterion can be interpreted as the minimization of the volume of the (asymptotic) confidence ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Prior Results In Radloff and Schwabe (2016) we stated results on equivariance and invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' By rotating the design space Bk — the k-dimensional unit ball — and the parameter space Rk+1 in an analogous way the linear predictor of the multiple regression problem reduces to f(x)⊤β = β0 + β1x1 and β1 ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='1) Using the rotation invariance with fixed x1, this means the invariance to all orthogonal transformations in O(k) which let the x1-component unchanged, the (locally) D-optimal (generalized) design ξ∗ can be decomposed (ξ∗ = ξ∗ 1 ⊗ η) in a marginal probability measure ξ∗ 1 on [−1, 1] for x1 and a probability kernel η given x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' For fixed x1 the kernel η(x1, ·) is 2 Martin Radloff, Rainer Schwabe Exact Designs on the Ball the uniform distribution on the surface of a (k − 1)-dimensional ball with radius � 1 − x2 1 — the orbit at position x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' As a consequence the multidimensional problem collapses to a one-dimensional marginal problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Only the positions of the orbits and their weights have to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' To get an exact design the uniform orbits have to be discretized, for example, by using regular simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In our first paper — Radloff and Schwabe (2019b) — we started with models where the intensity function belongs to the class of monotonous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Such models have already been investigated in one dimension, for example, by Konstantinou et al (2014) and on multidimensional cuboids or orthants by Schmidt and Schwabe (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' These authors gave the following four conditions on the intensity function λ: (A1) λ is positive on R and twice continuously differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' (A2) The first derivative λ′ is positive on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' (A3) The second derivative u′′ of u = 1 λ is injective on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' (A4) The function λ′ λ is non-increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Condition (A2) is the motivation for the name class of monotonous intensity functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The intensity functions of this class have to satisfy always (A1) to (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' (A4) is an extra condition to guarantee uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' For a concise notation q(x1) = λ(β0 + β1x1) is used and the properties (A1), (A2), (A3) and (A4) transfer to q for β1 > 0, respectively, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Poisson regression with intensity function qP(x1) = exp(β0 + β1x1) and negative binomial regression as well as special proportional hazard models with censoring, see Schmidt and Schwabe (2017), satisfy all four conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' If β1 = 0 then the intensity function q is always a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' This yields to a (locally) D-optimal design as it can be found in linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In Pukelsheim (1993, section 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='12) such a design consists of the equally weighted vertices of a regular simplex inscribed in the unit sphere, the boundary of the design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The orientation of the simplex is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The main result for β1 > 0 in Radloff and Schwabe (2019b) is recited for the readers’ convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' There is a (locally) D-optimal design for the multiple regression prob- lem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='1) with β1 > 0 and intensity function satisfying (A1)-(A3) which has one support point equal to (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' , 0)⊤ and the other k support points are the vertices of an arbi- trarily rotated (k − 1)-dimensional regular simplex which is maximally inscribed in the intersection of the k-dimensional unit ball and a hyperplane with x1 = x∗ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' For k ≥ 2 the position x∗ 12 ∈ (−1, 1) is solution of q′(x∗ 12) q(x∗ 12) = 2 (1 + kx∗ 12) k (1 − x∗ 2 12 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' If additionally (A4) is satisfied, the solution x∗ 12 is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' 3 Martin Radloff, Rainer Schwabe Exact Designs on the Ball For k = 1 the position x∗ 12 ∈ [−1, 1) is either solution of q′(x∗ 12) q(x∗ 12) = 2 1 − x∗ 12 , if such a solution exists in [−1, 1), or otherwise x∗ 12 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' If additionally (A4) is satisfied, the solution x∗ 12 is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The design is equally weighted with 1 k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' It should be noted, that for fixed β this theorem does not need (A1) to (A4) on the entire real line R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' It is enough to have it in the ball and so on x1 ∈ [−1, 1] for q and on [β0 − β1, β0 + β1] for λ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' But the model has to satisfy the conditions always on the whole real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In our second paper — Radloff and Schwabe (2019a) — the conditions (A2) and (A3) were replaced by (A2′) and (A3′) and a fifth property (A5) was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' (A2′) λ is unimodal with mode c(A2′) λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' (A3′) There exists a threshold c(A3′) λ ∈ R so that the second derivative u′′ of u = 1 λ is both injective on (−∞, c(A3′) λ ] and injective on [c(A3′) λ , ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' (A5) u = 1 λ dominates z2 asymptotically for z → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In this context condition (A2′) means that there exists a c(A2′) λ ∈ R so that λ′ is positive on (−∞, c(A2′) λ ) and negative on (c(A2′) λ , ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Hence, there is only one local maximum which is simultaneously the global maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' So the class of intensity functions, which satisfy (A1), (A2′) and (A3′), is called class of unimodal intensity functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Indeed (A2) or (A3) do not imply (A2′) or (A3′), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' As mentioned before, we only focus on the unit ball and the interval x1 ∈ [−1, 1] for q or [β0 − β1, β0 + β1] for λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' So in our special case (A2) and (A3) can be transferred to (A2′) and (A3′) by using an arbitrary cλ > β0 + β1, which means that cq lies outside the interval [−1, 1] and only one branch of the function is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Property (A5) means lim z→∞ ���� u(z) z2 ���� = ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' This means that u(z) = 1 λ(z) goes faster to (±) infinity than z2 for z → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' As (A1) to (A4) the conditions (A2′), (A3′) and (A5) transfer from the intensity function λ to the abbreviated form q for β1 > 0 and vice versa — analogously c(·) q = c(·) λ −β0 β1 with (·) is (A2′), (A3′) or empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The logit model has the intensity function qlogit(x1) = exp(β0 + β1x1) (1 + exp(β0 + β1x1))2 and probit model has qprobit(x1) = φ2(β0 + β1x1) Φ(β0 + β1x1) · (1 − Φ(β0 + β1x1)) 4 Martin Radloff, Rainer Schwabe Exact Designs on the Ball with the density function φ and cumulative distribution function Φ of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Both models satisfy all five conditions (A1), (A2′), (A3′), (A4), (A5) and share a common c(A2′) λ = c(A3′) λ = 0, say cλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Analogously cq = − β0 β1 for q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Beside these two models other models like the complementary log-log model, see Ford et al (1992), with intensity function λcomp log log(z) = exp(2z) exp(exp(z))−1 satisfy all five conditions with c(A2′) λ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='466011 and c(A3′) λ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='049084, but here mode c(A2′) λ and threshold c(A3′) λ do not coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' We showed that if the (concise) intensity function q satisfies (A1), (A2′), (A3′) and (A5) the (locally) D-optimal design ξ∗ = ξ∗ 1 ⊗η is concentrated on exactly two orbits, which are the support points of the marginal design ξ∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The idea of the proof is based on Biedermann et al (2006) and Konstantinou et al (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The next theorem is the main result of our second paper — Radloff and Schwabe (2019a) — and is reproduced for the readers’ convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' It characterizes the positions of the two support points of the optimal marginal design ξ∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' For k ≥ 2 the simplified problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='1) with β1 > 0 and intensity function q satisfying (A1), (A2′), (A3′) and (A5) has a (locally) D-optimal marginal design ξ∗ 1 with exactly 2 support points x∗ 11 and x∗ 12 with x∗ 11 > x∗ 12 and weights w1 = ξ∗ 1(x∗ 11) and w2 = ξ∗ 1(x∗ 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' There are 3 cases: (a) If c(A2′) q > 1 and c(A3′) q /∈ [−1, 1], then x∗ 11 = 1, w1 = 1 k+1, w2 = k k+1 and x∗ 12 ∈ (−1, 1) is solution of q′(x∗ 12) q(x∗ 12) = 2 (1 + kx∗ 12) k (1 − x∗ 2 12 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='2) If additionally (A4) is satisfied, the solution x∗ 12 is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' (b) If c(A2′) q < −1 and c(A3′) q /∈ [−1, 1], then x∗ 12 = −1, w1 = k k+1, w2 = 1 k+1 and x∗ 11 ∈ (−1, 1) is solution of q′(x∗ 11) q(x∗ 11) = 2 (−1 + kx∗ 11) k (1 − x∗ 2 11 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='3) If additionally (A4) is satisfied, the solution x∗ 11 is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' (c) Otherwise c(A2′) q ∈ [−1, 1] or c(A3′) q ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Let x, y ∈ R with x > y and α ∈ � − 1 2, 1 2 � be solution of the equation system: q′(x) q(x) + 2 x−y + (k−1) q′(x) (1−x2) ( 1 2 −α) + q(x) (−2 x) ( 1 2 −α) q(x) (1−x2) ( 1 2 −α) + q(y) (1−y2) ( 1 2 +α) = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='4) q′(y) q(y) − 2 x−y + (k−1) q′(y) (1−y2) ( 1 2 +α) + q(y) (−2 y) ( 1 2 +α) q(x) (1−x2) ( 1 2 −α) + q(y) (1−y2) ( 1 2 +α) = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='5) 1 1 2 −α − 1 1 2 +α + (k−1) q(x) (1−x2) − q(y) (1−y2) q(x) (1−x2) ( 1 2 −α) + q(y) (1−y2) ( 1 2 +α) = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='6) 5 Martin Radloff, Rainer Schwabe Exact Designs on the Ball Figure 1: Logit model for k = 3 and β1 = 1: Dependence of x∗ 11 and x∗ 12 (solid lines) and the corresponding weights w1 and w2 = 1 − w1 (dashed lines) on −β0 = − β0 β1 = cq ∈ [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' (c0) If x, y ∈ (−1, 1) with x > y and α ∈ (− 1 2, 1 2) is a solution of the equation system, the orbit positions are x∗ 11 = x, x∗ 12 = y with weights w1 = 1 2 − α and w2 = 1 2 + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' (c1) If x ≥ 1 and y ∈ (−1, 1), then x∗ 11 = 1, w1 = 1 k+1, w2 = k k+1 and x∗ 12 ∈ (−1, 1) is the solution of the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' (c2) If y ≤ −1 and x ∈ (−1, 1), then x∗ 12 = −1, w1 = k k+1, w2 = 1 k+1 and x∗ 11 ∈ (−1, 1) is the solution of the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Instead of reproducing the whole theorem for k = 1, only the two main changes in case (c) should be mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' So the weights are always w1 = w2 = 1 2 and the equation system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='4)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='6) is replaced by q′(x) q(x) + 2 x − y = 0 and q′(y) q(y) − 2 x − y = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='7) To illustrate this complex issue we revisit the logit model in dimension k = 3 with β1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' We (numerically) plot the orbit positions x∗ 11 and x∗ 12 and corresponding weights w1 and w2 = 1 − w1 depending on −β0 = − β0 β1 = cq, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The cases (a) and (b) go along with Theorem 1 and the results from Radloff and Schwabe (2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The cases (c1) and (c2) yield marginal extremum solutions which are identical to (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' So for these four cases there is always an exact minimally supported (locally) D-optimal design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' As described in Theorem 1, it consists of a pole point in x1 = −1 or else x1 = 1 and the k vertices of a (regular) simplex which is maximally inscribed in the non-degenerated orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' But the problematic case is (c0) because the (locally) D-optimal (generalized) design consists of two non-degenerated orbits and additionally the weights are rarely appropriate for a discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In Radloff and Schwabe (2019a) we showed two examples for the logit model (k = 3, β1 = 1) from which we derived (nearly) exact designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' For −β0 = 0 the two orbit positions are symmetrical around 0, that is x∗ 11 = −x∗ 12 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='52, and the weights are ξ∗ 1(x∗ 11) = ξ∗ 1(x∗ 12) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' These two orbits were discretized by two 6 Martin Radloff, Rainer Schwabe Exact Designs on the Ball 2-dimensional simplices — overall 6 equally weighted support points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' see Figure 2 (left image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' For −β0 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='1 it is x∗ 11 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='42, x∗ 12 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='62 and ξ∗ 1(x∗ 11) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='4297, while 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='4297 ≈ 3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' We took the rounded design ξ≈ with the same support points x∗ 11 and x∗ 12 but with the marginal design ξ≈ 1 (x∗ 11) = 3 7 and ξ≈ 1 (x∗ 12) = 4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' So it was possible to substitute one orbit by the vertices of a 2-dimensional simplex (3 points — an equilateral triangle) and one by the vertices of a 2-dimensional cube or cross polytope (4 points — a square).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Because of rounding the design ξ≈ is not optimal but exact and has a high D-efficiency, which compares the rounded design ξ≈ and the optimal design ξ∗ β0 with respect to β0 — here p = k + 1 = 4 and β0 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='1, 1, 0, 0)⊤: EffD(ξ≈, β0) = � det(M(ξ≈, β0)) det(M(ξ∗ β0, β0)) �1 p ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='999757 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' These designs are not very satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' On the one hand the number of support points is not minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' On the other hand only special cases have appropriate rational weights which allow a discretization or otherwise the optimality is lost by rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Therefore we want to establish minimal supported exact designs for the case (c0) in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Mostly these designs wont be optimal but (highly) efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' But we start with the reduction of the system of three equations in Theorem 2 to only one single equation for special unimodal intensity functions — symmetrical unimodal intensity functions — which can be found, for example, in binary response models with logit and probit link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Optimal Design for Symmetrical Unimodal Intensity Functions An interesting observation was made in the discussion section in Radloff and Schwabe (2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' For models with unimodal intensity function in which the mode and threshold coincide (c(A2′) λ = c(A3′) λ = cλ) and which are symmetrical, also the two orbit positions are symmetrical in a certain way, which we want to investigate here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' For one dimension this has been considered and shown in Ford et al (1992, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='6), but this proof cannot be extended to higher dimensions directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' An unimodal intensity function in which the mode and threshold coincide (c(A2′) λ = c(A3′) λ = cλ) will be called symmetrical to cλ if λ(cλ + z) = λ(cλ − z) for all z ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The intensity functions of the logit and probit models are symmetrical with cλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' But the unimodal intensity function of the complementary log-log model has c(A2′) λ ̸= c(A3′) λ and cannot be symmetrical for this reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Let the intensity function λ be symmetrical to cλ in the situation of Theo- rem 2 (c0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' 7 Martin Radloff, Rainer Schwabe Exact Designs on the Ball For given β0 ̸= cλ let r solve λ′(cλ+r) λ(cλ+r) = − −2 k r2 (β2 1 +c2−r2)+(β2 1 −c2−r2)2−4 c2 r2 +(β2 1 −c2+r2) � (β2 1 −c2−r2)2+4 (k2−1) c2 r2 (k+1) r (r+c−β1)(r+c+β1)(r−c+β1)(r−c−β1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='8) with c := cλ − β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Then x = c β1 + r β1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='9) y = c β1 − r β1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='10) α = −(β2 1 −c2−r2)+ � (β2 1 −c2−r2)2+4 (k2−1) c2 r2 4 (k+1) c r (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='11) is a solution of the equation system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='4)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' For given β0 = cλ it is x = r β1, y = − r β1 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Here r is the solution of λ′(cλ + r) λ(cλ + r) = − 2 (β2 1 − k r2) (k + 1) r (β2 1 − r2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='12) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' For k = 1, see Remark 1, let λ be symmetrical to cλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Then x = cλ−β0 β1 + r β1 and y = cλ−β0 β1 − r β1 with r is solution of λ′(cλ + r) λ(cλ + r) = −1 r (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='13) solve the equation system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Lemma 1, whose proof sketch can be found in Appendix B, and Remark 2 in combination with Theorem 2 give (locally) D-optimal designs for models with symmetrical unimodal intensity functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' As a result we reduced the system of equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='4)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='6) to only one single equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' But now there is the question if condition (A4) can guarantee a unique solution as in Theorem 1 or in Theorem 2 (a) and (b) because Theorem 2 (c), especially (c0), tells nothing about uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' But we want to add a remark about the values of r before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Since the system of equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='4)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='6) in Theorem 2 (c0) should have a solution with two inner support points for the marginal design, x, y ∈ (−1, 1) is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' So −1 < cλ − β0 β1 ± r β1 < 1 must be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' This leads with β1 > 0 to r ∈ (−(cλ − β0) − β1, −(cλ − β0) + β1) and r ∈ ((cλ − β0) − β1, (cλ − β0) + β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Consequently, both intervals must overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' This happens for cλ − β0 > 0 at 0 < cλ − β0 < β1 and for cλ − β0 < 0 at −β1 < cλ − β0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Thus cλ − β0 ∈ (−β1, β1) and in particular β2 1 > (cλ − β0)2 must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Then r is in the interval (|cλ − β0| − β1, −|cλ − β0| + β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' But Theorem 2 (c) need x > y and consequently r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Hence, r ∈ (0, −|cλ − β0| + β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' This remains valid in particular for β0 = cλ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' cλ − β0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' So r ∈ (−β1, β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' With r > 0 it is r ∈ (0, β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' 8 Martin Radloff, Rainer Schwabe Exact Designs on the Ball Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In situation of Lemma 1 let the intensity function λ additionally satisfy condition (A4), then equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='8), whose right hand side is continuously continued in −|cλ − β0| + β1, has a unique solution in r ∈ (0, |cλ − β0| + β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' This also holds for β0 = cλ and equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='12), which has exactly one solution in r ∈ (0, β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' For k = 1, see Remark 2, and for an intensity function satisfying (A4) there is only one solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The proof sketch of Lemma 2 can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Lemma 2 guarantees a unique solution in r ∈ (0, |cλ − β0| + β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' But Remark 3 points out that for Theorem 2 (c0) we need r ∈ (0, −|cλ − β0| + β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' This means that the unique solution can result in the two-orbit case or in the one-orbit one-pole case of Theorem 2 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Minimally Supported Designs In the situation of Theorem 1 and Theorem 2 (a), (b), (c1) and (c2) the designs have always the minimal number of support points to estimate the parameter vector β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' These are k + 1 support points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In Radloff and Schwabe (2019a) revisited here in the introductory section we indicated exemplarily a (locally) D-optimal design for the logit model on the 3-dimensional ball with −β0 = 0 and β1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' This design consists of six support points which are the vertices of two regular 2-dimensional simplices — equilateral triangles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' see Figure 2 (left image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' But this is not the minimum of support points to estimate the four parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' So the question arises whether it is possible to reduce the number of support points as it can be found in the concept of fractional factorial designs, see, for example, Pukelsheim (1993, section 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Instead of using all vertices of the hypercube [−1, 1]k as in the full factorial design the fractional factorial design picks only a special percentage of these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' For k = 3 (−1, −1, 1)⊤, (−1, 1, −1)⊤, (1, −1, −1)⊤, (1, 1, 1)⊤ represent a 23−1-fractional factorial design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In our issue we do not want to pick four of the six points, but we want to use the orthogonality of the spaces spanned by the points (without the x1-component) in the two orbits (x1 = −1 and x1 = 1) of the given 23−1-fractional factorial design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Here span{(−1, 1)⊤, (1, −1)⊤} ⊥ span{(−1, −1)⊤, (1, 1)⊤}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The idea for our problem is il- lustrated in Figure 2 (right image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The spanned spaces by points (without the x1- component) in the orbits are orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' And all points span a simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' As stated above a (generalized) design ξ which is rotation invariant with fixed x1 — invariant with respect to all orthogonal transformations in O(k) which do not change the x1-component — and which has all mass on the unit sphere can be decomposed into a marginal design ξ1 on [−1, 1] and a probability kernel η (conditional design), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' ξ = ξ1 ⊗ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' For fixed x1 the kernel η(x1, ·) is the uniform distribution on the surface of a (k − 1)-dimensional ball with radius � 1 − x2 1 — the orbit at position x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' If x1 ∈ {−1, 1}, the (k − 1)-dimensional ball with the uniform distribution reduces to a single point and represents only a one-point-measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Remembering q(x1) = λ(β0 + β1x1) the related 9 Martin Radloff, Rainer Schwabe Exact Designs on the Ball information matrix, see Radloff and Schwabe (2019b), is M(ξ1 ⊗ η, β0) = � � � � q dξ1 � q id dξ1 � q id dξ1 � q id2 dξ1 O2×(k−1) O(k−1)×2 1 k−1 � q (1 − id2) dξ1 Ik−1 � � � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='14) with β0 = (β0, β1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' , 0)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The information matrix for a design on the k-dimensional unit sphere Sk−1, which is based on exactly two orbits, can be determined analogously to this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Additionally the uniform distribution does not cover the the full orbits but only sub-spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Let ξ1 be the two-point-measure in x11 and x12 with ξ1(x11) = 1 2 − α and ξ1(x12) = 1 2 + α with α ∈ � − 1 2, 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Further let η(x11, ·) be a uniform distribution on Sm−2 �� 1 − x2 11 � × {0}k−m and likewise η(x12, ·) be a uniform distribution on {0}m−1 × Sk−m−1 �� 1 − x2 12 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Then the information matrix is M(ξ1 ⊗ η, β0) = � � � � � � q dξ1 � q id dξ1 � q id dξ1 � q id2 dξ1 O2×(k−1) O(k−1)×2 c1 Im−1 O(m−1)×(k−m) O(k−m)×(m−1) c2 Ik−m � � � � � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='15) with c1 = 1 m−1 q(x11) (1−x2 11) ( 1 2 −α) and c2 = 1 k−m q(x12) (1−x2 12) ( 1 2 +α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Now the optimality case in Theorem 2 (c0) on two orbits should be used to investigate when both information matrices (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='14) und (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='15) are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' With that both related (generalized) designs would be (locally) D-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Both information matrices (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='14) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='15) are identical in the situation of Theorem 2 (c0) if and only if α = 1 2 − m k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The proof can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Consequently both orbits need the weights ξ1(x11) = m k+1 and ξ1(x12) = k−m+1 k+1 to coincide both information matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' This allows an experimental design, which has the same value for the D-optimality criterion, consisting of two orbits with m and with k −m+1 support Figure 2: Logit model for k = 3 and β1 = 1 and −β0 = 0: discretized (locally) D-optimal designs with 6 or 4 support points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' 10 1Martin Radloff, Rainer Schwabe Exact Designs on the Ball Figure 3: D-efficiency for the logit model with k = 3 and β1 = 1: comparison of designs with exactly k+1 = 4 equally weighted support points in −β0 ∈ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='403, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='403) (rounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' This can be done by two regular simplices — one simplex in dimension m − 1 and one in dimension k − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' So the simplices are the discretizations of the uniform distributions on Sm−2 �� 1 − x2 11 � × {0}k−m and on {0}m−1 × Sk−m−1 �� 1 − x2 12 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Let Sm ∈ Rm×(m+1) be a matrix, where the columns represent the m + 1 vertices of an m-dimensional regular simplex (in Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Then the columns of the matrix � � � x111⊤ m x121⊤ k−m+1 R1 Sm−1 O(m−1)×(k−m+1) O(k−m)×m R2 Sk−m � � � with arbitrary orthogonal transformations R1 ∈ O(m − 1) and R2 ∈ O(k − m) represent the support points of such a minimal supported design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' �� m + 1 m Im + 1 − √m + 1 m√m 1m1⊤ m ����� − 1 √m 1m � ∈ Rm×(m+1) is an example for Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In this notation Im stands for the standard simplex which needs to be scaled and shifted appropriately so that it is in combination with the last vertex − 1 √m 1m (last column) a regular simplex on the unit sphere Sm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Finally, we want to look at the D-efficiency, here with β0 = (β0, β1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' , 0)⊤, EffD(ξ, β0) = � det(M(ξ, β0)) det(M(ξ∗ β0, β0)) �1 p ∈ [0, 1] for designs ξ with exactly p = k + 1 equally weighted support points in the region where two non-degenerated orbits occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' As an example, the logit model with β1 = 1 is used to determine the D-efficiency in dimensions k = 3 and k = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In Figure 3 and Figure 4 only the regions for −β0 with 11 Martin Radloff, Rainer Schwabe Exact Designs on the Ball two non-degenerated orbits in the optimal design (case (c0) in Theorem 2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' −β0 ∈ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='403, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='403) (rounded) for k = 3 and −β0 ∈ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='480, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='480) (rounded) for k = 6, are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' For this purpose, three different types of exact designs are compared with the (locally) D-optimal design ξ∗ β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The optimal design is a generalized design with real weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Therefore it cannot be discretized as an exact design in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' First, the two optimal exact designs with one pole and one orbit, which are discretized as a regular (k−1)-dimensional simplex, are used for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The orbit position remains unchanged and is determined at the boundary values −β0 ≈ ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='403 or −β0 ≈ ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' See the solid lines in both figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Second, the designs with the same orbit position as the associated design which is (locally) optimal for −β0 are the next alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Only the weights were rounded/shifted to integral multiples of 1 k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' See the dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Third, the designs with fixed design weights which are integral multiples of 1 k+1 represent the last model category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' So only the positions of the orbits have to be optimized with these fixed design weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' This can be done by solving only the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='5) with the selected weights in Theorem 2 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='6) is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' See the dashed lines in both plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The Figure 3 reveals for dimension k = 3 that there are only three positions in the range −β0 ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='403, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='403] (rounded) where (locally) D-optimal designs with the min- imal number of support points — four points — exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' For −β0 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='403 this is the design consisting of the pole x∗ 12 = −1 and one orbit at x∗ 11 with three points as vertices of an equilateral triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Then for −β0 = 0 there are two orbits with two points each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' And, at −β0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='403 the design consists of one orbit at x∗ 12 with three equally weighted support points and the pole x∗ 11 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In the span between these optimality positions the considered discretizations provide a fairly high efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Using the transition directly from pole and orbit to orbit and pole, the efficiency is always greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='988 (intersec- tion of the solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' If the two orbits are also discretized in between, the efficiency is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='993 (intersection of dotted line and solid lines) or even greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='997 (intersection of dashed line and solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' For dimension k = 6, see figure 4, an efficiency of more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='986 is possible by stepping directly from pole and orbit with six support points to orbit with six design points and pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' If the intermediate steps — two orbits with 2 and 5 points, 3 and 4 points, 4 and 3 points as well as 5 and 2 points — are used, then by simple rounding of the weights to integral multiples of 1 k+1 an efficiency greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='995 (dotted lines) and with additional optimization of the orbit positions even greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='999 (dashed lines) can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Conclusion In summary it can be postulated that very efficient designs are generated based on only k + 1 design points which is the minimal number of support points to estimate the pa- rameter vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' It seems that higher dimensions enable designs with higher D-efficiency, in particular using the third option of discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Here we only considered designs with exactly two orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Thus it cannot be excluded that there are designs with a better efficiency or even (locally) optimal designs which are supported by exactly k + 1 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Maybe these designs have support points which lie not on the orbit but are jittered a little bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' This as well as a potential lower efficiency bound needs further investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' 12 Martin Radloff, Rainer Schwabe Exact Designs on the Ball Figure 4: D-efficiency for the logit model with k = 6 and β1 = 1: comparison of designs with exactly k+1 = 7 equally weighted support points in −β0 ∈ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='480, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='480) (rounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' On the other side the reduction of the equation system to one single equation for deter- mining (locally) D-optimal design for symmetrical unimodal intensity functions is a nice feature and can help to decrease computing costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Also the question of optimal designs on the ball with respect to other optimality criteria should be considered in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Finally, we want to emphasize that the established designs do not only work for the unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' By using the concept of equivariance for linear transformations, say scaling, reflecting and rotating, the class of design spaces can be extended to k-dimensional balls with arbitrary radius or any k-dimensional ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Appendix A Notation Bk k-dimensional unit ball Bk(r) k-dimensional ball with radius r Sk−1 unit sphere, which is the surface of Bk Sk−1(r) sphere with radius r, which is the surface of Bk(r) Ok k-dimensional zero-vector Ok1×k2 (k1 × k2)-dimensional zero-matrix 1k k-dimensional one-vector Ik (k × k)-dimensional identity matrix id identity function 13 Martin Radloff, Rainer Schwabe Exact Designs on the Ball Appendix B Proofs Proof sketch of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' By plugging (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='9) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='10) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='6) and using the sym- metry to simplify, we get −2 α (4 c r α+(β2 1 −c2−r2))+4 (k−1) c r � 1 2 −α � � 1 2 +α � � 1 2 −α � � 1 2 +α � (4 c r α+(β2 1 −c2−r2)) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In the numerator there is a polynomial of degree two in α with the two roots α∓(r) depending on r: α∓(r) := − (β2 1 − c2 − r2) ∓ � (β2 1 − c2 − r2)2 + 4 (k + 1) (k − 1) c2 r2 4 (k + 1) c r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Now we examine the values of α∓(r) depending on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Only −|c| − β1, |c| − β1, −|c| + β1 or |c| + β1 can solve the expression α∓(r) = ± 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' But −|c| − β1 and |c| + β1 are not in the interesting region for r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' We have α− (±(|c| − β1)) = ±1 2 sign(c) and α+ (±(|c| − β1)) = ∓1 2 sign(c) k − 1 k + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Because of limr↗0 α− (r) = sign(c)∞ and limr↘0 α− (r) = − sign(c)∞ the root α−(r) has in the interval r ∈ [|c| − β1, −|c| + β1] only values outside (− 1 2, 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Hence, α−(r) is not a relevant root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Since limr→0 α+ (r) = 0 the discontinuity of the root α+(r) in r = 0 can be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' So α+(r) has only values in (− 1 2, 1 2) on the interval r ∈ [|c| − β1, −|c| + β1] and α+(r), which is (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='11), is the only relevant root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' After inserting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='9) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='10) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='4) as well as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='9) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='10) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='5) and sub- tracting both obtained equations and simplifying by using the symmetry, we get (k + 1) λ′(cλ + r) λ(cλ + r) = −(k − 1) −2 r + α · 4 c (β2 1 − c2 − r2) + α · 4 c r − 2 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='8) follows by plugging α+(r) as α into it and by some simplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' For β0 = cλ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' c = cλ − β0 = 0, we get directly α = 0 by inserting x = r β1 and y = − r β1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='6) and exploiting the symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' This is inserted in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='4) and in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The difference between these two equations results in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Proof sketch of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' This proof is a lot of curve sketching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' We start with β0 ̸= cλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The denominator of the right hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='8) has five roots in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' −|cλ −β0|−β1 < 0 and |cλ − β0| − β1 < 0 are not in the considered interval (0, |cλ − β0| + β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In r = −|cλ − β0| + β1 there is a discontinuity which can be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In r = 0 and in r = |cλ − β0| + β1 there are two poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Analyzing these poles for the considered interval we see that the values start from −∞ (r ↘ 0) and go up to +∞ (r ↗ |cλ − β0| + β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Sophisticated curve sketching shows that the right hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='8) is strictly monotonically increasing on (0, |cλ − β0| + β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' So it is strictly monotonically increasing and covers (−∞, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In combination with (A4) for the left hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='8) (monotonically decreasing) there is exactly one solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' For β0 = cλ we can mention that the right hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='12) is also strictly monotonically increasing on (0, β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Hence, there is only one solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' An analogue result holds for the situation in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' 14 Martin Radloff, Rainer Schwabe Exact Designs on the Ball Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Rearranging equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='6) equivalently in two ways gives q(x12) (1−x2 12) ( 1 2 +α) = q(x11) (1−x2 11) ( 1 2 −α) k ( 1 2 +α)−( 1 2 −α) k ( 1 2 −α)−( 1 2 +α) and q(x11) (1−x2 11) ( 1 2 −α) = q(x12) (1−x2 12) ( 1 2 +α) k ( 1 2 −α)−( 1 2 +α) k ( 1 2 +α)−( 1 2 −α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The two denominators are zero if and only if α = 1 2 − 1 k+1 and α = 1 2 − k k+1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' But this cannot happen to non-degenerated orbits because 1 2 − k k+1 < α < 1 2 − 1 k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Putting both equations into the diagonal entry of the information matrix (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='14) yield 1 k − 1 � q (1 − id2) dξ1 = q(x11) (1−x2 11) ( 1 2 −α) � 1 k − 1 + 1 k − 1 · k ( 1 2 +α)−( 1 2 −α) k ( 1 2 −α)−( 1 2 +α) � and 1 k − 1 � q (1 − id2) dξ1 = q(x12) (1−x2 12) ( 1 2 −α) � 1 k − 1 · k ( 1 2 −α)−( 1 2 +α) k ( 1 2 +α)−( 1 2 −α) + 1 k − 1 � They are identical to the diagonal entries of the information matrix (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content='15) in Lemma 3 if and only if 1 k−1 + 1 k−1 · k ( 1 2 +α)−( 1 2 −α) k ( 1 2 −α)−( 1 2 +α) = 1 m−1 and 1 k−1 · k ( 1 2 −α)−( 1 2 +α) k ( 1 2 +α)−( 1 2 −α) + 1 k−1 = 1 k−m which are both equivalent to α = 1 2 − m k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' References Biedermann S, Dette H, Zhu W (2006) Optimal designs for dose-response models with restricted design spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Journal of the American Statistical Association 101:747–759 Dette H, Melas VB, Pepelyshev A, et al (2005) Optimal designs for three-dimensional shape analysis with spherical harmonic descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' The Annals of Statistics 33:2758– 2788 Dette H, Melas VB, Pepelyshev A (2007) Optimal designs for statistical analysis with zernike polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Statistics 41:453–470 Farrell RH, Kiefer J, Walbran A (1967) Optimum multivariate designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' University of California Press, Berkeley, Calif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=', pp 113–138 Ford I, Torsney B, Wu C (1992) The use of a canonical form in the construction of locally optimal designs for non-linear problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series B (Statistical Methodology) 54:569–583 15 Martin Radloff, Rainer Schwabe Exact Designs on the Ball Hirao M, Sawa M, Jimbo M (2015) Constructions of φp-optimal rotatable designs on the ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Sankhya A : The Indian Journal of Statistics 77:211–236 Kiefer JC (1961) Optimum experimental designs v, with applications to systematic and rotatable designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Univ of California Press, pp 381–405 Konstantinou M, Biedermann S, Kimber A (2014) Optimal designs for two-parameter nonlinear models with application to survival models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Statistica Sinica 24:415–428 Lau TS (1988) d-optimal designs on the unit q-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Journal of statistical planning and inference 19:299–315 Pukelsheim F (1993) Optimal Design of Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Wiley Series in Probability and Statistics Radloff M, Schwabe R (2016) Invariance and equivariance in experimental design for nonlinear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' In: Kunert J, Müller CH, Atkinson AC (eds) mODa 11-Advances in Model-Oriented Design and Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Springer, p 217–224 Radloff M, Schwabe R (2019a) Locally d-optimal designs for a wider class of non-linear models on the k-dimensional ball with applications to logit and probit models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Statis- tical Papers 60:165–177 Radloff M, Schwabe R (2019b) Locally d-optimal designs for non-linear models on the k-dimensional ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Journal of Statistical Planning and Inference 203:106–116 Schmidt D, Schwabe R (2017) Optimal design for multiple regression with information driven by the linear predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} +page_content=' Statistica Sinica 27:1371–1384 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'} diff --git a/9NA0T4oBgHgl3EQfO_8B/content/tmp_files/2301.02167v1.pdf.txt b/9NA0T4oBgHgl3EQfO_8B/content/tmp_files/2301.02167v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3b37e3d17fb66ce236bdb3018cbd29e690040e46 --- /dev/null +++ b/9NA0T4oBgHgl3EQfO_8B/content/tmp_files/2301.02167v1.pdf.txt @@ -0,0 +1,2443 @@ +TRACE ENCODING IN PROCESS MINING: A SURVEY AND +BENCHMARKING +Sylvio Barbon Jr. +University of Trieste +Trieste, Italy +sylvio.barbonjunior@units.it +Paolo Ceravolo, Rafael S. Oyamada, Gabriel M. Tavares +University of Milan +Milan, Italy +{paolo.ceravolo, rafael.oyamada, gabriel.tavares}@unimi.it +ABSTRACT +Encoding methods are employed across several process mining tasks, including predictive process +monitoring, anomalous case detection, trace clustering, etc. These methods are usually performed +as preprocessing steps and are responsible for transforming complex information into a numerical +feature space. Most papers choose existing encoding methods arbitrarily or employ a strategy based +on a specific expert knowledge domain. Moreover, existing methods are employed by using their +default hyperparameters without evaluating other options. This practice can lead to several drawbacks, +such as suboptimal performance and unfair comparisons with the state-of-the-art. Therefore, this +work aims at providing a comprehensive survey on event log encoding by comparing 27 methods, +from different natures, in terms of expressivity, scalability, correlation, and domain agnosticism. To +the best of our knowledge, this is the most comprehensive study so far focusing on trace encoding +in process mining. It contributes to maturing awareness about the role of trace encoding in process +mining pipelines and sheds light on issues, concerns, and future research directions regarding the use +of encoding methods to bridge the gap between machine learning models and process mining. +Keywords Encoding Methods · Process Mining · Anomaly Detection +1 +Introduction +Encoding methods are responsible for transforming complex information into a representative feature space. In process +mining (PM), several tasks (e.g., predictive process monitoring, trace clustering, anomaly detection, etc.) must encode +data before feeding specific algorithms. This step is crucial to account for the goals of a user correctly. For instance, +if a problem demands a solution where interpretability and explainability are needed, the data should be encoded by +methods that tend to accomplish those objectives. On the other hand, if the most essential requirements are space or +time complexity, the user should agree to lose part of the previous benefits to match these ones. +In the PM literature, most of the efforts have been dedicated to designing new algorithms and analytical methods but +little attention has been given to the impact of encoding methods across the existing tasks. For instance, in predictive +process monitoring [1] used the word embedding method to map the cases of an event log into real-valued vectors, +whereas [2] used the one-hot. A custom function is adopted by [3], whereas the count2vec (occurrence frequencies +of activities) is employed by [4]. Thus, a researcher interested in comparing the results of these works is in front of a +factor she cannot control, as the impact of encoding is not documented and the methods used are different. Moreover, +in this work, we emphasize that very few alternative encoding methods have been employed by the community and +demonstrate that arbitrarily encoding data might bring suboptimal results and misalignment with the user’s goals. We +arXiv:2301.02167v1 [cs.LG] 5 Jan 2023 + +Trace Encoding in Process Mining +Barbon et al. +believe that a better understanding of the effect of encoding methods, according to the datasets’ characteristics, is +decisive in developing more interpretable, explainable, robust, and accurate PM solutions. +Using anomaly detection as a case study, we extend the results of our previous paper [5] by considering several aspects. +First, we increase the number of encoding methods and provide a new taxonomy to classify them according to different +dimensions. Second, we include more datasets, considering more types of anomalies, in order to increase the space +of characteristics and achieve a better understanding of how each encoding method behaves according to the data +properties. Third, we employ evaluation criteria that are valuable for PM practitioners and can support the choice of the +suitable encoding method according to their goals. Lastly, we provide a systematic review of encoding methods across +popular PM tasks: predictive monitoring, trace clustering, anomaly detection, online process mining, and security and +privacy in PM. +More specifically, we first highlight how difficult it is not just to choose a suitable encoding method but also its +parameters. Subsequently, we perform an extensive experimental evaluation of 27 encoding methods with different +parameters over 420 synthetic event logs. We also discuss how current PM literature is limiting their experiments by +not considering the impact that encoding methods have in any problem domain. Thus, we discuss our results and focus +the contribution of our work on answering the following research questions: +1. How expressive is an encoding method for separating the problems’ classes? +2. What is the demand of time and memory to reach a suitable encoding method? +3. Is there any correlation between the encoding method and the performance achieved by algorithms in PM +tasks? +4. How generic encoding methods are, i.e. can an encoding method be applied to any PM task? +We answer these questions by proposing specific evaluation metrics according to different criteria. Through an in-depth +analysis, we consider the criteria expressivity, which aims at capturing patterns across different characteristics of +the employed datasets; scalability, which measures the elapsed time and the memory usage of encoding methods; +correlation power, which maps the data characteristics to the algorithm performances; and the domain agnosticism, +which considers if the encoding method depends or not on the problem domain. We demonstrate through our extensive +experimental evaluation how difficult it might be to choose a suitable encoding method since each of the evaluated +metrics has a different best performing method. Thus, the main contributions of this work include: +• A systematic review of encoding methods in PM and a new taxonomy developed according to such review. +• The proposal of new evaluation metrics to measure the quality of encoding methods in PM tasks. +• A deep experimental evaluation of several encoding methods never employed before in PM. +• A discussion of insights into future research on encoding for PM. +We organize the presentation of our work as follows. First, in Section 2 we define the problem of choosing the right +encoding method and its parameters. In Section 3 we provide the necessary background to understand this work. In +Section 4 we first present a systematic review of encoding methods in different process mining tasks. Subsequently, we +introduce a new taxonomy for encoding event data, organize the encoding methods found by families of algorithms, +describe each method, and discuss related works. Section 5 describes the employed methodology to implement our +experimental evaluation and Section 6 presents the carried experiments and results. In Section 7 we discuss the main +insights obtained in this work and provide future directions. We conclude our discussion in Section 8. +2 +Problem Definition +In this section, we address the problem of how arbitrarily employing encoding methods in PM tasks leads to sub-optimal +performance and results in unfair evaluations. Due to the wide range of encoding methods available nowadays, choosing +one given a specific problem is challenging. This can be seen in the current literature, across different domains, with +several automated solutions that have been proposed to decrease human intervention in the design of algorithms and +data science pipelines [6, 7, 8]. In PM, we believe this is even more challenging due to the nature of event logs, where +events can be described by both numerical and categorical attributes, are aggregated by cases, and are constrained by +the control flow of the process. For example, the availability of a given amount of resources may be a precondition +to observe an event (e.g., the execution of an activity) with dependencies to other preceding or concurrent events. +Condensing all this information into a single encoding method is difficult, and, in practical terms, each method can only +capture specific aspects. +2 + +Trace Encoding in Process Mining +Barbon et al. +Usually, encoding methods for PM are adapted from other domains. Simple techniques are often considered, for +instance, the one-hot encoding scheme [2] or frequency-based encoding methods [9]. To capture the sequential +nature of event logs, methods originally proposed in the Natural Language Processing (NLP) community have been +employed [10, 11]. However, while we can take into consideration the similarity between the sequential nature of traces +and natural language sentences, there are also differences that must be discussed. For instance, NLP tasks usually handle +a very large vocabulary, i.e., a set of unique words or tokens, whereas processes are usually represented by considerably +small vocabularies (e.g., the business process activities). As an attempt of capturing additional complexity, graph neural +networks have been recently studied in the literature [12]. Convolutional neural networks have also been used for +feature extraction [13]. Image-like data engineering methods have been introduced by [14, 15, 12]. More recently, +several pipelines have approached domain-specific encoding methods, which we will further describe in Section 4.6, +that exploit derived features, such as the resource pool discovery algorithm used to encode event resources by [16]. +In the context of our work, we stress that adopting the right encoding method and selecting optimal hyperparameters +can directly impact the final performance of a given task. Moreover, evaluating a new algorithm, e.g., a trace clustering +algorithm, by comparing it with other solutions but employing different data inputs (i.e., different encoding steps +preceding the clustering), produces an unfair evaluation. [17] stress this problem, highlighting that a given model cannot +be compared with another if their implementations consider different feature spaces. A brief example illustrating this +issue can be found in [16], where the authors are approaching the problem of predictive monitoring. In their evaluation, +the authors employ baselines to compare their proposal with existing LSTM architectures, each one based on a different +preprocessing procedure. Regarding other predictive monitoring work, word embedding is employed in [16, 18, 19] +whereas a traditional one-hot encoding was used in [20, 21, 22], preventing the comparison between these studies. This +problem is exacerbated by the fact that the PM community lacks shared benchmarks to be used in algorithm evaluation +and comparison. +In order to briefly illustrate the impact of arbitrarily encoding an event log, we demonstrate in Figure 1 the following +scenario. We compare two encoding algorithms, vary the parameter of vector dimensionality, apply them to two datasets +with different characteristics, and measure the accuracy achieved by a Random Forest classifier regarding the anomaly +detection problem1. The datasets have different cardinalities, different types of anomalies, and different rates of anomaly +injection. The first one has 10k traces, 444k events, and a 20% rate of insertion anomaly (a random activity is inserted +in the trace), whereas the second has 5k traces, 221k events, and a 15% rate of rework anomaly (an activity is doubled +in the trace). As we can see in the Figure 1, for the first event log (Event log 1), encoding the data employing the +Walklets method performed better than using the NMF-ADMM with low dimensionality and worse for medium and high +dimensionality. In addition, the latter method presented a high accuracy variation for different dimensionalities. For +the second event log (Event log 2), the Walklets encoding presented a more stable accuracy, while the NMF-ADMM +achieved higher accuracy w.r.t. the other dataset but always performed worse than Walklets. This is just a brief example +to demonstrate that there is no best encoding method for every dataset or default parameterization to apply. +Considering the nuances found regarding encoding methods, we address this existing limitation by evaluating encoding +methods for PM tasks. In this work, we attempt to demonstrate in detail how different methods perform on several +datasets with distinct characteristics and properties. We first demonstrate through extensive experimental evaluation +that each algorithm has distinct performances across different event logs and pipelines. We employ several metrics in +order to summarize the overall behavior of each method and focus the general evaluation on expressivity, scalability, +correlation, and domain agnosticism, which will be further detailed in Section 5. +3 +Background Notions +PM can be defined as a set of techniques to extract knowledge from event logs [23]. The goal is to provide analysis +that uses event data to extract process-related insights, i.e. creating solutions that are specifically tailored for business +processes and their stakeholders. +Thus, let us first consider Σ a universe of events, i.e. the set of all possible event identifiers. Σ∗ denotes the set of all +sequences over Σ. +Definition 3.1 (Event, Attribute) Events may have various attributes, such as timestamp, activity, resource, +and others. Let AN be the set of attribute names. For any event e ∈ Σ and an attribute n ∈ AN, then #n(e) is the +value of attribute n for event e. Typically, values are restricted to a domain. For example, #activity ∈ A, where A is the +universe of the legal activities of a business process, e.g. {a, b, c, d, e}. +1A detailed description of the material and methods is provided in Section 6. +3 + +Trace Encoding in Process Mining +Barbon et al. +0 +50 +100 +150 +200 +250 +Encoding dimensionality +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Accuracy +Event log 1 +encoding +NMF-ADMM +Walklest +0 +50 +100 +150 +200 +250 +Encoding dimensionality +Event log 2 +encoding +NMF-ADMM +Walklest +Figure 1: A brief example of performances achieved for two different datasets regarding the anomaly detection problem. +For both datasets we fixed two graph-based encoding methods with different parametrizations regarding dimensionality. +With abuse of notation, we refer to the name of the activity of an event #activity(e) as the event itself. Thus ⟨a, b, d⟩ +denotes a trace of three subsequent events. An event can also be denoted by its position in the sequence as ei with en +the last event of this sequence. A sequence of events composes a trace t ∈ Σ∗ and it can be defined as follows. +Definition 3.2 (Trace, Subtrace) In a trace each event appears only once and time is non-decreasing, i.e. for 1 ≤ i < +j ≤ |t| : t(i) ̸= t(j). A trace can also be denoted as a function generating the corresponding event for each position of +its sequence: t(i → n) �→ ⟨ei, ..., en⟩. A subtrace is a sequence t(i → j) where 0 < i ≤ j < n. +Now let C be the case universe, that is, the set of all possible identifiers of a business case execution. C is the domain +of an attribute case ∈ AN. +Definition 3.3 (Case, Event Log) We denote a case ci ∈ C as ⟨a, b, d⟩ci, meaning that all events share the same case. +For example, for ci we have #case(e1) = #case(e2) = #case(e3). An event log L is a set of cases L ⊆ Σ∗ where each +event appears only once in the log, i.e. for any two different cases the intersection of their events is empty. +In PM, encoding is a crucial step for several tasks in order to project the information contained in an event log to another +feature space before combining it with posterior algorithms such as clustering. In the context of this work, we approach +the anomaly detection problem to benchmark encoding methods. Thus, let M be a process model representing the event +log and f a test function that indicates if a trace from a log L is an instance of a model M. Thus, we can define the +anomaly detection problem as follows: +Definition 3.4 (Anomaly detection) Let f : L → {R, A} be a test function that evaluates whether a trace is regular +(R) or anomalous (A). A trace is considered anomalous if it can not be completely parsed by M. Thus, +f(t) = +�Regular, +if it can be replayed by M +Anomalous, +otherwise +(1) +Considering the particular granularity of PM data, i.e., traces consisting of events containing numerical, categorical, +and time-like values, in this paper, we propose a new taxonomy of methods that handle event data. +4 + +Trace Encoding in Process Mining +Barbon et al. +4 +Encoding Methods +A literature review guided us in proposing a taxonomy of encoding methods, which will be discussed in the following +sections. To the best of our knowledge, this work is the first in the PM literature to propose a systematic review of +encoding methods for PM tasks. There are surveys and benchmarks for specific groups of algorithms, for example +regarding graph embedding [24] or text embedding [25], but they fall outside the scope of PM applications. In PM, +different tasks need to employ an encoding method; we focus our review on trace clustering, predictive monitoring, and +anomaly detection tasks. +4.1 +Systematic Review +We performed a systematic review by analyzing the methods adopted in the literature. The online repositories employed +are the ACM Digital Library2, the IEEE Xplore3, and the Scopus4. We did not include Google Scholar in order +to narrow our search since it usually captures the same papers as the other repositories, plus papers from unknown +databases. Moreover, we searched only for works from the last 10 years with respect to the date time this review was +performed, i.e. from 2012 to 2022. A base query was defined and it was partially modified according to each PM task: +"process mining" AND (“encoding” OR “encode”) AND < task >, where the keyword task might be “clustering”, +(“predictive monitoring” OR “Process monitoring”), (“anomaly detection” OR “conformance-checking”), (“online +process mining” OR “stream process mining”), or (“security” AND “privacy”). Notice that we are including the terms +conformance-checking and anomaly detection interchangeably since anomaly detection can be seen as a sub-task of +conformance-checking. +After filtering by including only conference and journal papers, and dropping duplicates, we examined the abstracts +of each retrieved document to eliminate irrelevant papers. We achieved a total of 616 papers, where 208 are included +as clustering (CLUS), 165 as predictive process monitoring (PPM), 144 as anomaly detection (AD), 51 as online +process mining (OPM), and 48 as security (SEC). We illustrate the number of publications for each task and per year +in Figure 2a and the total publications for each task in Figure 2b. All the retrieved works are public available5 +Publications over the past ten years +2012 +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +2022 +0 +5 +10 +15 +20 +25 +30 +35 +# publications +SEC +OPM +AD +PPM +CLUS +(a) +SEC +OPM +AD +PPM +CLUS +0 +25 +50 +75 +100 +125 +150 +175 +200 +(b) +Figure 2: (a) Number of publications each year. (b) The total number of publications over the past ten years. +2https://dl.acm.org/ +3https://ieeexplore.ieee.org/Xplore/home.jsp +4https://www.scopus.com/search +5shorturl.at/uwJNW +5 + +Trace Encoding in Process Mining +Barbon et al. +4.2 +Taxonomy +To guide our discussion, the methods reviewed are organized into a taxonomy, presented in Figure 3. First, in Figure 3, +we illustrate all existing classes on the encoding problem and the intersections among them. We can think about +encoding at the control-flow or data-flow level [26], where the former considers only the event data whereas the latter +analyzes the remaining data from the case [26]. Inter-case and intra-case terminologies have recently been proposed +[27] to partially cover this difference by encoding each flow type. More specifically, the inter-case level aims to capture +relationships among cases, e.g. classify case types and similarities between cases. One motivation behind this concept is +the need to distinguish, for example, two identical sequences of activities (prefixes) with different labels (next activity). +For intra-case encoding, the focus is to represent past executions by encoding individual activities or completed traces. +Despite being more relevant for real scenarios, the former one is less approached in existing machine learning-based +applications since it was recently proposed by [27]. On the other hand, intra-case encoding is mostly employed across +different tasks in PM, and for this reason, we focus our contribution on this encoding level. +E1 +T2 +D2 +W2 +E2 +T1 +D1 +W1 +A +B +Workflow +W1: Control-flow +W2: Data-flow +Dependency +D1: Intra-case +D2: Inter-case +Trace +T1: trace directly +T2: trace by +aggregating events +Event Attributes +E1: categorical +E2: numerical/time +A +W1: Control-flow +D1: Inter-case +R2: trace by +aggregating events +E1: categorical +B +W1: Control-flow +D1: Inter-case +R1: trace directly +E1: categorical +Figure 3: General taxonomy for event log encoding. The encoding can be at the event attribute level, where each +attribute is encoded independently, or at the trace level, where the event attributes encoded are aggregated to summarize +the entire trace. +Furthermore, at this level, there might be specific targets when encoding data. For instance, event attributes might need +to be encoded individually. This is a common setting in predictive process monitoring, where each activity is encoded +in order to predict the next one. On the other hand, a specific application (e.g. clustering tasks) might need to encode +the complete trace. In this scenario, the trace can be encoded in a straightforward fashion by the employed algorithm or +it can be encoded by aggregating the individual event attributes. +Following our systematic review of encoding methods in process mining, the plethora of alternative encoding methods +in the literature, and our proposed diagram of encoding, we are able to wrap the insights from this study and organize +the encoding methods into different families. The taxonomy proposed in Figure 3 illustrates the intersections among +each possible scenario. In the figure, we point to only two intersections (A and B) since those are the most common in +the literature, although we expect that future works might fill the other ones. Some of the intersections, if not logically +impossible are hard to be achieved, for example, a method using both control- and data-flow or intra- and inter-case is of +complex design. In any case the fact only two intersections are covering the entire set of encoding methods we surveyed +is significant of the potential for new methods to experiment in PM. For example, none of the encoding methods we +surveyed is exploiting the temporal dimension of cases. By leveraging our systematic review, we can extend the pointed +6 + +Trace Encoding in Process Mining +Barbon et al. +intersections to group the found encoding methods into families as illustrated in Figure 4. Thus, these families can be +based on or inspired by techniques derived from the following research fields: process mining, text mining, and graph +embedding. Moreover, Figure 4 also presents the difference between both intersections, i.e. how the trace encoding +should be performed: if it must be encoded directly by a given algorithm or by aggregating previous encoded event +attributes. +Taking into consideration the families of encoding methods found in the literature, in the following subsections, we +describe each encoding method according to its respective families. For this work, most of the methods employed have +never been used in PM tasks to the best of our knowledge. Furthermore, to motivate researchers and practitioners to +consider these alternative methods more often, we only include methods that have open-source implementations in this +study. + Encoding +PM +Text +Graph +Event attributes +Aggregation +Encoded attribute +Encoded trace +Figure 4: Different levels of data encoding. First, event attributes might be individually encoded. Second, the trace can +be encoded directly by a certain algorithm or it can be encoded by aggregating the encoded event attributes. +4.3 +PM-based Encoding +Given an event log, we retrieve its respective process model and perform conformance-checking techniques to measure +its adherence to the model. Each trace in the event log is evaluated. The results produced are employed as the encoded +representation of the trace. The methods considered in our survey are illustrated below. +Trace-replay: given a process model, traces are replayed in it to obtain values that measure its conformance [28]. More +specifically, the values accumulated at each step are the number of tokens correctly consumed (c), the number of tokens +correctly produced (p), the number of missing tokens to execute the event in the next step (m), and the number of +unconsumed tokens after the last event execution (r). Thus, the final measure defined by the trace-replay metric is given +by fitness = 1 +2(1 − m +c ) + 1 +2(1 − r +p). All the values produced, ⟨c, p, m, r, fitness⟩, are used as the feature vector of a +given trace. +Trace alignment: performs a comparison between the process model and a trace and relates the trace to valid execution +sequences, i.e., allowed by the model [29]. An alignment is a sequence of moves that can be synchronous, model- +dependent, or log-dependent. It is also important to note that more than one alignment between the log and model +is possible, and techniques aim at finding the optimal one. The final feature vector is composed of the cost of the +alignment, the number of visited states, the number of queued states, the number of traversed arcs, and the fitness value +produced. +7 + +Trace Encoding in Process Mining +Barbon et al. +Log skeleton: this technique aims at summarizing activity traces by capturing a set of constraints that apply to activities +throughout the log [30]. For example, the Req +L captures the equivalence relation between two activities, which exists if +both activities have the same frequency of occurrence in every trace. On the other hand, the Cdf +L counts the number of +directly-follows occurrences for every pair of activities. Other examples of measures to capture relations include the +always-after and never-together; examples of countermeasures include the sum of occurrences of a given activity in the +entire log and the min and max numbers of occurrences of an activity in any trace. In the implementation used for this +paper, six different constraints are used. +Position profile: this technique represents an event log through a matrix, where each position refers to the activity × +position regarding all traces [31]. It can be formally defined as a triple apf = (a, p, f) ∈ E, where a is the activity, p +is the position of the activity, f is the frequency of occurrence of the given activity, and E is the universe of events. +4.4 +Text-inspired Encoding +Many solutions used for trace encoding in PM are adapted from methods used in NLP. Exploiting the fact that words +in sentences are ordered in sequence and are constrained by dependencies, encoding methods applied to text capture +that information. Because traces are composed of sequences of activities the same information appears relevant to +characterize them. In particular, in our survey, we consider the following methods. +N-grams [32]: this method represents a given sequence of elements through sub-sequences of n items. Thus, considering +a sequence s = {s1, ..., si}, the n-grams representation of these sequences is given by n-grams = {(s1, ..., sn), +(s2, ..., sn+1), ...(si−n, ..., si)}. +One-hot [33]: given a variable containing n different values, the variable is transformed into an array where each +unique value is represented as a binary vector with the i-th position set to one and the rest set to zero. Clearly, the +dimension of the vector depends on the size n of the unique values in the vector space, easily reaching high dimensional +spaces. +CountVectorizer (count2vec) [33]: given a collection of categorical documents, this method produces a matrix of +token occurrences, where each line in the matrix represents a document and each column a token. The size of the vector +space depends on the n unique values in the vector space. +HashVectorizer (hash2vec) [33]: it does the same as count2vec. However, instead of storing tokens, it directly maps +each token to a column position in the matrix of occurrences. It is mainly useful for large datasets, and unlike one-hot +and count2vec, which have the same dimensionality as the vocabulary length, this method has the flexibility to hash +tokens in any dimensionality. +TF-IDF [34]: the term frequency (TF) captures the frequency of a particular token w.r.t. to a given document, whereas +the inverse document frequency (IDF) measures how common the token is in the corpus. TF can be simply the +number of times the token appears and the IDF is calculated as follows: idf(t, D) = log( +N +count(d∈D:t∈d)), where t is +the token and N is the number of documents d in the corpus D. Thus, the TF-IDF is obtained by multiplying both +TF-IDF(t, d, D) = tf(t, d) × idf(t, D). +Word2vec [35, 36]: the main contribution behind word2vec was learning distributed representations of words and +reducing the computational cost compared to the state of the art at the time. Although there are two original model +architectures for learning the word vectors, Continuous Bag-of-Words (CBOW) and Continuous Skip-gram Model (skip- +gram), the core characteristic of word2vec is the removal of the hidden layer of a simple Neural Net Language Model. +CBOW predicts the current word based on the t words around it, i.e., it predicts wt given (wt−i, ..., wt−1, wt+1, ...wt+i). +On the other hand, given wt, the skip-gram predicts the surrounding words (wt−i, ..., wt − 1, wt+1, ...wt+i). The +parameter i in both cases is a parameter representing a range surrounding the current word wt. +Doc2vec [37]: this algorithm is an extension of word2vec and learns the embeddings of documents (sentence, paragraph, +essay, etc.). The difference w.r.t. word2vec is given by the learning which is performed via the distributed memory and +distributed bag of words models and by adding another vector (document ID) to the input. +GloVe [38]: this is an unsupervised learning algorithm for obtaining vector representations for words. The main +intuition behind this model is the capturing ratios of word-word co-occurrence probabilities in order to capture both +local and global dependencies. This is expressed by F(wi, wj, +-wk) = Pik +Pjk , where Pik and Pjk are the probabilities that +the word k appears in the context of words i and j respectively. +8 + +Trace Encoding in Process Mining +Barbon et al. +4.5 +Graph-based Encoding +The intuition behind graph embedding methods is to represent nodes of a graph as low dimensional vectors, where such +vectors are representative enough to keep its original relations (edges) intact. We can formally define the general idea as +follows. A graph can be described as G = (V, E), where V = {v1, ..., vn} is a set of vertices (nodes) and E is a set of +edges e = (u, v) that connect a pair of vertices u, v ∈ V . Given a graph G, a graph embedding is a mapping function +f : vi → yi ∈ Rd, such that d << |v| and f preserves the original structure of their local neighborhood and minimizes +the information loss. In this section, we describe graph embedding methods for event log encoding. +DeepWalk [39]: it can be seen as a two-stage algorithm. First, a discovery of the local structure is performed through +random walks. There are two parameters here, the number of random walks α and the number of vertices to visit t for +each random walk. Second, similar to the word2vec, the skip-gram is performed to learn the embeddings. The intuition +behind this algorithm is learning embeddings close to each other if they often occur in a similar structural context. +Node2vec [40]: this algorithm is similar to DeepWalk, where the difference is a biased-random walk that aims at +employing a trade-off between breadth-first and depth-first searches. In practice, such balance is capable of providing +more informative embeddings than DeepWalk. +Walklets [41]: while DeepWalk and node2vec implicitly capture a certain level of dependencies by generating multiple +random walks through Ak +ij, this algorithm does explicitly by combining factorization approaches with random walks. It +preserves dependencies by sub-sampling short random walks on the vertices and by skipping over steps in each random +walk. This results in paths of fixed lengths composing sets of pairs of vertices. Thus, these sets are used to learn the +latent representations. +role2vec [42]: this is a framework that uses random walks to approximate the pointwise mutual information matrix, +which is obtained by multiplying a matrix of structural features with the pooled adjacency power matrix. +Laplacian Eigenmaps [43]: this algorithm intuitively keeps the embedding of two nodes close when the weight Wij +is high. Given a graph G, this algorithm computes eigenvalues and eigenvectors Ly = λDy, where D is a diagonal +weight matrix Dii = � +j Wji, and W is the weight matrix. Thus, L = D − W is the Laplacian matrix that can be used +to minimize the function ρ(Y ) = 1 +2 +� |Yi − Yj|2Wij = tr(Y T LY ). +GraRep [44]: this algorithms learns the latent representation W ∈ R|V |×d of the vertices of the weighted graphs. +It leverages global structural information to capture long-distance connections. The overall idea is first to calculate +the transition probability matrix A = D−1S for each k, where 1 <= k <= K is the maximum transition step. Sub- +sequently, obtain each k-step representation by factorizing the log probability matrix using singular value decomposition. +Finally, the k-step representations for each vertex on the graph are concatenated and used as latent representations. +Hope [45]: this embedding algorithm is similar to GraRep, but instead of using the transition probability matrix, +it employs a similarity matrix S. Thus, S can be obtained by using different similarity measures and consequently +preserves higher-order dependencies. +BoostNE [46]: this algorithm performs a non-negative matrix factorization to calculate the residuals generated by +previous embedding models. It assumes the same idea as the gradient boosting method in ensemble learning, where +multiple weak learners lead to a better one when aggregated. Given a connectivity matrix obtained through the adjacency +matrix of the graph, the algorithm calculates k residual matrices and uses each one as input to the next one using the +following equation: +Ri = +�X, +if i = 1 +max(Ri−1 − Ui−1Vi−1, 0), +if i ≥ 2 +(2) +where Ui ∈ Rn×ds ++ +and Vi ∈ Rn×ds ++ +intuitively act like the embedding representation of the center node and the context +node in the i − th level, respectively. Assuming the defined residual matrix, the embedding representation at the i − th +level is obtained by minimizing the loss function L = minUi,Vi,≥0 ||Ri − UiVi||2 +F , for 1 <= i <= k. +Diff2vec [47]: the overall idea of this algorithm is sub-sampling diffusion graphs for each node in a graph and generating +sequences of vertices through an Euler tour. Given a graph G, a graph G′ of l vertices is sub-sampled in a diffusion-like +random process. Then, from G′, sequences of vertices are generated by performing an Euler walk. In this process, G′ is +first converted to a multi-graph by doubling each edge. Thus, the Euler walk is employed instead of the random walk +since this algorithm can capture a more complete view in graphs with this characteristic. The generated sequences of +vertices are then used to create the graph embedding. +9 + +Trace Encoding in Process Mining +Barbon et al. +GLEE [48]: unlike most graph embedding algorithms that expect similar nodes to have their embeddings close to +each other, this algorithm uses the Laplacian matrix of a given graph to find an embedding with geometric properties. +Examples of such properties are dot product (angle), length (area or volume) of a line segment (or polygon), the convex +hull of a set of vectors, etc. Thus, given a graph G and its Laplacian matrix L, this procedure extracts eigenvectors +corresponding to the largest eigenvalues in L. These vectors are used as node embeddings. +NetMF [49]: this method is built on a theoretical analysis that shows the equivalence of different graph embedding +algorithms based on DeepWalk. In the original paper, the authors show that methods that use negative samplings, such +as DeepWalk and node2vec, implicitly perform matrix factorization. Thus, the framework NetMF is proposed to unify +existing methods and perform an explicit factorization. +NMF-ADMM [50]: given an adjacency matrix, the NMF-ADMM algorithm learns the embeddings by using the +alternating direction method of multipliers to solve the negative matrix factorization problem. +GraphWave [51]: given an undirected graph G = (V, E), an adjacency matrix A (binary or weighted), and a +degree matrix Dii = � +j Aij, this method learns a structural embedding of every vertex v ∈ V . The learning is +performed in an unsupervised fashion based on spectral graph wavelets. GraphWave is given by GraphWave = +U Diag(gs(λ1), ..., gs(λn))U T αv, where αv is the one-hot vector for the vertex v, U the decomposition of the +eigenvector from A, and gs is a kernel that modulates the eigenspectra. +NodeSketch [52]: this method recursively generates k-order node embeddings in a recursive manner. These embeddings +are categorized into low-order (k = 2) and high-order (k > 2). At each step k, a Self-Loop-Augmented (SLA) +adjacency matrix is generated to obtain the embeddings. Low-order SLA is obtained by simply adding the identity +matrix to the original adjacency matrix M ′ = M + I. On the other hand, high-order embeddings first sketch an +approximate k-order SLA adjacency of the current nodes and merge it with the (k − 1)-order SLA adjacency matrix in +a weighted manner. +4.6 +Related Works +Several tasks in PM, such as predictive monitoring, trace clustering, and anomaly detection, need to encode data to +feed algorithms that are applied down the pipeline. Although transforming event data into a reasonable feature space +is a sensible task, i.e., it might drastically impact algorithms’ performances, very little attention has been given to +encoding methods. Regarding the literature of other problem domains, there are surveys and benchmarks trying to +standardize and better investigate the behaviors of encoding methods according to different problems’ characteristics. +For instance, [24] surveyed several graph-based embedding methods on different datasets and discussed the main +challenges for future research in the field. Regardless of the approached task (e.g. link prediction, node classification, +etc.), the authors demonstrate the difficulty of choosing not only the right algorithm but also the right set of parameters +(mainly the dimensionality). Several trade-offs must always be taken into consideration, for instance increasing the +memory usage to achieve more precision or decreasing the dimensionality to decrease the computation time. On the +other hand, [53] covered a wide range of methods to encode textual information. The work focuses on methods based +on encoding methods to feed neural network architectures regarding different tasks and also provides historical notes +for each category of task. +The aforementioned works usually focus on representational learning, which employs neural networks to learn a +high-quality representation (encoding) of data. In the natural language literature, the word2vec [35, 36] can be seen +as one of the most important methods for this purpose, which has two architectures variants, one using the CBOW +algorithm and another one using the skip-gram model. From this perspective, several methods derived from it, for +instance, [37, 10]. The resulting feature vectors representing the original data are also called embeddings. +Recently, representational learning has been applied in PM as well. [10] proposes the act2vec, trace2vec, log2vec, +and model2vec. Each approach derives from existing encoding methods in the literature and leverages the previous +level information to enrich the learning. That is, the first level is act2vec, which extends the word2vec architecture to +learn the representation of activities. Subsequently, the trace2vec adopts the doc2vec concept and jointly learns the +representation of activities and traces. The log2vec architecture derives from the same idea as trace2vec where the log +representation is included in the architecture to be jointly learned. Finally, for model2vec, the authors extend graph +representation learning techniques to represent a process model discovered from the event log. The final architecture +also includes all the previous representations to be learned jointly. +In the literature, we also find “hand-crafted” methods, which are usually developed by following some expertise domain +knowledge. [12] proposes using graph convolutional networks for predictive monitoring. In their approach, the authors +first perform a feature engineering step to handle time features and then transforms each activity in an event log into +a matrix num_unique_activities × num_time_features. [16] employs a PM algorithm to encode resources in +10 + +Trace Encoding in Process Mining +Barbon et al. +event logs. In a nutshell, the applied algorithm is able to automatically discover resource pools and, hence, reduce the +dimensionality of categorical values by grouping them. In [14], the authors propose the use of convolutional neural +networks to perform predictive monitoring. Thus, they transform the data into an image-like structure in order to be +able to train the neural network. [54] presents a method for feature extraction that can be seen as an encoding method, +where seven different features are extracted from each activity given a Petri net. These features aim at capturing local +information for the activity with respect to its current case. +Although recent works in predictive process monitoring have explored more alternatives, it is noticeable that works +in PM often use a minimal variety of encoding methods. Most papers use naive techniques like one-hot encoding. +Moreover, other encoding approaches are usually combined with results from feature engineer procedures that handle +numerical and time-related information. In recent works, the most common encoding methods for different tasks include +the one-hot [2, 22, 55, 21, 19], counting the frequencies of categorical data [9], some type of embedding network +[18, 56, 16, 19], or hand-crafted representations [16, 12, 14]. Therefore, we motivate our work in order to fulfill this +limitation by exploring a wider range of encoding methods. +5 +Methodology +In this section, we describe the experimental analysis carried out to evaluate encoding methods. We provide details on +the software and materials and on the metrics used in order to assess the quality of the surveyed encoding methods. +5.1 +Implementation Overview +Our implementation can be organized into three steps: (i) dataset preparation, (ii) encoding generation, and (iii) +evaluation of the encoding methods from multiple perspectives. The source code is available online in this repository6. +First, we generated synthetic logs using the PLG2 tool [57]. Subsequently, the encoding of the generated logs was +performed using open-source libraries in Python as described in Table 1, which include Sklearn7, Karate Club8, PM4PY9, +NLTK10, Gensim11, GloVe12, and the position profile implementation on github13. We organize each method according +to the proposed taxonomy and provide the respective references for original papers and online implementations. +Moreover, we set as baselines the methods count2vec, one-hot, n-grams, and position profile, which implement the +most simple transformations. In the case of event-level encoding, the procedure was first performed at the activity level +and then the results were aggregated to obtain the trace representation (trace-level encoding). This aggregation takes the +resulting encoded information of each activity and averages it into a unique feature vector. For graph-based methods, +this aggregation was obtained in two different ways: from edges or from nodes. +5.2 +Evaluation Metrics +Assuming that encoding methods are used to map the original problem space into a different vector space, we observed +the quality of the new space based on several criteria. Moreover, each encoding method has particularities regarding +performance delivered, descriptive capability, computational cost, and complexity of hyperparameter space. Thus, to be +effective, an encoding method should meet the following criteria: +• Expressivity: the relative capacity of an encoding method to affect the complexity of the mapped space +regarding the original problem space. An encoding method should be able to map the event logs of varying +complexity, in which a straightforward representation regards a simple process and an intricate representation +concerns a complex process. +• Scalability: the property related to increasing or decreasing the encoding computational cost in response to +changes in the event log size. The encoding method should be able to map the event log quickly, without +compromising the PM pipeline run time. +6https://github.com/gbrltv/business_process_encoding +7https://github.com/scikit-learn/scikit-learn +8https://github.com/benedekrozemberczki/karateclub +9https://github.com/pm4py/pm4py-core +10https://github.com/nltk/nltk +11https://github.com/RaRe-Technologies/gensim +12https://github.com/maciejkula/glove-python +13https://github.com/gbrltv/meta_trace_clustering/blob/main/clustering.py#L64 +11 + +Trace Encoding in Process Mining +Barbon et al. +Algorithm +Year +Family +Implementation +count2vec [33] +- +Baseline +Sklearn +n-grams [32] +- +Baseline +NLTK +position profile [31] +2017 +Baseline +GitHub +one-hot [33] +- +Baseline +Sklearn +GraphWave [51] +2018 +Graph +Karate Club +Laplacian Eigenmaps [43] +2001 +Graph +Karate Club +NMF-ADMM [50] +2014 +Graph +Karate Club +DeepWalk [39] +2014 +Graph +Karate Club +GraRep [44] +2015 +Graph +Karate Club +node2vec [40] +2016 +Graph +Karate Club +Walklets [41] +2017 +Graph +Karate Club +role2vec [42] +2018 +Graph +Karate Club +NetMF [49] +2018 +Graph +Karate Club +NodeSketch [52] +2019 +Graph +Karate Club +BoostNE [46] +2019 +Graph +Karate Club +GLEE [48] +2020 +Graph +Karate Club +Hope [45] +2016 +Graph +Karate Club +diff2vec [47] +2018 +Graph +Karate Club +Log skeleton [30] +2018 +PM +PM4PY +token-replay [23] +2016 +PM +PM4PY +alignment [23] +2016 +PM +PM4PY +word2vec (skip-gram) [35] +2013 +Text +Gensim +hash2vec [33] +- +Text +Sklearn +GloVe [38] +2014 +Text +GloVe +doc2vec [37] +2014 +Text +Gensim +word2vec (CBOW) [36] +2013 +Text +Gensim +TF-IDF [34] +1958 +Text +Sklearn +Table 1: Encoding methods and related details. +• Correlation power: the capacity of an encoding method to improve the original problem space. The new +feature vector needs to be highly correlated to the PM task goal, i.e., the encoded feature vector should enhance +the performance of PM tasks. +• Domain agnosticism: refers to how well a given encoding method maps data from different domains. Encoding +methods that are non-agnostic can be used only in specific applications. +There are different strategies and metrics to assess encoding methods considering the presented criteria. In this work, we +exploit the followings. We exploited Principal Component Analysis (PCA) [58] to verify how well a vector space can be +compressed. Classification complexity metrics [59] to measure how well samples, i.e., encoded traces, are distributed +within classes. The F1-score [60] to observe the impact of encoding methods on accuracy. Time and space complexity +to assess the computational performances. Table 2 summarizes the contribution of each measure we exploited. +5.3 +Experimental Design +Our experimental design relies on labeled data for ground truth evaluation of the compared encoding methods, as an +extension of [5]. Synthetic event logs were generated based on standard PM research practices and anomalies were +injected into the generated traces, representing an anomaly detection PM task. Afterward, traces were labeled as +anomalous or normal, making our data set suitable for supervised learning. Our dataset was made more realistic by +adding heterogeneous behaviors to the event logs. +PLG2 [57] was used to create five different process models by performing a random generation of a process capable of +capturing several behaviors, such as sequential, parallel, and iterative control-flow. The rationale of PLG2 is based on +the combination of traditional control-flow patterns [61], e.g., sequence, parallel split, and synchronization. In order to +simulate real-world scenarios, the patterns are progressively combined according to predetermined rules. Each of the +five generated process models defines five different base scenarios based on the activities and gateways included in the +scenario. +12 + +Trace Encoding in Process Mining +Barbon et al. +Criteria +Analysis +Acronym +Description +Expressivity +Principal Component +Analysis +PCA +Using the 2D projection of a PCA space +it is possible to observe patterns across +scenarios from different complexities, +ranging from very low, low, average, +high and very high expressivity. +Ratio of the PCA +dimension to the +original dimension +T4 +This measure is related to the proportion +of relevant dimensions that the coded +feature vector is composed of. A larger +T4 value means more encoded features +are needed to describe data variability. +Scalability +Encoding Time +Time +Accumulated time in seconds during the +encoding task +Encoding Memory +Mem +Accumulated memory in megabytes +during the encoding task +Correlation power +Ratio of intra/extra +class near neighbor +distance +N2 +This measure is sensitive to how data +are distributed within classes and +labeling noise in the data. Low values +are indicative of simple problems. +F1-score +F1 +Average of F1-score obtained from the +PM classification task, representing the +predictive performance delivered by an +encoding method. +Domain agnosticism +General usage of +algorithm +DA +This is a binary evaluation (Agnostic or +Non-Agnostic) considering agnosticism +regarding the PM domain. +Table 2: List of criteria, strategies, and metrics to evaluate encoding methods in process mining. +Creating the log required simulating the process model. For that, we applied the ProM plug-in14 for the simulation of a +stochastic Petri net. We went through 10 thousand simulated cases, with a case arrival rate of about 30 minutes, and +kept the default values of the others hyperparameters. We injected anomalies, following [62], by perturbing regular +traces as proposed by [63], as in Table 3. +Anomaly +Description +skip +A sequence of 3 or fewer necessary +events are skipped +insert +3 or fewer random activities are +inserted in the case +rework +A sequence of 3 or fewer necessary +events is executed twice +early +A sequence of 2 or fewer events +executed too early, which is then +skipped later in the case +late +A sequence of 2 or fewer events +executed too late +all +Scenario where the event log is +affected by all anomalies listed +above +Table 3: Anomalies used to simulate the real-life event logs. +For each scenario, we injected different percentages of anomalies (5%, 10%, 15%, and 20%) by replacing normal traces. +A total of 420 event logs were generated given five process models, six scenarios of anomalies, and four anomaly +percentages using labels and descriptions as additional attributes. Labels regard a normal execution or an anomalous +one. The description attribute describes the anomaly and its impact on the case. A general overview is in Table 4. It is +important to note the different scenarios were created with increasing complexity and trace lengths and log sizes (1k, +5k, and 10k cases). +14http://www.promtools.org/doku.php +13 + +Trace Encoding in Process Mining +Barbon et al. +log +#gw +trace size +#acts +#cases (103) +#evts (103) +#vars (103) +scenario 1 +8 +9-13 +22 +1 +75 ± 1 +1 ± 0 +5 +378 ± 5 +2 ± 0 +10 +755 ± 9 +2 ± 1 +scenario 2 +12 +26-30 +41 +1 +186 ± 1 +1 ± 0 +5 +929 ± 5 +5 ± 0 +10 +1857 ± 10 +10 ± 0 +scenario 3 +22 +42-50 +64 +1 +308 ± 1 +1 ± 0 +5 +1538 ± 5 +5 ± 0 +10 +3077 ± 10 +10 ± 0 +scenario 4 +30 +3-30 +83 +1 +89 ± 3 +0 ± 0 +5 +439 ± 6 +2 ± 0 +10 +879 ± 12 +4 ± 0 +scenario 5 +34 +4-37 +103 +1 +133 ± 3 +1 ± 0 +5 +659 ± 7 +3 ± 0 +10 +1318 ± 13 +6 ± 0 +Table 4: Overview of five process models. For each scenario, we generated event logs by combining three different +cardinalities, injecting seven different anomalies, at four different rates of injection. This resulted in 84 event logs for +each scenario and 420 event logs in total. gw, acts, evts, and vars stand for the number of gateways, activities, events, +and variants, respectively. +6 +Benchmarking Process Mining Encoding +In this section, we report on the results achieved in our experiments for each family of encoding methods (Baseline, +Graph, Text, and Process Mining). +6.1 +Expressivity +In this work, the expressivity of encoding methods is based on PCA and T4 analysis. PCA models were calculated +using the encoded vector of all event logs for each encoding method, using a 2D sub-space projection of the first and the +second principal component to identify how complex is the mapped space. Since the original problem (i.e., the event +logs) is the same, differences in the distribution and density of the encoded events can lead to interpretations about +the mapping quality offered by each encoding method. In the PCA, each point represents a feature vector, and each +color represents a scenario. The depiction highlights the encoding capacity of generating feature vectors preserving +inter- and intra-traces similarities. This is demonstrated by the co-location of samples, i.e., encoded traces, of similar +scenarios, i.e., the same color points near to same color points in each 2D projection. A high level of expressivity +relies on non-overlapped clusters of samples from the same scenario with clusters sorted by scenarios’ complexity +occupying the whole sub-space projected. A low level of expressivity is associated with occluded samples with mixed +sparse distributions or dense overlapped allocation. An average expressivity is identified when the projected distribution +matches partially high and low characteristics. Very high and very low expressivity are obtained when an encoding +method completely matches the mentioned characteristics, positively or negatively. +Figure 5 shows the PCA projections of Baseline methods (count2vec, n-grams, position profile , and one-hot). Figure 5.a +and Figure 5.b, regarding count2vec and one-hot, look similar, assigning more or less the same areas to the same +scenarios, but keeping uncovered a significant part of the space. This is due to the methods being very related, with the +difference being that count2vec accounts for frequencies. Since most traces do not have a high number of repeated +activities, when reducing the dimensionality using PCA, the distances in the low-dimensional space are almost the +same. On the other hand, n-grams (Figure 5.c) and position profile (Figure 5.d) have an average expressivity as different +scenarios overlap on the same areas. +Figure 6 shows the projections of PM-based encoding methods. They can be assessed to very high and high expressivity +levels, respectively alignment (Figure 6.a) with the highest level, followed by token-replay (Figure 6.b) and Log skeleton +(Figure 6.c). It is worth observing that the distribution in the alignment projection follows the complexity of scenarios. +The Text encoding family presented average, low, and very low levels of expressivity, as shown in Figure 7. GloVe and +hash2vec are from average level, as observed in Figure 7.a and Figure 7.b. Low levels of expressivity were obtained +by TF-IDF (Figure 7.c), CBOW (Figure 7.d) and skip-gram (Figure 7.e). The worst expressivity level, very low, was +obtained by doc2vec (Figure 7.f). +Very high, high, and average were the levels observed when using the Graph encoding family (Figure 8). Average was +obtained by BoostNE (Figure 8.a) and role2vec (Figure 8.m). PCA projections that represented high expressivity were +computed from encoded vectors of DeepWalk, diff2vec, GLEE, GraRep, Hope, Laplacian Eigenmaps, NetMF, NMF- +ADMM, node2vec, NodeSketch and Walklets, respectively, Figure 8.b, Figure 8.c, Figure 8.d, Figure 8.f, Figure 8.g, +Figure 8.h, Figure 8.i, Figure 8.j, Figure 8.k, Figure 8.l, and Figure 8.n. Very high expressivity was observed using +14 + +Trace Encoding in Process Mining +Barbon et al. +−200 +−100 +0 +100 +200 +−2 +−1 +0 +1 +2 +3 +4 +Scenario +1 +2 +3 +4 +5 +0 +1 +−2 +0 +2 +−2 +−1 +0 +1 +2 +Scenario +1 +2 +3 +4 +5 +0 +1 +(a) count2vec +−2 +0 +2 +−2 +−1 +0 +1 +2 +Scenario +1 +2 +3 +4 +5 +0 +1 +(b) one-hot +0 +1 +2 +−0.5 +0 +0.5 +1 +1.5 +Scenario +1 +2 +3 +4 +5 +0 +1 +(c) n-grams +−500 +0 +500 +−200 +0 +200 +400 +Scenario +1 +2 +3 +4 +5 +0 +1 +(d) position profile +Figure 5: 2D projections based on PCA transformation of the feature vectors of Baseline Family encoding methods of +event logs from five different scenarios (1, 2, 3, 4, and 5). Each projection regards an encoding method, each point an +encoded trace and each color represents a scenario. +−200 +−100 +0 +100 +200 +−2 +−1 +0 +1 +2 +3 +4 +Scenario +1 +2 +3 +4 +5 +0 +1 +0 +50k +100k +−5k +0 +5k +10k +15k +Scenario +1 +2 +3 +4 +5 +0 +1 +(a) alignment +−50 +0 +50 +100 +−2 +0 +2 +4 +6 +Scenario +1 +2 +3 +4 +5 +0 +1 +(b) token-replay +−2000 +−1000 +0 +1000 +2000 +0 +5 +10 +15 +20 +25 +Scenario +1 +2 +3 +4 +5 +0 +1 +(c) log skeleton +Figure 6: 2D projections based on PCA transformation of the feature vectors of Process Mining encoding family +methods of event logs from five different scenarios (1, 2, 3, 4, and 5). Each projection regards an encoding method, +each point an encoded event log and each color represents a scenario. +−200 +−100 +0 +100 +200 +−2 +−1 +0 +1 +2 +3 +4 +Scenario +1 +2 +3 +4 +5 +0 +1 +0 +1 +−1 +−0.5 +0 +0.5 +1 +Scenario +1 +2 +3 +4 +5 +0 +1 +(a) GloVe +−0.5 +0 +0.5 +−0.4 +−0.2 +0 +0.2 +0.4 +0.6 +Scenario +1 +2 +3 +4 +5 +0 +1 +(b) hash2vec +−0.4 +−0.2 +0 +0.2 +0.4 +−0.4 +−0.2 +0 +0.2 +0.4 +0.6 +Scenario +1 +2 +3 +4 +5 +0 +1 +(c) TF-IDF +−0.005 +0 +0.005 +0.01 +0.015 +−0.005 +0 +0.005 +Scenario +1 +2 +3 +4 +5 +0 +1 +(d) CBOW +−0.005 +0 +0.005 +0.01 +−0.005 +0 +0.005 +0.01 +Scenario +1 +2 +3 +4 +5 +0 +1 +(e) skip-gram +−0.02 +0 +0.02 +−0.01 +0 +0.01 +0.02 +0.03 +Scenario +1 +2 +3 +4 +5 +0 +1 +(f) doc2vec +Figure 7: The figure illustrates the 2D projections based on PCA transformation of the feature vectors of Text encoding +family methods of event logs from five different scenarios (1, 2, 3, 4, and 5). Each projection regards an encoding +method, each point an encoded event log and each color represents a scenario. +15 + +:.Trace Encoding in Process Mining +Barbon et al. +GraphWave (Figure 8.e), where it is possible to observe an organized gradient by scenario complexity, all different +event logs are identified spread in the 2D projection. This result is confirmed by the measurements obtained with the T4 +measure. +−200 +−100 +0 +100 +200 +−2 +−1 +0 +1 +2 +3 +4 +Scenario +1 +2 +3 +4 +5 +0 +1 +0 +50k +−10k +0 +10k +20k +30k +Scenario +1 +2 +3 +4 +5 +0 +1 +(a) BoostNE +0 +2 +4 +−3 +−2 +−1 +0 +1 +2 +3 +Scenario +1 +2 +3 +4 +5 +0 +1 +(b) DeepWalk +0 +2 +−2 +−1 +0 +1 +2 +Scenario +1 +2 +3 +4 +5 +0 +1 +(c) diff2vec +0 +0.1 +0.2 +−0.05 +0 +0.05 +Scenario +1 +2 +3 +4 +5 +0 +1 +(d) GLEE +−0.05 +0 +0.05 +0.1 +−0.006 +−0.004 +−0.002 +0 +0.002 +0.004 +0.006 +Scenario +1 +2 +3 +4 +5 +0 +1 +(e) GraphWave +−20 +0 +20 +−10 +0 +10 +20 +Scenario +1 +2 +3 +4 +5 +0 +1 +(f) GraRep +−200 +0 +200 +−50 +0 +50 +100 +150 +200 +Scenario +1 +2 +3 +4 +5 +0 +1 +(g) Hope +0 +0.1 +0.2 +−0.05 +0 +0.05 +Scenario +1 +2 +3 +4 +5 +0 +1 +(h) Laplacian +−10 +0 +10 +−10 +−5 +0 +5 +10 +Scenario +1 +2 +3 +4 +5 +0 +1 +(i) NetMF +−2 +0 +2 +4 +−1 +0 +1 +2 +Scenario +1 +2 +3 +4 +5 +0 +1 +(j) NMF-ADMM +0 +2 +−2 +−1 +0 +1 +2 +3 +4 +Scenario +1 +2 +3 +4 +5 +0 +1 +(k) node2vec +−200 +0 +200 +400 +−50 +0 +50 +100 +150 +200 +Scenario +1 +2 +3 +4 +5 +0 +1 +(l) NodeSketch +−2 +0 +2 +4 +−2 +−1 +0 +1 +2 +Scenario +1 +2 +3 +4 +5 +0 +1 +(m) role2vec +−2 +0 +2 +4 +−2 +0 +2 +4 +Scenario +1 +2 +3 +4 +5 +0 +1 +(n) Walklets +Figure 8: 2D projections based on PCA transformation of the feature vectors of Graph encoding family methods of +event logs from five different scenarios (1, 2, 3, 4, and 5). Each projection regards an encoding method, each point an +encoded event log and each color represents a scenario. +T4 gives a rough measure, from 0 to 1, of the proportion of relevant dimensions used by the encoding vector to map the +event log [5]. Relevance is determined according to the PCA criterion, which strives to describe most of the variability in +the data with uncorrelated linear functions of the features [59]. A higher T4 value indicates a more complex relationship +between the input variables, indicating a larger number of original features are required to describe the data variability. +Graph-based methods obtained the best expressivity, followed by the Baseline family, as shown in Figure 9. In terms of +T4 value, doc2vec reached the lowest expressivity level, 0.92. +6.2 +Scalability +We drive the discussion of scalability considering the time (seconds) and memory (KB) consumption accumulated +during the whole encoding. Since costly methods are prohibitive to real-life event logs with huge volumes of data, their +time and memory costs can directly influence the choice of an encoding method. In our experiments, we considered +16 + +Trace Encoding in Process Mining +Barbon et al. +������� +��������� +������ +��������� +������ +����������� +����� +������� +��������� +����� +�������� +������������ +���� +���������� +������������� +�������� +�������� +�������� +����������������� +���������������� +�������� +�������� +������ +����� +���� +������������������ +������� +�������� +��� +��� +��� +��� +��� +�� +������ +�� +����� +�������� +���� +s +asd +PM +Graph +Baseline +Text +Figure 9: T4 value obtained from encoding methods for expressivity evaluation. Low T4 values are correlated to good +expressivity and high T4 values indicate poor expressivity. +the time and memory consumed only during the encoding task. By comparing three different versions of event log +size (1k, 5k, and 10k), we can observe how the costs are affected when encoding the same group of problems using +different methods. Figures 10, 11, 12 and 13 were used to demonstrate the scalability of time and memory, limiting the +y-axis according to the higher observed value (time on left-side and memory on right-side) over all experiments of each +encoding method. +Baseline family analyses are supported by Figure 10. In terms of memory cost, the Baseline encoding family showed +that n-grams was the most expensive method with a high space complexity. On the other hand, position profile was the +worst method in terms of time complexity. One-hot demonstrated that memory costs increase more than time as the +problem is scaled. In terms of scalability, count2vec presented the best scalability results of the Baseline family. +1000 +5000 +10000 +Event log size +0 +20 +40 +60 +80 +100 +Encoding Time (s) +Time Count2vec +Time N-grams +Time One-hot +Time. Position Profile +0 +1 +2 +3 +4 +5 +6 +7 +8 +Encoding Memory (KB) +1e8 +Mem. Count2vec +Mem. N-grams +Mem. One-hot +Mem. Position Profile +Figure 10: Time and memory costs across different event log sizes (1k, 5k and 10k) when using Baseline encoding +family. +17 + +Trace Encoding in Process Mining +Barbon et al. +The PM encoding family presented high memory costs and average time costs, but with good scalability in both +measures, as visible when the dataset increased and the performances slightly grew, as in Figure 11. Alignment method +was the cheapest one in terms of memory, token-replay presented a good balance in terms of time, and Log skeleton the +most costly in both, memory and time. +1000 +5000 +10000 +Event log size +0 +1000 +2000 +3000 +4000 +5000 +6000 +Encoding Time (s) +Time Alignment +Time Log skeleton +Time Token-replay +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Encoding Memory (KB) +1e8 +Mem. Alignment +Mem. Log skeleton +Mem. Token-replay +Figure 11: Time and memory costs across different event log sizes (1k, 5k and 10k) when using Process Mining +encoding family. +The Text encoding family presented the fastest methods with low memory consumption. However, presented time +scalability issues by the majority of methods (GloVe, hash2vec, CBOW, skip-gram and doc2vec), as represented by +Figure 12. The least scalable was doc2vec. A notable exception was TF-IDF, which presented reduced memory usage +even with increasing problem size. Note that the scalability was evaluated considering the ability to not suffer from data +growth, and the average usage of time and memory in the Text encoding family is the lowest. +1000 +5000 +10000 +Event log size +0 +2 +4 +6 +8 +Encoding Time (s) +Time Doc2Vec +Time GloVe +Time Hash2vec +Time TF-IDF +Mem. Cbow +Mem. SkipGram +0 +1 +2 +3 +4 +Encoding Memory (KB) +1e7 +Mem. Doc2Vec +Mem. GloVe +Mem. Hash2vec +Mem. TF-IDF +Mem. Cbow +Mem. SkipGram +Figure 12: Time and memory costs across different event log sizes (1k, 5k and 10k) when using Text encoding family. +18 + +Trace Encoding in Process Mining +Barbon et al. +The Graph encoding family presented a heterogeneous usage of memory and an average time cost, as Figure 13 shows. +GLEE, Hope and NetMF were the fastest methods of this family. These methods presented average scalability. The best +scalability was demonstrated by NodeSketch. The higher memory cost from the graph encoding family was achieved by +GraRep with a cost comparable to token-replay (PM family), but with average scalability regarding time. The slowest +method was node2vec, using the smallest event log it was presented 4 times the average of the other methods of the +same family. When dealing with the larger event log the time difference reached 7 times the other methods. +1000 +5000 +10000 +Event log size +0 +10 +20 +30 +40 +50 +60 +Encoding Time (s) +Time BoostNE +Time DeepWalk +Time Diff2vec +Time GLEE +Time GraphWave +Time GraRep +Time Hope +Time Laplacian Eigenmaps +Time NetMF +Time NMF-ADMM +Time Node2Vec +Time NodeSketch +Time Role2vec +Time Walklets +0 +1 +2 +3 +4 +5 +6 +7 +Encoding Memory (KB) +1e7 +Mem. BoostNE +Mem. DeepWalk +Mem. Diff2vec +Mem. GLEE +Mem. GraphWave +Mem. GraRep +Mem. Hope +Mem. Laplacian Eigenmaps +Mem. NetMF +Mem. NMF-ADMM +Mem. Node2Vec +Mem. NodeSketch +Mem. Role2vec +Mem. Walklets +Figure 13: Time and memory costs across different event log sizes (1k, 5k and 10k) when using Graph encoding family. +We organized the results of scalability and consumption of both time and memory analysis as a heat map (Figure 14). +In the figure, it is possible to observe that general low memory and little time consumption count2vec, do not reflects +the scalability, i.e., increasing event log sizes, some methods compromise their costs with quadratic complexity costs of +time and memory. Alternatively, Log skeleton, token-replay, and NodeSketch are very scalable but have a high memory +and time cost. +alignment +boostne +count2vec +deepwalk +diff2vec +doc2vec +glee +glove +graphwave +grarep +hash2vec +hope +laplacianeigenmaps +log_skeleton +netmf +ngrams +nmfadmm +node2vec +nodesketch +onehot +position_profile +role2vec +tfidf +tokenreplay +walklets +word2vec_cbow +word2vec_skipgram +encoding +Time Scalability +Memory Scalability +Time Consumption +Memory Consumption +7 14 24 11 12 27 21 17 5 +8 26 22 20 2 23 4 13 6 +1 16 25 10 15 3 +9 18 19 +21 10 25 13 14 12 24 7 17 2 19 20 23 3 22 11 6 15 5 +4 27 16 26 1 18 9 +8 +26 14 2 16 21 20 12 6 17 13 1 +9 11 27 10 5 15 23 22 4 24 18 3 25 19 8 +7 +20 22 8 +9 10 21 3 17 16 24 5 +4 +2 25 1 27 19 11 18 26 6 12 7 23 15 13 14 +Ranking +5 +10 +15 +20 +25 +Figure 14: Ranking of Time Scalability, Memory Scalability, Time Consumption, and Memory Consumption. The +better approaches are positioned in the first ranking position, colored by white. The most costly and less scalable are the +last potions with dark colors. +6.3 +Correlation power +Correlation is an important analysis perspective since it reflects how correlated an encoding method is to the executed +task in terms of performance. In other words, how the encoding positively contributed to the final performance. In +19 + +Trace Encoding in Process Mining +Barbon et al. +our benchmark, we evaluated the correlation based on an anomalous trace detection task. In particular, we evaluated +the F1-score obtained to detect anomalous behavior and the N2 from the mapped space. N2 is a ratio that computes +distances between an example and its closest neighbor within a particular class (intra-class) and between an example +and its closest neighbor from a different class (extra-class). The N2 value, which ranges from 0 to 1, is low when there +is a greater distance between examples of different classes than between examples from the same class. Thus, a mapped +space with a lower N2 refers to a representation that is better equipped to distinguish classes and support supervised +learning. In our paper, we followed the N2 calculation as described by [59]. +Figure 15 represents the obtained N2 values from encoded space from all encoding methods. The figure presents each +family with a particular color and results are sorted by N2, from the best one to the worst N2 value. Position profile +achieved the best N2 values, mean of 0.48, and was the only method with less than 0.50 in terms of this measure. +The top 5 N2 values were obtained by Graph and Text families. The PM encoding family, particularly alignment and +token-replay reached high N2 values, superior to 0.85. In contrast, Log skeleton obtained less than 0.70, ranking in the +bottom 10 encoding methods, but supported the best F1-score in the anomaly detection task. +���������������� +������� +����� +��������� +������ +�������� +�������� +����� +������������ +������� +�������� +�������� +������� +�������� +���� +������������������ +������������� +����� +����������������� +���� +������ +������ +��������� +���������� +�������� +��������� +����������� +�������� +��� +��� +��� +��� +��� +��� +�� +������ +�� +����� +�������� +���� +PM +Graph +Baseline +Text +Figure 15: Ratio of Intra/extra class nearest neighbor distance (N2), provided by the same group of tasks considering 27 +different encoding methods. The bars are sorted from the best score (left) to the worst one (right). Each bar is colored +according to the coding family. +When evaluating the F1-score, Log skeleton provided the best performance (mean of 0.942), followed by position +profile (mean of 0.935) and all encoding from the Graph encoding family (above 0.915). Text encoding family provided +results above 0.903, except doc2vec that lead to an average F1-score of about 0.845. Surprisingly, the most traditional +encoding used in PM, one-hot, obtained the worst results, i.e., inferior to 0.840 of the F1-score. The obtained F1-scores +are sorted by the performance from the best to the worst one in Figure 16. +In order to provide a fair and statistically grounded comparison in terms of predictive performance, we used Friedman’s +statistical test and the post-hoc test of Nemenyi. Both tests were employed to verify the statistically significant difference +between the performance of the F1-score of each encoding method. The result of the statistical comparison can be +observed as a Critical Difference chart, as illustrated in Figure 17. Critical Difference test allows checking when +there were statistical differences between the segmenters, each diagram and method’s average ranks are placed on +the horizontal axis, with the best ranked to the right. The solid line connects encoding methods with no significant +performance difference. Thus, Figure 17 demonstrates no statistical difference among Log skeleton, position profile, +20 + +Trace Encoding in Process Mining +Barbon et al. +������������ +���������������� +���������� +��������� +������� +������� +����� +���� +������������������ +���� +�������� +������ +�������� +�������� +�������� +�������� +������������� +����������������� +����� +����� +�������� +��������� +����������� +��������� +������ +������� +������ +�������� +����� +����� +����� +����� +����� +����� +����� +����� +�������� +������ +�� +�������� +���� +����� +PM +Baseline +Text +Graph +Figure 16: F1-score obtained by several classification tasks performed using a Random Forest algorithm, considering +27 different encoding tasks. The bars are sorted from the best score (left) to the worst one (right). Each bar is colored +according to the coding family. +NodeSketch, NMF-ADMM, GraphWave, BoostNE, NetMF, Walklets, Hope, GLEE, Laplacian Eigenmaps, node2vec, +all of them supporting high predictive performance. On the other hand, there is no statistical difference as encoding +methods that provided lesser predictive models for one-hot, doc2vec, n-grams, count2vec, token-replay, alignment, +hash2vec, GloVe, skip-gram, CBOW and TF-IDF. +6.4 +Domain agnosticism +The last comparison criterion regards Domain Agnosticism. We consider this criterion as important as Expressivity, +Scalability, and Correlation to comprehend the obtained results. Since a method valid for multiple applications is +instrumental to the construction of adaptive data science pipelines. +The methods comprising the baseline family are traditionally used in different PM tasks. They realize straightforward +transformations mapping traces into feature vectors. Their usage was not limited to PM pipelines, indeed, they were +created in other data mining areas. For these reasons, we consider the Baseline family as domain agnostic. As well, +the Text and Graph families have been used for a wide range of purposes. These last families are good examples of +domain agnosticism and have been used with simple encoding scenarios as well as with complex and highly structured +representations. +Considering our criteria, PM-based solutions are not domain agnostic. This family of encoding methods was conceived +to represent event logs and has been used exclusively for this purpose. It should not be viewed as a limitation, but rather +as a characteristic of specialized methods demonstrating a high correlation power, even at the cost of high computational +costs, e.g., Log skeleton provided encoded spaces to induce models that obtained high F1-scores. +An overview of domain agnosticism and some implications of encoding method performance and resource consumption +could be observed in Figure 18. The most notable observation is that non-agnostic methods (token-replay, alignment, +and Log skeleton) share high N2 values and high costs of space and time complexity. Among the agnostic methods, the +best N2 and F1 performances are obtained with average space and time complexity, e,g, with GraphWave or BoostNE. +21 + +Trace Encoding in Process Mining +Barbon et al. +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +25 +26 +27 +one-hot +doc2vec +count2vec +token-replay +n-grams +alignment +hash2vec +GloVe +skip-gram +TF-IDF +CBOW +role2vec +diff2vec +GraRep +DeepWalk +node2vec +Laplacian Eigenmaps +GLEE +Hope +Walklets +NetMF +BoostNE +GraphWave +NMF-ADMM +NodeSketch +position profile +Log skeleton +CD +Figure 17: Nemenyi post-hoc test (significance of α = 0.05 and critical distance of 10.71) considering the F1-score +(predictive performance) accuracy obtained when performing the classification task using all encoding methods over 10 +scenarios (1k, 5k and 10k traces from five different scenarios). Statically similar methods are linked by the solid line. +7 +Issues, Concerns and Future Directions +Research comparisons about encoding methods focused on PM are still embryonic. After classifying the papers in +Section 4, we observed that the proposed solutions do not investigate the strong and weak points of each method. Thus, +we have proposed a study involving a large set of methods from different families. +Investigating the pros and cons of encoding methods based on expressivity, scalability, correlation power, and domain +agnosticism over different encoding families and hundreds of event logs with various complexities, we were able to +gain insights and share some assumptions for future directions and possible novel PM encoding methods. We believe +the criteria employed in this work to assess the effectiveness are the most important aspects to take into consideration in +order to achieve significant accomplishments for any PM task that needs to encode event logs. +Moreover, the proposed metrics can also serve as user requirements when deciding which encoding method should +be employed for the specific problem. The expressivity of an encoding method can measure how straightforward the +representation of an event log is according to its complexity. The scalability is also an important concern since real-life +event logs consist of a large volume of data which can lead to high computational costs regarding elapsed time and +memory usage. Understanding the correlation power between the nature of an encoding method and the performance of +the executed task is also a relevant analysis that allows practitioners to estimate the complexity of analyzing a specific +event log. Lastly, the domain agnosticism is a novel and important discussion introduced in this work to consider if +encoding methods can be adapted for different problem domains. +Addressing further issues on encoding methods in PM, online PM may introduce new challenges for the current +encoding methods. As emphasized by [64], measures such as accuracy and memory consumption need to drive the +creation of methods to match online PM goals, as the encoding methods used in such solutions. The concern of +limitations posed by an online PM task was also highlighted by [65], mentioning also the demand of adapting when +dealing with concept drifts and focusing on inter-activity time implications. STARDUST [66] is an example of online +PM, particularly on trace streams. The authors discussed issues regarding approaches to handling traces recorded +without the final activity. It is therefore essential to investigate encoding methods that support online PM tasks coping +with new challenges, such as reduced memory consumption and the ability to map partial traces. In our experiments, +22 + +Trace Encoding in Process Mining +Barbon et al. +10 +1 +100 +101 +102 +103 +Time (log) +106 +107 +108 +Memory (log) +alignment +BoostNE +count2vec +DeepWalk +diff2vec +doc2vec +GLEE +GloVe +GraphWave +GraRep +hash2vec +Hope +Laplacian Eigenmaps +Log skeleton +NetMF +n-grams +NMF-ADMM +node2vec +NodeSketch +one-hot +position profile +role2vec +TF-IDF +token-replay +Walklets +CBOWskip-gram +F1 +0.84 +0.86 +0.88 +0.90 +0.92 +0.94 +N2 +0.45 +0.60 +0.75 +0.90 +non-agnostic +agnostic +Figure 18: Mean values of time, memory, and predictive performance (F1) projected by time and memory in a +logarithmic scale. The marker size represents the N2 and their color gradient performance in terms of F1. The marker +symbol represents the domain agnosticism evaluation. +we used scalability measure to support insights and discussions regarding this topic. A crucial indicator of scalability +was the encoding families. +Currently, the predictive process monitoring problem also faces this issue regarding the lack of encoding methods +specific to PM. Representation learning or feature learning is a learning paradigm that has been recently introduced in +the community by [10]. We believe this is a promising path for improving the encoding procedure in process mining +tasks. In the mentioned work, the authors derived their new proposals from the word2vec. However, we believe PM +requires a specialized method or a sufficiently generic one regardless of the problem domain since the event data might +be considered more complex than sequential text. We claim that since the nature of event data, in general, contains +sequential rules, relational information, concurrency of resources, parallel activities, etc. +Encoding regards mapping data into another representation for different goals. The new space could not allow +interpretations and explainability as previously supported by the original data. Moreover, practitioners need to trust the +generated mapped space, as mentioned by [67]. In particular, the predictive model presented in a great part of predictive +monitoring tasks does not explain why it provided wrong predictions, so the reason why a prediction model made a +mistake cannot be understood. Shedding some light on this topic, [68] presented post-hoc explainers and different +encoding methods for identifying the important features. On a general note, our experiments using expressivity and +domain agnosticism confirmed the wide range of representations provided by different encoding methods, even from +23 + +Trace Encoding in Process Mining +Barbon et al. +the same family. Regarding this point of eXplainable Artificial Intelligence (XAI), as a tendency, encoding methods +able to provide high explainability levels should be integrated into the PM pipeline as a key component, not a separate +step follow-up effort for particular applications. +The great number of encoding methods and reduced availability of experts pose an additional challenge to selecting +and properly setting the encoding method. Strategies focused on accuracy or time performance are applied when +selecting a method, but the cost of testing different setups and costly tuning strategies could impact the PM pipeline +conception. This problem has been addressed by promising strategies based on meta-learning [69, 70, 71], but the +current solutions require creating a meta-database containing the history of possible solutions. Also, the criteria to +recommend a particular algorithm are still limited to simple performance functions. Therefore, Automatic Machine +Learning (AutoML) [6, 8] proves to be an important research area that impacts the aforementioned concerns regarding +encoding methods used in PM tasks. Alternatively, another learning paradigm that could be explored for this nature of +data is self-supervised learning (SSL). The general idea behind SSL is learning a set of possible outcomes given an +input, instead of predicting a unique value as traditional methods. For instance, the data2vec was recently presented by +[72], where the authors propose a generic framework for encoding any type of data, although only the domains of image, +speech, and language have been considered. Intuitively, this might be interesting for capturing mutual dependencies in +event data. +8 +Conclusion +The main contributions presented in this work include a systematic review of process mining tasks using encoding +methods, a new taxonomy to categorize each type of method into families, and an extensive experimental evaluation +and benchmark assessing relevant evaluation metrics to measure the effectiveness of an encoding method. We believe +this work can support researchers and practitioners to achieve significant accomplishments in different application areas +in PM. Furthermore, we stress current challenges and issues in the literature regarding the difficulty of choosing the +right algorithm and its parameters. We also discuss how arbitrarily selecting algorithms lead to unfair evaluation and +sub-optimal solutions. This is the first work that focuses on a detailed analysis for preprocessing event logs instead of +focusing only on the task algorithm itself (i.e. a clustering or learning algorithm). +We also highlighted the need for a better understanding of how each method behaves according to different scenarios +of event logs and different PM tasks. Thus, to fill this gap we simulated such scenarios by employing the PLG2 +tool to generate synthetic processes with distinct properties and presented the results as a benchmark. In total, 27 +encoding methods were evaluated throughout 420 different event logs containing different anomalies. We considered +four different evaluation criteria to measure the effectiveness of encoding methods for process mining tasks. We limited +our evaluation to only one task, anomaly detection, but the analysis and insights presented in this work can be leveraged +for other applications, such as predictive monitoring and clustering. +We conclude this work by stressing the difficulty of choosing suitable algorithms and their parameters according to the +user’s preferences since each pipeline setting performs differently according to the event log characteristics. This might +be a direction to novel automated solutions whether for entire pipelines or preprocessing steps only. We also believe +that an encoding method that specifically handles the nature of processes is essential for advancing the state-of-the-art. +This is claimed by considering that most methods are adapted or adopted from other areas. Therefore, this research line +is promising and has several opportunities for work to be developed. +References +[1] Joerg Evermann, Jana-Rebecca Rehse, and Peter Fettke. 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PMLR, 2022. +28 + diff --git a/9NA0T4oBgHgl3EQfO_8B/content/tmp_files/load_file.txt b/9NA0T4oBgHgl3EQfO_8B/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0408a1ceccf97d2ec64558dc13874c77fa0c4e58 --- /dev/null +++ b/9NA0T4oBgHgl3EQfO_8B/content/tmp_files/load_file.txt @@ -0,0 +1,1661 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf,len=1660 +page_content='TRACE ENCODING IN PROCESS MINING: A SURVEY AND BENCHMARKING Sylvio Barbon Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' University of Trieste Trieste, Italy sylvio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='barbonjunior@units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='it Paolo Ceravolo, Rafael S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Oyamada, Gabriel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Tavares University of Milan Milan, Italy {paolo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='ceravolo, rafael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='oyamada, gabriel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='tavares}@unimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='it ABSTRACT Encoding methods are employed across several process mining tasks, including predictive process monitoring, anomalous case detection, trace clustering, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' These methods are usually performed as preprocessing steps and are responsible for transforming complex information into a numerical feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Most papers choose existing encoding methods arbitrarily or employ a strategy based on a specific expert knowledge domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Moreover, existing methods are employed by using their default hyperparameters without evaluating other options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' This practice can lead to several drawbacks, such as suboptimal performance and unfair comparisons with the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Therefore, this work aims at providing a comprehensive survey on event log encoding by comparing 27 methods, from different natures, in terms of expressivity, scalability, correlation, and domain agnosticism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' To the best of our knowledge, this is the most comprehensive study so far focusing on trace encoding in process mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' It contributes to maturing awareness about the role of trace encoding in process mining pipelines and sheds light on issues, concerns, and future research directions regarding the use of encoding methods to bridge the gap between machine learning models and process mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Keywords Encoding Methods · Process Mining · Anomaly Detection 1 Introduction Encoding methods are responsible for transforming complex information into a representative feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In process mining (PM), several tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', predictive process monitoring, trace clustering, anomaly detection, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=') must encode data before feeding specific algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' This step is crucial to account for the goals of a user correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For instance, if a problem demands a solution where interpretability and explainability are needed, the data should be encoded by methods that tend to accomplish those objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' On the other hand, if the most essential requirements are space or time complexity, the user should agree to lose part of the previous benefits to match these ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In the PM literature, most of the efforts have been dedicated to designing new algorithms and analytical methods but little attention has been given to the impact of encoding methods across the existing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For instance, in predictive process monitoring [1] used the word embedding method to map the cases of an event log into real-valued vectors, whereas [2] used the one-hot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' A custom function is adopted by [3], whereas the count2vec (occurrence frequencies of activities) is employed by [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, a researcher interested in comparing the results of these works is in front of a factor she cannot control, as the impact of encoding is not documented and the methods used are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Moreover, in this work, we emphasize that very few alternative encoding methods have been employed by the community and demonstrate that arbitrarily encoding data might bring suboptimal results and misalignment with the user’s goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='02167v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='LG] 5 Jan 2023 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' believe that a better understanding of the effect of encoding methods, according to the datasets’ characteristics, is decisive in developing more interpretable, explainable, robust, and accurate PM solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Using anomaly detection as a case study, we extend the results of our previous paper [5] by considering several aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' First, we increase the number of encoding methods and provide a new taxonomy to classify them according to different dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Second, we include more datasets, considering more types of anomalies, in order to increase the space of characteristics and achieve a better understanding of how each encoding method behaves according to the data properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Third, we employ evaluation criteria that are valuable for PM practitioners and can support the choice of the suitable encoding method according to their goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Lastly, we provide a systematic review of encoding methods across popular PM tasks: predictive monitoring, trace clustering, anomaly detection, online process mining, and security and privacy in PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' More specifically, we first highlight how difficult it is not just to choose a suitable encoding method but also its parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Subsequently, we perform an extensive experimental evaluation of 27 encoding methods with different parameters over 420 synthetic event logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We also discuss how current PM literature is limiting their experiments by not considering the impact that encoding methods have in any problem domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, we discuss our results and focus the contribution of our work on answering the following research questions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' How expressive is an encoding method for separating the problems’ classes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' What is the demand of time and memory to reach a suitable encoding method?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Is there any correlation between the encoding method and the performance achieved by algorithms in PM tasks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' How generic encoding methods are, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' can an encoding method be applied to any PM task?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We answer these questions by proposing specific evaluation metrics according to different criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Through an in-depth analysis, we consider the criteria expressivity, which aims at capturing patterns across different characteristics of the employed datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' scalability, which measures the elapsed time and the memory usage of encoding methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' correlation power, which maps the data characteristics to the algorithm performances;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' and the domain agnosticism, which considers if the encoding method depends or not on the problem domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We demonstrate through our extensive experimental evaluation how difficult it might be to choose a suitable encoding method since each of the evaluated metrics has a different best performing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, the main contributions of this work include: A systematic review of encoding methods in PM and a new taxonomy developed according to such review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The proposal of new evaluation metrics to measure the quality of encoding methods in PM tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' A deep experimental evaluation of several encoding methods never employed before in PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' A discussion of insights into future research on encoding for PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We organize the presentation of our work as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' First, in Section 2 we define the problem of choosing the right encoding method and its parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In Section 3 we provide the necessary background to understand this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In Section 4 we first present a systematic review of encoding methods in different process mining tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Subsequently, we introduce a new taxonomy for encoding event data, organize the encoding methods found by families of algorithms, describe each method, and discuss related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Section 5 describes the employed methodology to implement our experimental evaluation and Section 6 presents the carried experiments and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In Section 7 we discuss the main insights obtained in this work and provide future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We conclude our discussion in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 2 Problem Definition In this section, we address the problem of how arbitrarily employing encoding methods in PM tasks leads to sub-optimal performance and results in unfair evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Due to the wide range of encoding methods available nowadays, choosing one given a specific problem is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' This can be seen in the current literature, across different domains, with several automated solutions that have been proposed to decrease human intervention in the design of algorithms and data science pipelines [6, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In PM, we believe this is even more challenging due to the nature of event logs, where events can be described by both numerical and categorical attributes, are aggregated by cases, and are constrained by the control flow of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For example, the availability of a given amount of resources may be a precondition to observe an event (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', the execution of an activity) with dependencies to other preceding or concurrent events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Condensing all this information into a single encoding method is difficult, and, in practical terms, each method can only capture specific aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 2 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Usually, encoding methods for PM are adapted from other domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Simple techniques are often considered, for instance, the one-hot encoding scheme [2] or frequency-based encoding methods [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' To capture the sequential nature of event logs, methods originally proposed in the Natural Language Processing (NLP) community have been employed [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' However, while we can take into consideration the similarity between the sequential nature of traces and natural language sentences, there are also differences that must be discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For instance, NLP tasks usually handle a very large vocabulary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', a set of unique words or tokens, whereas processes are usually represented by considerably small vocabularies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', the business process activities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' As an attempt of capturing additional complexity, graph neural networks have been recently studied in the literature [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Convolutional neural networks have also been used for feature extraction [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Image-like data engineering methods have been introduced by [14, 15, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' More recently, several pipelines have approached domain-specific encoding methods, which we will further describe in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='6, that exploit derived features, such as the resource pool discovery algorithm used to encode event resources by [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In the context of our work, we stress that adopting the right encoding method and selecting optimal hyperparameters can directly impact the final performance of a given task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Moreover, evaluating a new algorithm, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', a trace clustering algorithm, by comparing it with other solutions but employing different data inputs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', different encoding steps preceding the clustering), produces an unfair evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' [17] stress this problem, highlighting that a given model cannot be compared with another if their implementations consider different feature spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' A brief example illustrating this issue can be found in [16], where the authors are approaching the problem of predictive monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In their evaluation, the authors employ baselines to compare their proposal with existing LSTM architectures, each one based on a different preprocessing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Regarding other predictive monitoring work, word embedding is employed in [16, 18, 19] whereas a traditional one-hot encoding was used in [20, 21, 22], preventing the comparison between these studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' This problem is exacerbated by the fact that the PM community lacks shared benchmarks to be used in algorithm evaluation and comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In order to briefly illustrate the impact of arbitrarily encoding an event log, we demonstrate in Figure 1 the following scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We compare two encoding algorithms, vary the parameter of vector dimensionality, apply them to two datasets with different characteristics, and measure the accuracy achieved by a Random Forest classifier regarding the anomaly detection problem1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The datasets have different cardinalities, different types of anomalies, and different rates of anomaly injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The first one has 10k traces, 444k events, and a 20% rate of insertion anomaly (a random activity is inserted in the trace), whereas the second has 5k traces, 221k events, and a 15% rate of rework anomaly (an activity is doubled in the trace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' As we can see in the Figure 1, for the first event log (Event log 1), encoding the data employing the Walklets method performed better than using the NMF-ADMM with low dimensionality and worse for medium and high dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In addition, the latter method presented a high accuracy variation for different dimensionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For the second event log (Event log 2), the Walklets encoding presented a more stable accuracy, while the NMF-ADMM achieved higher accuracy w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' the other dataset but always performed worse than Walklets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' This is just a brief example to demonstrate that there is no best encoding method for every dataset or default parameterization to apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Considering the nuances found regarding encoding methods, we address this existing limitation by evaluating encoding methods for PM tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In this work, we attempt to demonstrate in detail how different methods perform on several datasets with distinct characteristics and properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We first demonstrate through extensive experimental evaluation that each algorithm has distinct performances across different event logs and pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We employ several metrics in order to summarize the overall behavior of each method and focus the general evaluation on expressivity, scalability, correlation, and domain agnosticism, which will be further detailed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 3 Background Notions PM can be defined as a set of techniques to extract knowledge from event logs [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The goal is to provide analysis that uses event data to extract process-related insights, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' creating solutions that are specifically tailored for business processes and their stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, let us first consider Σ a universe of events, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' the set of all possible event identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Σ∗ denotes the set of all sequences over Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 (Event, Attribute) Events may have various attributes, such as timestamp, activity, resource, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Let AN be the set of attribute names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For any event e ∈ Σ and an attribute n ∈ AN, then #n(e) is the value of attribute n for event e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Typically, values are restricted to a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For example, #activity ∈ A, where A is the universe of the legal activities of a business process, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' {a, b, c, d, e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 1A detailed description of the material and methods is provided in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 3 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 0 50 100 150 200 250 Encoding dimensionality 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='00 Accuracy Event log 1 encoding NMF-ADMM Walklest 0 50 100 150 200 250 Encoding dimensionality Event log 2 encoding NMF-ADMM Walklest Figure 1: A brief example of performances achieved for two different datasets regarding the anomaly detection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For both datasets we fixed two graph-based encoding methods with different parametrizations regarding dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' With abuse of notation, we refer to the name of the activity of an event #activity(e) as the event itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus ⟨a, b, d⟩ denotes a trace of three subsequent events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' An event can also be denoted by its position in the sequence as ei with en the last event of this sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' A sequence of events composes a trace t ∈ Σ∗ and it can be defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2 (Trace, Subtrace) In a trace each event appears only once and time is non-decreasing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' for 1 ≤ i < j ≤ |t| : t(i) ̸= t(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' A trace can also be denoted as a function generating the corresponding event for each position of its sequence: t(i → n) �→ ⟨ei, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', en⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' A subtrace is a sequence t(i → j) where 0 < i ≤ j < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Now let C be the case universe, that is, the set of all possible identifiers of a business case execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' C is the domain of an attribute case ∈ AN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='3 (Case, Event Log) We denote a case ci ∈ C as ⟨a, b, d⟩ci, meaning that all events share the same case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For example, for ci we have #case(e1) = #case(e2) = #case(e3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' An event log L is a set of cases L ⊆ Σ∗ where each event appears only once in the log, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' for any two different cases the intersection of their events is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In PM, encoding is a crucial step for several tasks in order to project the information contained in an event log to another feature space before combining it with posterior algorithms such as clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In the context of this work, we approach the anomaly detection problem to benchmark encoding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, let M be a process model representing the event log and f a test function that indicates if a trace from a log L is an instance of a model M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, we can define the anomaly detection problem as follows: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='4 (Anomaly detection) Let f : L → {R, A} be a test function that evaluates whether a trace is regular (R) or anomalous (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' A trace is considered anomalous if it can not be completely parsed by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, f(t) = �Regular, if it can be replayed by M Anomalous, otherwise (1) Considering the particular granularity of PM data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', traces consisting of events containing numerical, categorical, and time-like values, in this paper, we propose a new taxonomy of methods that handle event data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 4 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 4 Encoding Methods A literature review guided us in proposing a taxonomy of encoding methods, which will be discussed in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' To the best of our knowledge, this work is the first in the PM literature to propose a systematic review of encoding methods for PM tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' There are surveys and benchmarks for specific groups of algorithms, for example regarding graph embedding [24] or text embedding [25], but they fall outside the scope of PM applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In PM, different tasks need to employ an encoding method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' we focus our review on trace clustering, predictive monitoring, and anomaly detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 Systematic Review We performed a systematic review by analyzing the methods adopted in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The online repositories employed are the ACM Digital Library2, the IEEE Xplore3, and the Scopus4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We did not include Google Scholar in order to narrow our search since it usually captures the same papers as the other repositories, plus papers from unknown databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Moreover, we searched only for works from the last 10 years with respect to the date time this review was performed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' from 2012 to 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' A base query was defined and it was partially modified according to each PM task: "process mining" AND (“encoding” OR “encode”) AND < task >, where the keyword task might be “clustering”, (“predictive monitoring” OR “Process monitoring”), (“anomaly detection” OR “conformance-checking”), (“online process mining” OR “stream process mining”), or (“security” AND “privacy”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Notice that we are including the terms conformance-checking and anomaly detection interchangeably since anomaly detection can be seen as a sub-task of conformance-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' After filtering by including only conference and journal papers, and dropping duplicates, we examined the abstracts of each retrieved document to eliminate irrelevant papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We achieved a total of 616 papers, where 208 are included as clustering (CLUS), 165 as predictive process monitoring (PPM), 144 as anomaly detection (AD), 51 as online process mining (OPM), and 48 as security (SEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We illustrate the number of publications for each task and per year in Figure 2a and the total publications for each task in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' All the retrieved works are public available5 Publications over the past ten years 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 0 5 10 15 20 25 30 35 # publications SEC OPM AD PPM CLUS (a) SEC OPM AD PPM CLUS 0 25 50 75 100 125 150 175 200 (b) Figure 2: (a) Number of publications each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' (b) The total number of publications over the past ten years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 2https://dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='org/ 3https://ieeexplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='org/Xplore/home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='jsp 4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='scopus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='com/search 5shorturl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='at/uwJNW 5 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2 Taxonomy To guide our discussion, the methods reviewed are organized into a taxonomy, presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' First, in Figure 3, we illustrate all existing classes on the encoding problem and the intersections among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We can think about encoding at the control-flow or data-flow level [26], where the former considers only the event data whereas the latter analyzes the remaining data from the case [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Inter-case and intra-case terminologies have recently been proposed [27] to partially cover this difference by encoding each flow type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' More specifically, the inter-case level aims to capture relationships among cases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' classify case types and similarities between cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' One motivation behind this concept is the need to distinguish, for example, two identical sequences of activities (prefixes) with different labels (next activity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For intra-case encoding, the focus is to represent past executions by encoding individual activities or completed traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Despite being more relevant for real scenarios, the former one is less approached in existing machine learning-based applications since it was recently proposed by [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' On the other hand, intra-case encoding is mostly employed across different tasks in PM, and for this reason, we focus our contribution on this encoding level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' E1 T2 D2 W2 E2 T1 D1 W1 A B Workflow W1: Control-flow W2: Data-flow Dependency D1: Intra-case D2: Inter-case Trace T1: trace directly T2: trace by aggregating events Event Attributes E1: categorical E2: numerical/time A W1: Control-flow D1: Inter-case R2: trace by aggregating events E1: categorical B W1: Control-flow D1: Inter-case R1: trace directly E1: categorical Figure 3: General taxonomy for event log encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The encoding can be at the event attribute level, where each attribute is encoded independently, or at the trace level, where the event attributes encoded are aggregated to summarize the entire trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Furthermore, at this level, there might be specific targets when encoding data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For instance, event attributes might need to be encoded individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' This is a common setting in predictive process monitoring, where each activity is encoded in order to predict the next one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' On the other hand, a specific application (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' clustering tasks) might need to encode the complete trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In this scenario, the trace can be encoded in a straightforward fashion by the employed algorithm or it can be encoded by aggregating the individual event attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Following our systematic review of encoding methods in process mining, the plethora of alternative encoding methods in the literature, and our proposed diagram of encoding, we are able to wrap the insights from this study and organize the encoding methods into different families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The taxonomy proposed in Figure 3 illustrates the intersections among each possible scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In the figure, we point to only two intersections (A and B) since those are the most common in the literature, although we expect that future works might fill the other ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Some of the intersections, if not logically impossible are hard to be achieved, for example, a method using both control- and data-flow or intra- and inter-case is of complex design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In any case the fact only two intersections are covering the entire set of encoding methods we surveyed is significant of the potential for new methods to experiment in PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For example, none of the encoding methods we surveyed is exploiting the temporal dimension of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' By leveraging our systematic review, we can extend the pointed 6 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' intersections to group the found encoding methods into families as illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, these families can be based on or inspired by techniques derived from the following research fields: process mining, text mining, and graph embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Moreover, Figure 4 also presents the difference between both intersections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' how the trace encoding should be performed: if it must be encoded directly by a given algorithm or by aggregating previous encoded event attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Taking into consideration the families of encoding methods found in the literature, in the following subsections, we describe each encoding method according to its respective families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For this work, most of the methods employed have never been used in PM tasks to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Furthermore, to motivate researchers and practitioners to consider these alternative methods more often, we only include methods that have open-source implementations in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Encoding PM Text Graph Event attributes Aggregation Encoded attribute Encoded trace Figure 4: Different levels of data encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' First, event attributes might be individually encoded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Second, the trace can be encoded directly by a certain algorithm or it can be encoded by aggregating the encoded event attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='3 PM-based Encoding Given an event log, we retrieve its respective process model and perform conformance-checking techniques to measure its adherence to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Each trace in the event log is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The results produced are employed as the encoded representation of the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The methods considered in our survey are illustrated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Trace-replay: given a process model, traces are replayed in it to obtain values that measure its conformance [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' More specifically, the values accumulated at each step are the number of tokens correctly consumed (c), the number of tokens correctly produced (p), the number of missing tokens to execute the event in the next step (m), and the number of unconsumed tokens after the last event execution (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, the final measure defined by the trace-replay metric is given by fitness = 1 2(1 − m c ) + 1 2(1 − r p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' All the values produced, ⟨c, p, m, r, fitness⟩, are used as the feature vector of a given trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Trace alignment: performs a comparison between the process model and a trace and relates the trace to valid execution sequences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', allowed by the model [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' An alignment is a sequence of moves that can be synchronous, model- dependent, or log-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' It is also important to note that more than one alignment between the log and model is possible, and techniques aim at finding the optimal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The final feature vector is composed of the cost of the alignment, the number of visited states, the number of queued states, the number of traversed arcs, and the fitness value produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 7 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Log skeleton: this technique aims at summarizing activity traces by capturing a set of constraints that apply to activities throughout the log [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For example, the Req L captures the equivalence relation between two activities, which exists if both activities have the same frequency of occurrence in every trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' On the other hand, the Cdf L counts the number of directly-follows occurrences for every pair of activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Other examples of measures to capture relations include the always-after and never-together;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' examples of countermeasures include the sum of occurrences of a given activity in the entire log and the min and max numbers of occurrences of an activity in any trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In the implementation used for this paper, six different constraints are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Position profile: this technique represents an event log through a matrix, where each position refers to the activity × position regarding all traces [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' It can be formally defined as a triple apf = (a, p, f) ∈ E, where a is the activity, p is the position of the activity, f is the frequency of occurrence of the given activity, and E is the universe of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='4 Text-inspired Encoding Many solutions used for trace encoding in PM are adapted from methods used in NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Exploiting the fact that words in sentences are ordered in sequence and are constrained by dependencies, encoding methods applied to text capture that information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Because traces are composed of sequences of activities the same information appears relevant to characterize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In particular, in our survey, we consider the following methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' N-grams [32]: this method represents a given sequence of elements through sub-sequences of n items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, considering a sequence s = {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', si}, the n-grams representation of these sequences is given by n-grams = {(s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', sn), (s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', sn+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='(si−n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', si)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' One-hot [33]: given a variable containing n different values, the variable is transformed into an array where each unique value is represented as a binary vector with the i-th position set to one and the rest set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Clearly, the dimension of the vector depends on the size n of the unique values in the vector space, easily reaching high dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' CountVectorizer (count2vec) [33]: given a collection of categorical documents, this method produces a matrix of token occurrences, where each line in the matrix represents a document and each column a token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The size of the vector space depends on the n unique values in the vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' HashVectorizer (hash2vec) [33]: it does the same as count2vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' However, instead of storing tokens, it directly maps each token to a column position in the matrix of occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' It is mainly useful for large datasets, and unlike one-hot and count2vec, which have the same dimensionality as the vocabulary length, this method has the flexibility to hash tokens in any dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' TF-IDF [34]: the term frequency (TF) captures the frequency of a particular token w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' to a given document, whereas the inverse document frequency (IDF) measures how common the token is in the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' TF can be simply the number of times the token appears and the IDF is calculated as follows: idf(t, D) = log( N count(d∈D:t∈d)), where t is the token and N is the number of documents d in the corpus D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, the TF-IDF is obtained by multiplying both TF-IDF(t, d, D) = tf(t, d) × idf(t, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Word2vec [35, 36]: the main contribution behind word2vec was learning distributed representations of words and reducing the computational cost compared to the state of the art at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Although there are two original model architectures for learning the word vectors, Continuous Bag-of-Words (CBOW) and Continuous Skip-gram Model (skip- gram), the core characteristic of word2vec is the removal of the hidden layer of a simple Neural Net Language Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' CBOW predicts the current word based on the t words around it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', it predicts wt given (wt−i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', wt−1, wt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='wt+i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' On the other hand, given wt, the skip-gram predicts the surrounding words (wt−i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', wt − 1, wt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='wt+i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The parameter i in both cases is a parameter representing a range surrounding the current word wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Doc2vec [37]: this algorithm is an extension of word2vec and learns the embeddings of documents (sentence, paragraph, essay, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The difference w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' word2vec is given by the learning which is performed via the distributed memory and distributed bag of words models and by adding another vector (document ID) to the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' GloVe [38]: this is an unsupervised learning algorithm for obtaining vector representations for words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The main intuition behind this model is the capturing ratios of word-word co-occurrence probabilities in order to capture both local and global dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' This is expressed by F(wi, wj, wk) = Pik Pjk , where Pik and Pjk are the probabilities that the word k appears in the context of words i and j respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 8 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 Graph-based Encoding The intuition behind graph embedding methods is to represent nodes of a graph as low dimensional vectors, where such vectors are representative enough to keep its original relations (edges) intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We can formally define the general idea as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' A graph can be described as G = (V, E), where V = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', vn} is a set of vertices (nodes) and E is a set of edges e = (u, v) that connect a pair of vertices u, v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Given a graph G, a graph embedding is a mapping function f : vi → yi ∈ Rd, such that d << |v| and f preserves the original structure of their local neighborhood and minimizes the information loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In this section, we describe graph embedding methods for event log encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' DeepWalk [39]: it can be seen as a two-stage algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' First, a discovery of the local structure is performed through random walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' There are two parameters here, the number of random walks α and the number of vertices to visit t for each random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Second, similar to the word2vec, the skip-gram is performed to learn the embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The intuition behind this algorithm is learning embeddings close to each other if they often occur in a similar structural context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Node2vec [40]: this algorithm is similar to DeepWalk, where the difference is a biased-random walk that aims at employing a trade-off between breadth-first and depth-first searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In practice, such balance is capable of providing more informative embeddings than DeepWalk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Walklets [41]: while DeepWalk and node2vec implicitly capture a certain level of dependencies by generating multiple random walks through Ak ij, this algorithm does explicitly by combining factorization approaches with random walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' It preserves dependencies by sub-sampling short random walks on the vertices and by skipping over steps in each random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' This results in paths of fixed lengths composing sets of pairs of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, these sets are used to learn the latent representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' role2vec [42]: this is a framework that uses random walks to approximate the pointwise mutual information matrix, which is obtained by multiplying a matrix of structural features with the pooled adjacency power matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Laplacian Eigenmaps [43]: this algorithm intuitively keeps the embedding of two nodes close when the weight Wij is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Given a graph G, this algorithm computes eigenvalues and eigenvectors Ly = λDy, where D is a diagonal weight matrix Dii = � j Wji, and W is the weight matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, L = D − W is the Laplacian matrix that can be used to minimize the function ρ(Y ) = 1 2 � |Yi − Yj|2Wij = tr(Y T LY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' GraRep [44]: this algorithms learns the latent representation W ∈ R|V |×d of the vertices of the weighted graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' It leverages global structural information to capture long-distance connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The overall idea is first to calculate the transition probability matrix A = D−1S for each k, where 1 <= k <= K is the maximum transition step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Sub- sequently, obtain each k-step representation by factorizing the log probability matrix using singular value decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Finally, the k-step representations for each vertex on the graph are concatenated and used as latent representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Hope [45]: this embedding algorithm is similar to GraRep, but instead of using the transition probability matrix, it employs a similarity matrix S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, S can be obtained by using different similarity measures and consequently preserves higher-order dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' BoostNE [46]: this algorithm performs a non-negative matrix factorization to calculate the residuals generated by previous embedding models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' It assumes the same idea as the gradient boosting method in ensemble learning, where multiple weak learners lead to a better one when aggregated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Given a connectivity matrix obtained through the adjacency matrix of the graph, the algorithm calculates k residual matrices and uses each one as input to the next one using the following equation: Ri = �X, if i = 1 max(Ri−1 − Ui−1Vi−1, 0), if i ≥ 2 (2) where Ui ∈ Rn×ds + and Vi ∈ Rn×ds + intuitively act like the embedding representation of the center node and the context node in the i − th level, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Assuming the defined residual matrix, the embedding representation at the i − th level is obtained by minimizing the loss function L = minUi,Vi,≥0 ||Ri − UiVi||2 F , for 1 <= i <= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Diff2vec [47]: the overall idea of this algorithm is sub-sampling diffusion graphs for each node in a graph and generating sequences of vertices through an Euler tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Given a graph G, a graph G′ of l vertices is sub-sampled in a diffusion-like random process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Then, from G′, sequences of vertices are generated by performing an Euler walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In this process, G′ is first converted to a multi-graph by doubling each edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, the Euler walk is employed instead of the random walk since this algorithm can capture a more complete view in graphs with this characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The generated sequences of vertices are then used to create the graph embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 9 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' GLEE [48]: unlike most graph embedding algorithms that expect similar nodes to have their embeddings close to each other, this algorithm uses the Laplacian matrix of a given graph to find an embedding with geometric properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Examples of such properties are dot product (angle), length (area or volume) of a line segment (or polygon), the convex hull of a set of vectors, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, given a graph G and its Laplacian matrix L, this procedure extracts eigenvectors corresponding to the largest eigenvalues in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' These vectors are used as node embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' NetMF [49]: this method is built on a theoretical analysis that shows the equivalence of different graph embedding algorithms based on DeepWalk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In the original paper, the authors show that methods that use negative samplings, such as DeepWalk and node2vec, implicitly perform matrix factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, the framework NetMF is proposed to unify existing methods and perform an explicit factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' NMF-ADMM [50]: given an adjacency matrix, the NMF-ADMM algorithm learns the embeddings by using the alternating direction method of multipliers to solve the negative matrix factorization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' GraphWave [51]: given an undirected graph G = (V, E), an adjacency matrix A (binary or weighted), and a degree matrix Dii = � j Aij, this method learns a structural embedding of every vertex v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The learning is performed in an unsupervised fashion based on spectral graph wavelets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' GraphWave is given by GraphWave = U Diag(gs(λ1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', gs(λn))U T αv, where αv is the one-hot vector for the vertex v, U the decomposition of the eigenvector from A, and gs is a kernel that modulates the eigenspectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' NodeSketch [52]: this method recursively generates k-order node embeddings in a recursive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' These embeddings are categorized into low-order (k = 2) and high-order (k > 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' At each step k, a Self-Loop-Augmented (SLA) adjacency matrix is generated to obtain the embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Low-order SLA is obtained by simply adding the identity matrix to the original adjacency matrix M ′ = M + I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' On the other hand, high-order embeddings first sketch an approximate k-order SLA adjacency of the current nodes and merge it with the (k − 1)-order SLA adjacency matrix in a weighted manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='6 Related Works Several tasks in PM, such as predictive monitoring, trace clustering, and anomaly detection, need to encode data to feed algorithms that are applied down the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Although transforming event data into a reasonable feature space is a sensible task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', it might drastically impact algorithms’ performances, very little attention has been given to encoding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Regarding the literature of other problem domains, there are surveys and benchmarks trying to standardize and better investigate the behaviors of encoding methods according to different problems’ characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For instance, [24] surveyed several graph-based embedding methods on different datasets and discussed the main challenges for future research in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Regardless of the approached task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' link prediction, node classification, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' ), the authors demonstrate the difficulty of choosing not only the right algorithm but also the right set of parameters (mainly the dimensionality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Several trade-offs must always be taken into consideration, for instance increasing the memory usage to achieve more precision or decreasing the dimensionality to decrease the computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' On the other hand, [53] covered a wide range of methods to encode textual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The work focuses on methods based on encoding methods to feed neural network architectures regarding different tasks and also provides historical notes for each category of task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The aforementioned works usually focus on representational learning, which employs neural networks to learn a high-quality representation (encoding) of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In the natural language literature, the word2vec [35, 36] can be seen as one of the most important methods for this purpose, which has two architectures variants, one using the CBOW algorithm and another one using the skip-gram model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' From this perspective, several methods derived from it, for instance, [37, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The resulting feature vectors representing the original data are also called embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Recently, representational learning has been applied in PM as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' [10] proposes the act2vec, trace2vec, log2vec, and model2vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Each approach derives from existing encoding methods in the literature and leverages the previous level information to enrich the learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' That is, the first level is act2vec, which extends the word2vec architecture to learn the representation of activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Subsequently, the trace2vec adopts the doc2vec concept and jointly learns the representation of activities and traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The log2vec architecture derives from the same idea as trace2vec where the log representation is included in the architecture to be jointly learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Finally, for model2vec, the authors extend graph representation learning techniques to represent a process model discovered from the event log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The final architecture also includes all the previous representations to be learned jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In the literature, we also find “hand-crafted” methods, which are usually developed by following some expertise domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' [12] proposes using graph convolutional networks for predictive monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In their approach, the authors first perform a feature engineering step to handle time features and then transforms each activity in an event log into a matrix num_unique_activities × num_time_features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' [16] employs a PM algorithm to encode resources in 10 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' event logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In a nutshell, the applied algorithm is able to automatically discover resource pools and, hence, reduce the dimensionality of categorical values by grouping them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In [14], the authors propose the use of convolutional neural networks to perform predictive monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, they transform the data into an image-like structure in order to be able to train the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' [54] presents a method for feature extraction that can be seen as an encoding method, where seven different features are extracted from each activity given a Petri net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' These features aim at capturing local information for the activity with respect to its current case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Although recent works in predictive process monitoring have explored more alternatives, it is noticeable that works in PM often use a minimal variety of encoding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Most papers use naive techniques like one-hot encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Moreover, other encoding approaches are usually combined with results from feature engineer procedures that handle numerical and time-related information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In recent works, the most common encoding methods for different tasks include the one-hot [2, 22, 55, 21, 19], counting the frequencies of categorical data [9], some type of embedding network [18, 56, 16, 19], or hand-crafted representations [16, 12, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Therefore, we motivate our work in order to fulfill this limitation by exploring a wider range of encoding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 5 Methodology In this section, we describe the experimental analysis carried out to evaluate encoding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We provide details on the software and materials and on the metrics used in order to assess the quality of the surveyed encoding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 Implementation Overview Our implementation can be organized into three steps: (i) dataset preparation, (ii) encoding generation, and (iii) evaluation of the encoding methods from multiple perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The source code is available online in this repository6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' First, we generated synthetic logs using the PLG2 tool [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Subsequently, the encoding of the generated logs was performed using open-source libraries in Python as described in Table 1, which include Sklearn7, Karate Club8, PM4PY9, NLTK10, Gensim11, GloVe12, and the position profile implementation on github13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We organize each method according to the proposed taxonomy and provide the respective references for original papers and online implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Moreover, we set as baselines the methods count2vec, one-hot, n-grams, and position profile, which implement the most simple transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In the case of event-level encoding, the procedure was first performed at the activity level and then the results were aggregated to obtain the trace representation (trace-level encoding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' This aggregation takes the resulting encoded information of each activity and averages it into a unique feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For graph-based methods, this aggregation was obtained in two different ways: from edges or from nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2 Evaluation Metrics Assuming that encoding methods are used to map the original problem space into a different vector space, we observed the quality of the new space based on several criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Moreover, each encoding method has particularities regarding performance delivered, descriptive capability, computational cost, and complexity of hyperparameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, to be effective, an encoding method should meet the following criteria: Expressivity: the relative capacity of an encoding method to affect the complexity of the mapped space regarding the original problem space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' An encoding method should be able to map the event logs of varying complexity, in which a straightforward representation regards a simple process and an intricate representation concerns a complex process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Scalability: the property related to increasing or decreasing the encoding computational cost in response to changes in the event log size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The encoding method should be able to map the event log quickly, without compromising the PM pipeline run time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 6https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='com/gbrltv/business_process_encoding 7https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='com/scikit-learn/scikit-learn 8https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='com/benedekrozemberczki/karateclub 9https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='com/pm4py/pm4py-core 10https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='com/nltk/nltk 11https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='com/RaRe-Technologies/gensim 12https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='com/maciejkula/glove-python 13https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='com/gbrltv/meta_trace_clustering/blob/main/clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='py#L64 11 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Family ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Implementation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='count2vec [33] Baseline ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Sklearn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='n-grams [32] Baseline ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='NLTK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='position profile [31] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2017 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Baseline ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='GitHub ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='one-hot [33] Baseline ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Sklearn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='GraphWave [51] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2018 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Karate Club ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Laplacian Eigenmaps [43] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2001 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Karate Club ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='NMF-ADMM [50] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2014 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Karate Club ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='DeepWalk [39] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2014 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Karate Club ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Walklets [41] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2017 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Karate Club ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='role2vec [42] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2018 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Karate Club ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='NetMF [49] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2018 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Karate Club ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='NodeSketch [52] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2019 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Karate Club ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='BoostNE [46] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2019 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Karate Club ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='GLEE [48] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2020 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Karate Club ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Hope [45] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2016 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Karate Club ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='diff2vec [47] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2018 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Karate Club ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Log skeleton [30] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2018 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='PM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='PM4PY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='token-replay [23] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2016 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='PM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='PM4PY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='alignment [23] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2016 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='PM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='PM4PY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='word2vec (skip-gram) [35] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2013 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Gensim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='hash2vec [33] Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Sklearn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='GloVe [38] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2014 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='GloVe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='doc2vec [37] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2014 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Gensim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='word2vec (CBOW) [36] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2013 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Gensim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='TF-IDF [34] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1958 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Sklearn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Table 1: Encoding methods and related details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Correlation power: the capacity of an encoding method to improve the original problem space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The new feature vector needs to be highly correlated to the PM task goal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', the encoded feature vector should enhance the performance of PM tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Domain agnosticism: refers to how well a given encoding method maps data from different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Encoding methods that are non-agnostic can be used only in specific applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' There are different strategies and metrics to assess encoding methods considering the presented criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In this work, we exploit the followings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We exploited Principal Component Analysis (PCA) [58] to verify how well a vector space can be compressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Classification complexity metrics [59] to measure how well samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', encoded traces, are distributed within classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The F1-score [60] to observe the impact of encoding methods on accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Time and space complexity to assess the computational performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Table 2 summarizes the contribution of each measure we exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='3 Experimental Design Our experimental design relies on labeled data for ground truth evaluation of the compared encoding methods, as an extension of [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Synthetic event logs were generated based on standard PM research practices and anomalies were injected into the generated traces, representing an anomaly detection PM task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Afterward, traces were labeled as anomalous or normal, making our data set suitable for supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Our dataset was made more realistic by adding heterogeneous behaviors to the event logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' PLG2 [57] was used to create five different process models by performing a random generation of a process capable of capturing several behaviors, such as sequential, parallel, and iterative control-flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The rationale of PLG2 is based on the combination of traditional control-flow patterns [61], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', sequence, parallel split, and synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In order to simulate real-world scenarios, the patterns are progressively combined according to predetermined rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Each of the five generated process models defines five different base scenarios based on the activities and gateways included in the scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 12 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Criteria Analysis Acronym Description Expressivity Principal Component Analysis PCA Using the 2D projection of a PCA space it is possible to observe patterns across scenarios from different complexities, ranging from very low, low, average, high and very high expressivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Ratio of the PCA dimension to the original dimension T4 This measure is related to the proportion of relevant dimensions that the coded feature vector is composed of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' A larger T4 value means more encoded features are needed to describe data variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Scalability Encoding Time Time Accumulated time in seconds during the encoding task Encoding Memory Mem Accumulated memory in megabytes during the encoding task Correlation power Ratio of intra/extra class near neighbor distance N2 This measure is sensitive to how data are distributed within classes and labeling noise in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Low values are indicative of simple problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' F1-score F1 Average of F1-score obtained from the PM classification task, representing the predictive performance delivered by an encoding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Domain agnosticism General usage of algorithm DA This is a binary evaluation (Agnostic or Non-Agnostic) considering agnosticism regarding the PM domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Table 2: List of criteria, strategies, and metrics to evaluate encoding methods in process mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Creating the log required simulating the process model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For that, we applied the ProM plug-in14 for the simulation of a stochastic Petri net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We went through 10 thousand simulated cases, with a case arrival rate of about 30 minutes, and kept the default values of the others hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We injected anomalies, following [62], by perturbing regular traces as proposed by [63], as in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Anomaly Description skip A sequence of 3 or fewer necessary events are skipped insert 3 or fewer random activities are inserted in the case rework A sequence of 3 or fewer necessary events is executed twice early A sequence of 2 or fewer events executed too early,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' which is then skipped later in the case late A sequence of 2 or fewer events executed too late all Scenario where the event log is affected by all anomalies listed above Table 3: Anomalies used to simulate the real-life event logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For each scenario, we injected different percentages of anomalies (5%, 10%, 15%, and 20%) by replacing normal traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' A total of 420 event logs were generated given five process models, six scenarios of anomalies, and four anomaly percentages using labels and descriptions as additional attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Labels regard a normal execution or an anomalous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The description attribute describes the anomaly and its impact on the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' A general overview is in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' It is important to note the different scenarios were created with increasing complexity and trace lengths and log sizes (1k, 5k, and 10k cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 14http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='promtools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='org/doku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='php 13 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='#gw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='trace size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='#acts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='#cases (103) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='#evts (103) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='#vars (103) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='scenario 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='9-13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='75 ± 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 ± 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='378 ± 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2 ± 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='755 ± 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2 ± 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='scenario 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='26-30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='186 ± 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 ± 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='929 ± 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 ± 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1857 ± 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='10 ± 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='scenario 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='42-50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='308 ± 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 ± 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1538 ± 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 ± 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='3077 ± 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='10 ± 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='scenario 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='3-30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='83 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='89 ± 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='0 ± 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='439 ± 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2 ± 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='879 ± 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='4 ± 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='scenario 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='4-37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='133 ± 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 ± 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='659 ± 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='3 ± 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1318 ± 13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='6 ± 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Table 4: Overview of five process models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For each scenario, we generated event logs by combining three different cardinalities, injecting seven different anomalies, at four different rates of injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' This resulted in 84 event logs for each scenario and 420 event logs in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' gw, acts, evts, and vars stand for the number of gateways, activities, events, and variants, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 6 Benchmarking Process Mining Encoding In this section, we report on the results achieved in our experiments for each family of encoding methods (Baseline, Graph, Text, and Process Mining).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 Expressivity In this work, the expressivity of encoding methods is based on PCA and T4 analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' PCA models were calculated using the encoded vector of all event logs for each encoding method, using a 2D sub-space projection of the first and the second principal component to identify how complex is the mapped space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Since the original problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', the event logs) is the same, differences in the distribution and density of the encoded events can lead to interpretations about the mapping quality offered by each encoding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In the PCA, each point represents a feature vector, and each color represents a scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The depiction highlights the encoding capacity of generating feature vectors preserving inter- and intra-traces similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' This is demonstrated by the co-location of samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', encoded traces, of similar scenarios, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', the same color points near to same color points in each 2D projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' A high level of expressivity relies on non-overlapped clusters of samples from the same scenario with clusters sorted by scenarios’ complexity occupying the whole sub-space projected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' A low level of expressivity is associated with occluded samples with mixed sparse distributions or dense overlapped allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' An average expressivity is identified when the projected distribution matches partially high and low characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Very high and very low expressivity are obtained when an encoding method completely matches the mentioned characteristics, positively or negatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Figure 5 shows the PCA projections of Baseline methods (count2vec, n-grams, position profile , and one-hot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='a and Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='b, regarding count2vec and one-hot, look similar, assigning more or less the same areas to the same scenarios, but keeping uncovered a significant part of the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' This is due to the methods being very related, with the difference being that count2vec accounts for frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Since most traces do not have a high number of repeated activities, when reducing the dimensionality using PCA, the distances in the low-dimensional space are almost the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' On the other hand, n-grams (Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='c) and position profile (Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='d) have an average expressivity as different scenarios overlap on the same areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Figure 6 shows the projections of PM-based encoding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' They can be assessed to very high and high expressivity levels, respectively alignment (Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='a) with the highest level, followed by token-replay (Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='b) and Log skeleton (Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' It is worth observing that the distribution in the alignment projection follows the complexity of scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The Text encoding family presented average, low, and very low levels of expressivity, as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' GloVe and hash2vec are from average level, as observed in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='a and Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Low levels of expressivity were obtained by TF-IDF (Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='c), CBOW (Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='d) and skip-gram (Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The worst expressivity level, very low, was obtained by doc2vec (Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Very high, high, and average were the levels observed when using the Graph encoding family (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Average was obtained by BoostNE (Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='a) and role2vec (Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' PCA projections that represented high expressivity were computed from encoded vectors of DeepWalk, diff2vec, GLEE, GraRep, Hope, Laplacian Eigenmaps, NetMF, NMF- ADMM, node2vec, NodeSketch and Walklets, respectively, Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='b, Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='c, Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='d, Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='f, Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='g, Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='h, Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='i, Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='j, Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='k, Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='l, and Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Very high expressivity was observed using 14 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' −200 −100 0 100 200 −2 −1 0 1 2 3 4 Scenario 1 2 3 4 5 0 1 −2 0 2 −2 −1 0 1 2 Scenario 1 2 3 4 5 0 1 (a) count2vec −2 0 2 −2 −1 0 1 2 Scenario 1 2 3 4 5 0 1 (b) one-hot 0 1 2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 Scenario 1 2 3 4 5 0 1 (c) n-grams −500 0 500 −200 0 200 400 Scenario 1 2 3 4 5 0 1 (d) position profile Figure 5: 2D projections based on PCA transformation of the feature vectors of Baseline Family encoding methods of event logs from five different scenarios (1, 2, 3, 4, and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Each projection regards an encoding method, each point an encoded trace and each color represents a scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='−200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='−100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Scenario ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='(b) token-replay ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='−2000 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='(c) log skeleton ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Figure 6: 2D projections based on PCA transformation of the feature vectors of Process Mining encoding family ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='methods of event logs from five different scenarios (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Each projection regards an encoding method, each point an encoded event log and each color represents a scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' −200 −100 0 100 200 −2 −1 0 1 2 3 4 Scenario 1 2 3 4 5 0 1 0 1 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 1 Scenario 1 2 3 4 5 0 1 (a) GloVe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='4 −0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='6 Scenario 1 2 3 4 5 0 1 (c) TF-IDF −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='005 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='015 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='005 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='005 Scenario 1 2 3 4 5 0 1 (d) CBOW −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='005 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='005 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='01 Scenario 1 2 3 4 5 0 1 (e) skip-gram −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='03 Scenario 1 2 3 4 5 0 1 (f) doc2vec Figure 7: The figure illustrates the 2D projections based on PCA transformation of the feature vectors of Text encoding family methods of event logs from five different scenarios (1, 2, 3, 4, and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Each projection regards an encoding method, each point an encoded event log and each color represents a scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 15 :.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' GraphWave (Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e), where it is possible to observe an organized gradient by scenario complexity, all different event logs are identified spread in the 2D projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' This result is confirmed by the measurements obtained with the T4 measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' −200 −100 0 100 200 −2 −1 0 1 2 3 4 Scenario 1 2 3 4 5 0 1 0 50k −10k 0 10k 20k 30k Scenario 1 2 3 4 5 0 1 (a) BoostNE 0 2 4 −3 −2 −1 0 1 2 3 Scenario 1 2 3 4 5 0 1 (b) DeepWalk 0 2 −2 −1 0 1 2 Scenario 1 2 3 4 5 0 1 (c) diff2vec 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='05 Scenario 1 2 3 4 5 0 1 (d) GLEE −0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Scenario ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='(n) Walklets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Figure 8: 2D projections based on PCA transformation of the feature vectors of Graph encoding family methods of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='event logs from five different scenarios (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Each projection regards an encoding method, each point an encoded event log and each color represents a scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' T4 gives a rough measure, from 0 to 1, of the proportion of relevant dimensions used by the encoding vector to map the event log [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Relevance is determined according to the PCA criterion, which strives to describe most of the variability in the data with uncorrelated linear functions of the features [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' A higher T4 value indicates a more complex relationship between the input variables, indicating a larger number of original features are required to describe the data variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Graph-based methods obtained the best expressivity, followed by the Baseline family, as shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In terms of T4 value, doc2vec reached the lowest expressivity level, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2 Scalability We drive the discussion of scalability considering the time (seconds) and memory (KB) consumption accumulated during the whole encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Since costly methods are prohibitive to real-life event logs with huge volumes of data, their time and memory costs can directly influence the choice of an encoding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In our experiments, we considered 16 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' ������� ��������� ������ ��������� ������ ����������� ����� ������� ��������� ����� �������� ������������ ���� ���������� ������������� �������� �������� �������� ����������������� ���������������� �������� �������� ������ ����� ���� ������������������ ������� �������� ��� ��� ��� ��� ��� �� ������ �� ����� �������� ���� s asd PM Graph Baseline Text Figure 9: T4 value obtained from encoding methods for expressivity evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Low T4 values are correlated to good expressivity and high T4 values indicate poor expressivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' the time and memory consumed only during the encoding task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' By comparing three different versions of event log size (1k, 5k, and 10k), we can observe how the costs are affected when encoding the same group of problems using different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Figures 10, 11, 12 and 13 were used to demonstrate the scalability of time and memory, limiting the y-axis according to the higher observed value (time on left-side and memory on right-side) over all experiments of each encoding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Baseline family analyses are supported by Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In terms of memory cost, the Baseline encoding family showed that n-grams was the most expensive method with a high space complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' On the other hand, position profile was the worst method in terms of time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' One-hot demonstrated that memory costs increase more than time as the problem is scaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In terms of scalability, count2vec presented the best scalability results of the Baseline family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 1000 5000 10000 Event log size 0 20 40 60 80 100 Encoding Time (s) Time Count2vec Time N-grams Time One-hot Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Position Profile 0 1 2 3 4 5 6 7 8 Encoding Memory (KB) 1e8 Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Count2vec Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' N-grams Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' One-hot Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Position Profile Figure 10: Time and memory costs across different event log sizes (1k, 5k and 10k) when using Baseline encoding family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 17 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The PM encoding family presented high memory costs and average time costs, but with good scalability in both measures, as visible when the dataset increased and the performances slightly grew, as in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Alignment method was the cheapest one in terms of memory, token-replay presented a good balance in terms of time, and Log skeleton the most costly in both, memory and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 1000 5000 10000 Event log size 0 1000 2000 3000 4000 5000 6000 Encoding Time (s) Time Alignment Time Log skeleton Time Token-replay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='0 Encoding Memory (KB) 1e8 Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Alignment Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Log skeleton Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Token-replay Figure 11: Time and memory costs across different event log sizes (1k, 5k and 10k) when using Process Mining encoding family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The Text encoding family presented the fastest methods with low memory consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' However, presented time scalability issues by the majority of methods (GloVe, hash2vec, CBOW, skip-gram and doc2vec), as represented by Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The least scalable was doc2vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' A notable exception was TF-IDF, which presented reduced memory usage even with increasing problem size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Note that the scalability was evaluated considering the ability to not suffer from data growth, and the average usage of time and memory in the Text encoding family is the lowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 1000 5000 10000 Event log size 0 2 4 6 8 Encoding Time (s) Time Doc2Vec Time GloVe Time Hash2vec Time TF-IDF Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Cbow Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' SkipGram 0 1 2 3 4 Encoding Memory (KB) 1e7 Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Doc2Vec Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' GloVe Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Hash2vec Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' TF-IDF Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Cbow Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' SkipGram Figure 12: Time and memory costs across different event log sizes (1k, 5k and 10k) when using Text encoding family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 18 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The Graph encoding family presented a heterogeneous usage of memory and an average time cost, as Figure 13 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' GLEE, Hope and NetMF were the fastest methods of this family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' These methods presented average scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The best scalability was demonstrated by NodeSketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The higher memory cost from the graph encoding family was achieved by GraRep with a cost comparable to token-replay (PM family), but with average scalability regarding time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The slowest method was node2vec, using the smallest event log it was presented 4 times the average of the other methods of the same family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' When dealing with the larger event log the time difference reached 7 times the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 1000 5000 10000 Event log size 0 10 20 30 40 50 60 Encoding Time (s) Time BoostNE Time DeepWalk Time Diff2vec Time GLEE Time GraphWave Time GraRep Time Hope Time Laplacian Eigenmaps Time NetMF Time NMF-ADMM Time Node2Vec Time NodeSketch Time Role2vec Time Walklets 0 1 2 3 4 5 6 7 Encoding Memory (KB) 1e7 Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' BoostNE Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' DeepWalk Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Diff2vec Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' GLEE Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' GraphWave Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' GraRep Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Hope Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Laplacian Eigenmaps Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' NetMF Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' NMF-ADMM Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Node2Vec Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' NodeSketch Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Role2vec Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Walklets Figure 13: Time and memory costs across different event log sizes (1k, 5k and 10k) when using Graph encoding family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We organized the results of scalability and consumption of both time and memory analysis as a heat map (Figure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In the figure, it is possible to observe that general low memory and little time consumption count2vec, do not reflects the scalability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', increasing event log sizes, some methods compromise their costs with quadratic complexity costs of time and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Alternatively, Log skeleton, token-replay, and NodeSketch are very scalable but have a high memory and time cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='alignment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='boostne ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='count2vec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='deepwalk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='diff2vec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='doc2vec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='glee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='glove ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='graphwave ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='grarep ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='hash2vec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='hope ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='laplacianeigenmaps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='log_skeleton ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='netmf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='ngrams ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='nmfadmm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='node2vec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='nodesketch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='onehot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='position_profile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='role2vec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='tfidf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='tokenreplay ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='walklets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='word2vec_cbow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='word2vec_skipgram ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='encoding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Time Scalability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Memory Scalability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Time Consumption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Memory Consumption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='7 14 24 11 12 27 21 17 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='8 26 22 20 2 23 4 13 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='1 16 25 10 15 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='9 18 19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='21 10 25 13 14 12 24 7 17 2 19 20 23 3 22 11 6 15 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='4 27 16 26 1 18 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='26 14 2 16 21 20 12 6 17 13 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='9 11 27 10 5 15 23 22 4 24 18 3 25 19 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='20 22 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='9 10 21 3 17 16 24 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='2 25 1 27 19 11 18 26 6 12 7 23 15 13 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Ranking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Figure 14: Ranking of Time Scalability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Memory Scalability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Time Consumption,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' and Memory Consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The better approaches are positioned in the first ranking position, colored by white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The most costly and less scalable are the last potions with dark colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='3 Correlation power Correlation is an important analysis perspective since it reflects how correlated an encoding method is to the executed task in terms of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In other words, how the encoding positively contributed to the final performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In 19 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' our benchmark, we evaluated the correlation based on an anomalous trace detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In particular, we evaluated the F1-score obtained to detect anomalous behavior and the N2 from the mapped space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' N2 is a ratio that computes distances between an example and its closest neighbor within a particular class (intra-class) and between an example and its closest neighbor from a different class (extra-class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The N2 value, which ranges from 0 to 1, is low when there is a greater distance between examples of different classes than between examples from the same class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, a mapped space with a lower N2 refers to a representation that is better equipped to distinguish classes and support supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In our paper, we followed the N2 calculation as described by [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Figure 15 represents the obtained N2 values from encoded space from all encoding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The figure presents each family with a particular color and results are sorted by N2, from the best one to the worst N2 value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Position profile achieved the best N2 values, mean of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='48, and was the only method with less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='50 in terms of this measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The top 5 N2 values were obtained by Graph and Text families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The PM encoding family, particularly alignment and token-replay reached high N2 values, superior to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In contrast, Log skeleton obtained less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='70, ranking in the bottom 10 encoding methods, but supported the best F1-score in the anomaly detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' ���������������� ������� ����� ��������� ������ �������� �������� ����� ������������ ������� �������� �������� ������� �������� ���� ������������������ ������������� ����� ����������������� ���� ������ ������ ��������� ���������� �������� ��������� ����������� �������� ��� ��� ��� ��� ��� ��� �� ������ �� ����� �������� ���� PM Graph Baseline Text Figure 15: Ratio of Intra/extra class nearest neighbor distance (N2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' provided by the same group of tasks considering 27 different encoding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The bars are sorted from the best score (left) to the worst one (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Each bar is colored according to the coding family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' When evaluating the F1-score, Log skeleton provided the best performance (mean of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='942), followed by position profile (mean of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='935) and all encoding from the Graph encoding family (above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='915).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Text encoding family provided results above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='903, except doc2vec that lead to an average F1-score of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='845.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Surprisingly, the most traditional encoding used in PM, one-hot, obtained the worst results, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', inferior to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='840 of the F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The obtained F1-scores are sorted by the performance from the best to the worst one in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In order to provide a fair and statistically grounded comparison in terms of predictive performance, we used Friedman’s statistical test and the post-hoc test of Nemenyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Both tests were employed to verify the statistically significant difference between the performance of the F1-score of each encoding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The result of the statistical comparison can be observed as a Critical Difference chart, as illustrated in Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Critical Difference test allows checking when there were statistical differences between the segmenters, each diagram and method’s average ranks are placed on the horizontal axis, with the best ranked to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The solid line connects encoding methods with no significant performance difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, Figure 17 demonstrates no statistical difference among Log skeleton, position profile, 20 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='���������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='���������� ' metadata={'source': 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+page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='�������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='PM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Baseline ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='Figure 16: F1-score obtained by several classification tasks performed using a Random Forest algorithm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' considering 27 different encoding tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The bars are sorted from the best score (left) to the worst one (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Each bar is colored according to the coding family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' NodeSketch, NMF-ADMM, GraphWave, BoostNE, NetMF, Walklets, Hope, GLEE, Laplacian Eigenmaps, node2vec, all of them supporting high predictive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' On the other hand, there is no statistical difference as encoding methods that provided lesser predictive models for one-hot, doc2vec, n-grams, count2vec, token-replay, alignment, hash2vec, GloVe, skip-gram, CBOW and TF-IDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='4 Domain agnosticism The last comparison criterion regards Domain Agnosticism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We consider this criterion as important as Expressivity, Scalability, and Correlation to comprehend the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Since a method valid for multiple applications is instrumental to the construction of adaptive data science pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The methods comprising the baseline family are traditionally used in different PM tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' They realize straightforward transformations mapping traces into feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Their usage was not limited to PM pipelines, indeed, they were created in other data mining areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For these reasons, we consider the Baseline family as domain agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' As well, the Text and Graph families have been used for a wide range of purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' These last families are good examples of domain agnosticism and have been used with simple encoding scenarios as well as with complex and highly structured representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Considering our criteria, PM-based solutions are not domain agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' This family of encoding methods was conceived to represent event logs and has been used exclusively for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' It should not be viewed as a limitation, but rather as a characteristic of specialized methods demonstrating a high correlation power, even at the cost of high computational costs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=', Log skeleton provided encoded spaces to induce models that obtained high F1-scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' An overview of domain agnosticism and some implications of encoding method performance and resource consumption could be observed in Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The most notable observation is that non-agnostic methods (token-replay, alignment, and Log skeleton) share high N2 values and high costs of space and time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Among the agnostic methods, the best N2 and F1 performances are obtained with average space and time complexity, e,g, with GraphWave or BoostNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 21 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 one-hot doc2vec count2vec token-replay n-grams alignment hash2vec GloVe skip-gram TF-IDF CBOW role2vec diff2vec GraRep DeepWalk node2vec Laplacian Eigenmaps GLEE Hope Walklets NetMF BoostNE GraphWave NMF-ADMM NodeSketch position profile Log skeleton CD Figure 17: Nemenyi post-hoc test (significance of α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='05 and critical distance of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='71) considering the F1-score (predictive performance) accuracy obtained when performing the classification task using all encoding methods over 10 scenarios (1k, 5k and 10k traces from five different scenarios).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Statically similar methods are linked by the solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 7 Issues, Concerns and Future Directions Research comparisons about encoding methods focused on PM are still embryonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' After classifying the papers in Section 4, we observed that the proposed solutions do not investigate the strong and weak points of each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, we have proposed a study involving a large set of methods from different families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Investigating the pros and cons of encoding methods based on expressivity, scalability, correlation power, and domain agnosticism over different encoding families and hundreds of event logs with various complexities, we were able to gain insights and share some assumptions for future directions and possible novel PM encoding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We believe the criteria employed in this work to assess the effectiveness are the most important aspects to take into consideration in order to achieve significant accomplishments for any PM task that needs to encode event logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Moreover, the proposed metrics can also serve as user requirements when deciding which encoding method should be employed for the specific problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The expressivity of an encoding method can measure how straightforward the representation of an event log is according to its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The scalability is also an important concern since real-life event logs consist of a large volume of data which can lead to high computational costs regarding elapsed time and memory usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Understanding the correlation power between the nature of an encoding method and the performance of the executed task is also a relevant analysis that allows practitioners to estimate the complexity of analyzing a specific event log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Lastly, the domain agnosticism is a novel and important discussion introduced in this work to consider if encoding methods can be adapted for different problem domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Addressing further issues on encoding methods in PM, online PM may introduce new challenges for the current encoding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' As emphasized by [64], measures such as accuracy and memory consumption need to drive the creation of methods to match online PM goals, as the encoding methods used in such solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The concern of limitations posed by an online PM task was also highlighted by [65], mentioning also the demand of adapting when dealing with concept drifts and focusing on inter-activity time implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' STARDUST [66] is an example of online PM, particularly on trace streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The authors discussed issues regarding approaches to handling traces recorded without the final activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' It is therefore essential to investigate encoding methods that support online PM tasks coping with new challenges, such as reduced memory consumption and the ability to map partial traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In our experiments, 22 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 10 1 100 101 102 103 Time (log) 106 107 108 Memory (log) alignment BoostNE count2vec DeepWalk diff2vec doc2vec GLEE GloVe GraphWave GraRep hash2vec Hope Laplacian Eigenmaps Log skeleton NetMF n-grams NMF-ADMM node2vec NodeSketch one-hot position profile role2vec TF-IDF token-replay Walklets CBOWskip-gram F1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='94 N2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='90 non-agnostic agnostic Figure 18: Mean values of time, memory, and predictive performance (F1) projected by time and memory in a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The marker size represents the N2 and their color gradient performance in terms of F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The marker symbol represents the domain agnosticism evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' we used scalability measure to support insights and discussions regarding this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' A crucial indicator of scalability was the encoding families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Currently, the predictive process monitoring problem also faces this issue regarding the lack of encoding methods specific to PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Representation learning or feature learning is a learning paradigm that has been recently introduced in the community by [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We believe this is a promising path for improving the encoding procedure in process mining tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In the mentioned work, the authors derived their new proposals from the word2vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' However, we believe PM requires a specialized method or a sufficiently generic one regardless of the problem domain since the event data might be considered more complex than sequential text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We claim that since the nature of event data, in general, contains sequential rules, relational information, concurrency of resources, parallel activities, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Encoding regards mapping data into another representation for different goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The new space could not allow interpretations and explainability as previously supported by the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Moreover, practitioners need to trust the generated mapped space, as mentioned by [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In particular, the predictive model presented in a great part of predictive monitoring tasks does not explain why it provided wrong predictions, so the reason why a prediction model made a mistake cannot be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Shedding some light on this topic, [68] presented post-hoc explainers and different encoding methods for identifying the important features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' On a general note, our experiments using expressivity and domain agnosticism confirmed the wide range of representations provided by different encoding methods, even from 23 Trace Encoding in Process Mining Barbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' the same family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Regarding this point of eXplainable Artificial Intelligence (XAI), as a tendency, encoding methods able to provide high explainability levels should be integrated into the PM pipeline as a key component, not a separate step follow-up effort for particular applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The great number of encoding methods and reduced availability of experts pose an additional challenge to selecting and properly setting the encoding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Strategies focused on accuracy or time performance are applied when selecting a method, but the cost of testing different setups and costly tuning strategies could impact the PM pipeline conception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' This problem has been addressed by promising strategies based on meta-learning [69, 70, 71], but the current solutions require creating a meta-database containing the history of possible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Also, the criteria to recommend a particular algorithm are still limited to simple performance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Therefore, Automatic Machine Learning (AutoML) [6, 8] proves to be an important research area that impacts the aforementioned concerns regarding encoding methods used in PM tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Alternatively, another learning paradigm that could be explored for this nature of data is self-supervised learning (SSL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' The general idea behind SSL is learning a set of possible outcomes given an input, instead of predicting a unique value as traditional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' For instance, the data2vec was recently presented by [72], where the authors propose a generic framework for encoding any type of data, although only the domains of image, speech, and language have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Intuitively, this might be interesting for capturing mutual dependencies in event data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 8 Conclusion The main contributions presented in this work include a systematic review of process mining tasks using encoding methods, a new taxonomy to categorize each type of method into families, and an extensive experimental evaluation and benchmark assessing relevant evaluation metrics to measure the effectiveness of an encoding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We believe this work can support researchers and practitioners to achieve significant accomplishments in different application areas in PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Furthermore, we stress current challenges and issues in the literature regarding the difficulty of choosing the right algorithm and its parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We also discuss how arbitrarily selecting algorithms lead to unfair evaluation and sub-optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' This is the first work that focuses on a detailed analysis for preprocessing event logs instead of focusing only on the task algorithm itself (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' a clustering or learning algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We also highlighted the need for a better understanding of how each method behaves according to different scenarios of event logs and different PM tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Thus, to fill this gap we simulated such scenarios by employing the PLG2 tool to generate synthetic processes with distinct properties and presented the results as a benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' In total, 27 encoding methods were evaluated throughout 420 different event logs containing different anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We considered four different evaluation criteria to measure the effectiveness of encoding methods for process mining tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We limited our evaluation to only one task, anomaly detection, but the analysis and insights presented in this work can be leveraged for other applications, such as predictive monitoring and clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We conclude this work by stressing the difficulty of choosing suitable algorithms and their parameters according to the user’s preferences since each pipeline setting performs differently according to the event log characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' This might be a direction to novel automated solutions whether for entire pipelines or preprocessing steps only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' We also believe that an encoding method that specifically handles the nature of processes is essential for advancing the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' This is claimed by considering that most methods are adapted or adopted from other areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' Therefore, this research line is promising and has several opportunities for work to be developed.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' PMLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} +page_content=' 28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NA0T4oBgHgl3EQfO_8B/content/2301.02167v1.pdf'} diff --git a/ANAzT4oBgHgl3EQfvv6Q/content/tmp_files/2301.01712v1.pdf.txt b/ANAzT4oBgHgl3EQfvv6Q/content/tmp_files/2301.01712v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b8c4f1c782c7da9967b39411d43c180239cdccde --- /dev/null +++ b/ANAzT4oBgHgl3EQfvv6Q/content/tmp_files/2301.01712v1.pdf.txt @@ -0,0 +1,2499 @@ +arXiv:2301.01712v1 [math.PR] 4 Jan 2023 +1 +Mesoscopic eigenvalue statistics for Wigner-type matrices +Volodymyr Riabov∗ +Institute of Science and Technology Austria +volodymyr.riabov@ist.ac.at +Abstract. We prove a universal mesoscopic central limit theorem for linear eigenvalue statistics of a Wigner- +type matrix inside the bulk of the spectrum with compactly supported twice continuously differentiable test +functions. The main novel ingredient is an optimal local law for the two-point function T(z, ζ) and a general +class of related quantities involving two resolvents at nearby spectral parameters. +Date: January 5, 2023 +Keywords and phrases: Wigner-type matrix, mesoscopic eigenvalue statistics, central limit theorem +2010 Mathematics Subject Classification: 60B20, 15B52 +1 +Introduction +In the study of the eigenvalue distribution of large random matrices, the most celebrated analog of +the Law of Large Numbers is the Wigner semicircle law [25]. It states that the empirical density of +eigenvalues converges to a deterministic limit known as the semicircle distribution ρsc. More explicitly, +if H is an N ×N Wigner matrix and f is a sufficiently smooth test function, then the linear eigenvalue +statistics N −1 Tr f(H) converge in probability to +� +R f(x)ρsc(x)dx in the large N limit. +The corresponding Central Limit Theorem (CLT) asserts that the asymptotic fluctuations of the +linear eigenvalue statistics Tr f(H)−E [Tr f(H)] are Gaussian. The absence of the N −1/2 normalization +factor, appearing in the classical CLT, can be viewed as a manifestation of the strongly-correlated +nature of the eigenvalues. For the special case of f(x) = (x − z)−1 with Im z ̸= 0, this result was +obtained by Khorunzhy, Khoruzhenko and Pastur [16]. Johansson obtained the CLT for invariant +ensembles with arbitrary polynomial potentials in [15]. +In [4], Bai and Yao used martingale CLT +to establish the result for Wigner matrices with analytic test functions. The proof for bounded test +functions f with bounded derivatives appeared in the work of Lytova and Pastur [22]. In subsequent +works, different moment conditions on the matrix and regularity conditions on the test function were +studied extensively by many authors, e.g., [6, 18, 23, 24]. +While fixed test functions represent macroscopic averaging in the spectrum, one can introduce N- +dependent scaling and consider scaled test functions of the form f(x) = g(η−1 +0 (x − E0)), where E0 +is a fixed reference energy in the bulk, η0 ≡ η0(N) ≪ 1 is a scaling parameter, and g is compactly +supported. Then Tr f(H) involves only about Nη0 eigenvalues of H. In particular, on mesoscopic +scales, corresponding to N −1 ≪ η0 ≪ 1, the limiting variance is given by the square of the +˙H1/2 +norm of g. +Mesoscopic test functions were first studied by Boutet de Monvel and Khorunzhy in +[7] for the Gaussian Orthogonal Ensemble, with subsequent extension to real Wigner matrices in [8] +with N −1/8 ≪ η0 ≪ 1. In [13], He and Knowles proved the CLT for Wigner matrices with general +mesoscopic test functions for all scaling parameters N −1 ≪ η0 ≪ 1. +∗Supported by the ERC Advanced Grant ”RMTBeyond” No. 101020331 + +The result was extended to ensembles of greater generality in the more recent works, see, e.g., [5] +and [20]. In particular, Li and Xu obtained mesoscopic CLT for generalized Wigner matrices 1 in the +bulk and at the spectral edge with C2 +c test functions in the full range of scales [21]. +Finally, Landon, Lopatto, and Sosoe proved the bulk CLT for the much more general ensemble +of Wigner-type matrices in [17] for two classes of C∞ test functions. For a special class of globally +supported regularized bump functions , the proof is performed via resolvent techniques for large scales +and extended to the entire mesoscopic range using Dyson Brownian motion (DBM) dynamics. For +the more conventional compactly supported scaled test functions, the bulk CLT is established on all +mesoscopic scales N −1 ≪ η0 ≪ 1 using a combination of DBM and Green’s function comparison. +Wigner-type matrices were first introduced in [2]; they have centered entries Hjk independent up +to the symmetry constraint H = H∗. The matrix of variances S, defined by Sjk := E +� +|Hjk|2� +, is +assumed to be flat, i.e., Sjk ∼ N −1 and satisfy a piece-wise H¨older regularity condition (see (B)). +As the main step towards CLT in the present paper, we prove the optimal averaged and entry-wise +local laws (Corollary 3.3) for the two-point function T , defined by +Txy(z, ζ) := +� +a̸=y +SxaGay(z)Gya(ζ), +x, y ∈ {1, . . ., N}, +(1.1) +where G(z) is the resolvent of H. The corresponding result in the simpler setting of generalized Wigner +matrices was obtained in [21]. Using the optimal local law for T (z, ζ), we prove the bulk mesoscopic +CLT for Wigner-type matrices in the full range of scales N −1 ≪ η0 ≪ 1 for compactly supported C2 +scaled test functions (Theorem 2.2). Our proof relies entirely on resolvent methods, circumventing the +DBM dynamics used in [17]. +Understanding T (z, ζ) is the crucial ingredient for the CLT as it was realized in [17]. In fact, a +suboptimal entry-wise local law for Txy(z, ζ) was proved in Proposition 5.1 of [17]. If one relies solely +on resolvent methods, this local law provides sufficient control for mesoscopic CLT only on scales +η0 ≫ N −1/5. The main reason for this limitation is that the error term in [17] contains the norm of +the inverted stability operator (defined in (4.5)). In the present paper, we show that this factor can be +removed by separating the destabilizing eigendirection corresponding to the smallest eigenvalue of the +stability operator. Using this method, we prove a local law for a general class of quantities involving +two resolvents (Theorem 3.2) and deduce the optimal averaged and entry-wise local laws for T (z, ζ). +In particular, this allows us to obtain the CLT on all mesoscopic scales without relying on DBM. +The main difficulty lies in the fact that the deterministic approximation of the resolvent for Wigner- +type matrices is not a multiple of the identity matrix, contrary to the generalized Wigner case [21]. +Consequently, the destabilizing direction is no longer parallel to the vector of ones, and generally, no +closed-form expression is known for the corresponding eigenprojector. It is important to note that for +the deformed Wigner matrices studied in [20], the deterministic approximation is also not a multiple +of the identity, but Sjk = N −1. Therefore, the two-point function can be expressed as the square of +the resolvent and can be studied using the local law, similarly to the standard Wigner case. +Instead of approximating the destabilizing direction to circumvent this difficulty, we use a contour +integral representation for the eigenprojector. It allows us to extend the decomposition approach of +[21] to the Wigner-type ensembles. This method benefits from yielding an integral representation for +the variance on all mesoscopic scales, under weaker regularity conditions on the test function than in +[17], and relying only on resolvent methods. +The paper is organized in the following way. Section 2 contains the precise definition of the model +and the statement of our main mesoscopic CLT result, Theorem 2.2. In Section 3, we present our main +technical result, the optimal local law for two-point functions in Theorem 3.2. In Section 4, we collect +notations and preliminary results to which we refer throughout the paper. In Section 5, we deduce +Theorem 2.2 from Propositions 5.1 and 5.2, and prove Proposition 5.1 using a local law for T (z, ζ) +(Corollary 3.3) as an input. The proofs of Theorem 3.2 and Corollary 3.3 are presented in Section 6. +In Section 7, we prove Proposition 5.2, which relates the variance of the linear eigenvalue statistics to +the ˙H1/2-norm. +Acknowledgments. +I would like to express my gratitude to L´aszl´o Erd˝os for suggesting the project and supervising my +work. I am also thankful to Yuanyuan Xu and Oleksii Kolupaiev for many helpful discussions. +1Generalized Wigner matrices are characterized by a flat doubly-stochastic matrix of variances S. Unlike the Wigner +case, the entries Sjk are not assumed to be equal. The limiting eigenvalue distribution remains semicircular. +2 + +2 +Model and Main Result +We begin with the definition of Wigner-type matrices originally introduced in Section 1.1 of [2]. +Definition 2.1 (Wigner-type matrices). Let H = (Hjk)N +j,k=1 be an N × N matrix with independent +entries up to the Hermitian symmetry condition H = H∗ satisfying +E [Hjk] = 0. +(2.1) +We consider both real and complex Wigner-type matrices. In case the matrix H is complex we assume +additionally that Re Hjk and Im Hjk are independent and E[H2 +jk] = 0 for k ̸= j. +Denote by S the matrix of variances Sjk := E[|Hjk|2], and assume it satisfies +cinf +N +≤ Sjk ≤ Csup +N , +(A) +for all j, k ∈ {1, . . ., N} and some strictly positive constants Csup, cinf. +We assume a uniform bound on all other moments of +√ +NHjk, that is, for any p ∈ N there exists +a positive constant Cp such that +E +� +| +√ +NHjk|p� +≤ Cp +(2.2) +holds for all j, k ∈ {1, . . ., N}. +Additionally, we assume that S satisfies a H¨older regularity condition1, that is, +|Sjk − Sj′k′| ≤ L +N +�|j − j′| + |k − k′| +N +�1/2 +, +(B) +for all j, j′, k, k′ ∈ {1, . . ., N} and some positive constant L. The constants cinf, Csup, Cp and L are +independent of N. +2.1 +Central Limit Theorem for Mesoscopic Linear Eigenvalue Statistics +Theorem 2.2. (c.f. Theorem 2.5 in [17]) Let g be a C2 +c (R) test function. Let ε0 be a small fixed +constant and let N −1+ε0 ≤ η0 ≤ N −ε0, and let E0 be a fixed reference energy in the bulk of the +spectrum, that is, ρ(E0) ≥ ε0 (here ρ is the density of states to be defined in (3.3) below ). Define the +scaled test function f to be +f(x) := g +�x − E0 +η0 +� +, +(2.3) +then +Tr f(H) − E [Tr f(H)] +d−→ N +� +0, +1 +2βπ2 ∥g∥2 +˙H1/2 +� +, +(2.4) +where β = 1 and β = 2 corresponds to real symmetric and complex Hermitian H, respectively. +Remark 2.3. We remark that the universal limiting variance in (2.4) coincides with the corre- +sponding formulas for standard Wigner matrices [13], where Sjk = N −1, mj(z) = msc(z) for all +j, k ∈ {1, . . . , N}, and msc(z) is the Stieltjes transform of the semicircle law. +1As stated in [2], assumption (B) can be weakened to piece-wise 1/2-H¨older regularity condition for some positive +constant L on finitely many intervals, in the sense that +max +a,b +max +j,j′∈(NIb) +max +k,k′∈(NIa) N3/2 +|Sjk − Sj′k′| +|j − j′|1/2 + |k − k′|1/2 ≤ L, +where {Ia}n +a=1 is a fixed finite partition of [0, 1] into smaller intervals, and (NIa) denotes the set of positive integers j +such that j/N lies in Ia. +3 + +3 +Local Laws for the Two-point Functions +In this section, we introduce our main technical result, local laws for quantities that involve two +resolvents of a Wigner-type matrix. Our prime motivation is to study the function T (z, ζ) defined in +(1.1), but our methods allow us to estimate a more general class of quantities, namely +� +a̸=y +waGαa(z)Gaβ(ζ), +� +b +� +a̸=b +WabGba(z)Gab(ζ), +(3.1) +for fixed indices α, β, y, and deterministic weights wa, Wab satisfying |wa|, |Wab| ≤ cN −1 for some +constant c > 0. +Here G(z) := (H − z)−1 denotes the resolvent of H. +Objects of this type were +first studied in [11] in the setting of random band matrices. We obtain the estimates in the sense of +stochastic domination. +Definition 3.1. (Definition 2.1 in [12]) Let X = X (N)(u) and Y = Y(N)(u) be two families of random +variables possibly depending on a parameter u ∈ U (N). We say that Y stochastically dominates X +uniformly in u if for any ε > 0 and D > 0 there exists N0(ε, D) such that for any N ≥ N0(ε, D), +sup +u∈U(N) P +� +X (N)(u) > N εY(N)(u) +� +< N −D. +We denote this relation by X ≺ Y or X = O≺(Y). +We consider spectral parameters z lying in the domain D, defined by +D := {z ∈ C : N −1+τ ≤ | Im z| ≤ τ −1, | Re z| ≤ τ −1}, +(3.2) +for a fixed τ > 0. As in Theorem 2.2, our analysis is limited to the bulk of the spectrum, which we +define via the self-consistent density of states ρ(E) ≡ ρN(E). The density ρ(E) is recovered by the +Stieltjes inversion formula, +ρ(E) := π−1 lim +η→+0 Im m(E + iη), +(3.3) +where m(z) := N −1 �N +j=1 mj(z), and m(z) = (mj(z))N +j=1 is the unique (Theorem 4.1 in [2]) solution +to the vector Dyson equation +−1 +m(z) = z + Sm(z), +Im m(z) Im z > 0. +(3.4) +Let I be the set on which ρ(E) is positive. Theorem 4.1 of [2] guarantees that I consists of a finite +union of open intervals (a(j), b(j)). Then for κ > 0, we define the bulk domain by +Dκ := {z ∈ D : Re z ∈ Iκ}, +Iκ := +� +j +[a(j) + κ, b(j) − κ]. +(3.5) +In particular, for all z ∈ Dκ, ρ(z) ≥ C(κ) for some constant C(κ) > 0. Given E0 as in Theorem 2.2, +we choose κ so that E0 ∈ I2κ. +Theorem 3.2. There exists a positive constant ǫ = ǫκ which is independent of N, such that for all +z, ζ in Dκ with | Re ζ − Re z| ≤ ǫ, and deterministic vectors w ∈ CN satisfying ∥w∥∞ ≤ cN −1, the +following estimate holds, +� +a̸=y +waGαa(z)Gaβ(ζ) = δαβ +� +m(z)m(ζ) +� +1 − Sm(z)m(ζ) +�−1w +� +α − δαβδαy[m(z)m(ζ)w]α ++ O≺ +� +(Ψ(z) + Ψ(ζ))(Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0} min{Θ(z), Θ(ζ)}) +� +, +(3.6) +where the vector m is identified with the diagonal operator diag (m). +Under the same conditions on z, ζ, for any deterministic N ×N matrix W satisfying |Wab| ≤ cN −1 +for all a, b, the following estimate holds, +� +b +� +a̸=b +WabGba(z)Gab(ζ) = Tr +� +m(z)m(ζ)Sm(z)m(ζ) +� +1 − Sm(z)m(ζ) +�−1W +� ++ NO≺ +� +(Ψ(z) + Ψ(ζ))Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0}Θ(z)Θ(ζ) +� +. +(3.7) +4 + +Here Ψ(z) and Θ(z) denote control parameters defined as +Ψ(z) := +� +| Im m(z)| +N|η| ++ +1 +N|η|, +Θ(z) := +1 +N|η|, +z = E + iη ∈ C\R. +(3.8) +Theorem 3.2 implies the following averaged and entry-wise local laws for T (z, ζ) from (1.1) . +Corollary 3.3. Let z, ζ satisfy the assumptions of Theorem 3.2. +The entries Txy(z, ζ) admit the +estimate +Txy(z, ζ) = +� +(Sm(z)m(ζ))2 � +1 − Sm(z)m(ζ) +�−1� +xy ++ O≺ +� +(Ψ(z) + Ψ(ζ)) +� +Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0} min{Θ(z), Θ(ζ)} +�� +. +(3.9) +Furthermore, for all deterministic N × N matrices A, the following equality holds +Tr[A T (z, ζ)] = Tr[A +� +1 − Sm(z)m(ζ) +�−1� +Sm(z)m(ζ) +�2] ++ N ∥A∥ℓ∞→ℓ∞ O≺ +� +(Ψ(z) + Ψ(ζ))Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0}Θ(z)Θ(ζ) +� +. +(3.10) +Remark 3.4. The error estimates in the entry-wise local law (3.6), and hence in (3.9) are optimal. +Indeed, for Sjk := N −1, which corresponds to the standard Wigner matrices, and ζ = ¯z, a simple +calculation using the Ward identity shows that +Txy(z, ¯z) = N −1| Im z|−1 Im msc(z) − N −1|msc(z)|2 + O≺ +� +Θ(z)Ψ(z) +� +. +(3.11) +The error estimate in (3.7) is not optimal; it can be improved to +O≺ +� +N(Ψ(z) + Ψ(ζ))2� +Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0}NΘ(z)Θ(ζ) +�� +(3.12) +However, (3.7) is sufficient for establishing the CLT, so for the sake of brevity, we do not present the +proof of (3.12) in full detail. We only indicate the necessary ingredients in Remark 6.8 below. +4 +Notations and Preliminaries +4.1 +Notations +For a vector x = (xj)N +j=1 ∈ CN we use the standard definitions of ℓ2 and ℓ∞ norms, namely, +∥x∥2 = +� N +� +j=1 +|xj|2 +�1/2 +, +∥x∥∞ = max +j +|xj|. +For a linear operator T : CN → CN, we denote its matrix norms induced by ℓ2 and ℓ∞ norms, +respectively, by +∥T ∥ℓ2→ℓ2 = +sup +∥x∥2=1 +∥T x∥2 , +∥T ∥ℓ∞→ℓ∞ = +sup +∥x∥∞=1 +∥T x∥∞ . +For two vectors x, y ∈ CN we use angle brackets to denote the ℓ2 scalar product, while for a single +vector x ∈ CN angle brackets denote the average of its coordinates +⟨x, y⟩ = +N +� +j=1 +¯xjyj, +⟨x⟩ = 1 +N +N +� +j=1 +xj. +We use xy to denote a coordinate-wise product of vectors x and y, +(xy)j = xjyj, +j ∈ {1, . . . , N}. +Similarly, for a given vector x with non-zero entries, 1x denotes a coordinate-wise multiplicative inverse +� 1 +x +� +j += 1 +xj +, +j ∈ {1, . . ., N}. +5 + +We use 1 to denote the vector of ones (1, . . . , 1)t in CN. +For a measurable function f : R → R we use the standard definition of the Lp norms for p ≥ 1, +and the following definition of the ˙H1/2 norm +∥f∥ ˙H1/2 = + + +�� +R2 +|f(x) − f(y)|2 +|x − y|2 +dxdy + + +1/2 +. +For two deterministic quantities X, Y ∈ R depending on N, we write X ≪ Y if there exists ε, N0 > 0 +such that |X| ≤ N −ε|Y | for all N ≥ N0. Similarly, we write X ≲ Y if there exists a constant C, N0 > 0 +such that |X| ≤ C|Y | for all N ≥ N0, and X ∼ Y if both X ≲ Y and Y ≲ X hold. +We use C and c to denote constants, the precise value of which is irrelevant and may change from +line to line. +4.2 +Local Law for the Resolvent +In this subsection, we summarize the facts on Wigner-type matrices that we use throughout our proofs. +Majority of these results were obtained in [1] (see also [3]), but we refer to their concise versions from +[2] adapted for the Wigner-type setting. +Lemma 4.1. (Theorem 4.1 in [2]) The solution m(z) of (3.4) satisfies the following properties: +(1) For every j ∈ {1, . . ., N} there exists a generating probability measure νj(dx) such that +mj(z) = +� +R +νj(dx) +x − z . +(4.1) +(2) If the matrix of variances S satisfies conditions (A) and (B), then for all z ∈ C\R, the solution +admits the following bounds +∥m(z)∥∞ ≤ +c +1 + |z|, +���� +1 +m(z) +���� +∞ +≤ C(1 + |z|). +(4.2) +We now state the optimal averaged and isotropic local laws for Wigner-type matrices. +Theorem 4.2. (Corollary 1.8 in [2]) Let w, x, y be deterministic vectors in CN satisfying ∥w∥∞ = 1 +and ∥x∥2 = ∥y∥2 = 1. Then the following estimates hold uniformly in z ∈ D: +N −1��Tr +� +w(G(z) − m(z)) +��� ≺ Θ(z), +��⟨x, (G(z) − m(z))y⟩ +�� ≺ Ψ(z), +(4.3) +where vectors m and w are associated with corresponding diagonal matrices. +In particular, it follows from the isotropic local law (4.3) that for any j, k ∈ {1, . . ., N}, +|Gjk(z) − δjkmj(z)| ≺ Ψ(z). +(4.4) +4.3 +Preliminary Bounds on the Stability Operator +A significant part of our proof revolves around the stability operator, originally introduced in [1], that +emerges when studying the two-point function T (z, ζ) defined in (1.1). In this subsection, we collect +the known bounds on the stability and related operators. +The stability operator (1 − Sm(z)m(ζ)) is defined by the matrix with entries +(1 − Sm(z)m(ζ))jk := δjk − Sjkmk(z)mk(ζ), +j, k ∈ {1, . . ., N}, +z, ζ ∈ C\R. +(4.5) +Throughout this paper we use m (and various functions of m, such as Im m, |m|, m−1, m′) to +denote both a vector (mj)N +j=1 and the corresponding multiplication operator, i.e., diag +� +(mj)N +j=1 +� +. Note +that this notation agrees with the point-wise multiplication of two vectors if the first multiplicand is +interpreted as an operator. We stress which interpretation is used whenever ambiguity may arise. +6 + +The analysis of the stability operator relies on the corresponding saturated self-energy operator F, +studied in [17], that depends on two spectral parameters z, ζ, and is defined as +Fjk(z, ζ) := |mj(z)mj(ζ)|1/2Sjk|mk(z)mk(ζ)|1/2, +j, k ∈ {1, . . ., N}, +z, ζ ∈ C\R. +(4.6) +The following statements encompass the main properties of F and preliminary bounds on the stability +operator. +Proposition 4.3. (Proposition 4.3 in [17], c.f. Proposition 7.2.9 and Lemma 7.4.4 in [9]) For any +z, ζ ∈ C, the principal eigenvalue of F defined in (4.6) is positive and simple, the corresponding ℓ2- +normalized eigenvector v(z, ζ) has strictly positive entries. The norm of F admits the following upper +bound +∥F(z, ζ)∥ℓ2→ℓ2 ≤ 1 − 1 +2 +� +| Im z| ⟨v(z, z), |m(z)|⟩ +⟨v(z, z), | Im m(z)| +|m(z)| ⟩ ++ | Im ζ| ⟨v(ζ, ζ), |m(ζ)|⟩ +⟨v(ζ, ζ), | Im m(ζ)| +|m(ζ)| ⟩ +� +. +(4.7) +If |z|, |ζ| ≲ 1, then the entries of v(z, ζ) are comparable in size, that is +cκ ≤ +√ +Nvj(z, ζ) ≤ Cκ, +j ∈ {1, . . ., N}, +(4.8) +and moreover, let Gap (F) denote the difference between the two largest eigenvalues of |F| = +√ +FF ∗, +then Gap (F) admits the bound +Gap (F) ≥ �δ, +(4.9) +where �δ is a constant that depends only on the constants in conditions (A), (B) and κ. +Furthermore, for a fixed κ > 0 and z, ζ ∈ Dκ there exists a positive constant �cκ such that +∥F(z, ζ)∥ℓ2→ℓ2 ≤ 1 − �cκ (| Im z| + | Im ζ|) , +(4.10) +Proposition 4.4. (Proposition 4.6 and Lemma 4.7 in [17]) Let z, ζ ∈ C, such that |z|, |ζ| ≲ 1 and +Re z, Re ζ ∈ Iκ, then +��(1 − Sm(z)m(ζ))−1�� +ℓ2→ℓ2 + +��(1 − Sm(z)m(ζ))−1�� +ℓ∞→ℓ∞ ≲ +1 +| Im z| + | Im ζ|. +(4.11) +If additionally Im z Im ζ > 0, the estimate is improved to +��(1 − Sm �m)−1�� +ℓ∞→ℓ∞ ≤ Cκ, +(4.12) +where Cκ > 0 is a positive constants dependent on κ. +Finally, we state the bounds on the stability operator in the special case of ζ = z, which is related +to the derivative of m via the (vector) identity m′(z) = (1 − m2(z)S)−1m2(z), obtained by taking the +derivative of (3.4). +Lemma 4.5. (Lemma 5.9 in [1], Lemma 7.3.2 in [9]) Let C > 0 be a positive constant, then for +z ∈ C\R with |z| ≤ C we have +��(1 − m2(z)S)−1�� +ℓ2→ℓ2 + +��(1 − m2(z)S)−1�� +ℓ∞→ℓ∞ ≲ |ρ(z)|−2, +(4.13) +where ρ(z) = π−1⟨Im m(z)⟩ is the harmonic extension of ρ(E) defined in (3.3). +Therefore for all z ∈ C\R with Re z ∈ Iκ we have +∥m′(z)∥∞ ≲ 1. +(4.14) +4.4 +Cumulant Expansion Formula +Lemma 4.6. (Section II in [7], Lemma 3.1 in [13]) Let h be a real-valued random variable with finite +moments, let f be a C∞(R) function. Then for any ℓ ∈ N the following expansion holds, +E [h · f(h)] = +ℓ +� +j=0 +1 +j!c(j+1)(h) E +� dj +dhj f(h) +� ++ Rℓ+1, +(4.15) +7 + +where c(j) is the j-th cumulant of h defined by +c(j)(h) = (−i)j dj +dtj +� +log E +� +eith������ +t=0 +, +and the remainder term Rℓ+1 satisfies +|Rℓ+1| ≤ Cl E +� +|h|ℓ+2� +sup +|x|≤M +|f (ℓ+1)(x)| + Cl E +� +|h|ℓ+2 · 1|h|>M +� ���f (ℓ+1)(x) +��� +∞ , +(4.16) +for any M > 0. +We apply formula (4.15) with h equal to the matrix element Hjk. Correspondingly, in the real +case (β = 1), C(p) denotes the matrix of p-th cumulants of H, C(p) +jk := C(p)(Hjk). In the complex case +(β = 2), C(p) is used as a notational shortcut and denotes the sum of matrices of p-th cumulants of +real and imaginary parts of H, that is C(p) +jk := C(p)(Re Hjk) + C(p)(Im Hjk). +5 +Proof of the Main Result +Proof of Theorem 2.2. We divide the proof into two parts contained in the following propositions. We +indicate their analogs in the settings of [21] and [17] in parenthesis. +Proposition 5.1. (c.f. Theorem 2.2 in [21] and (5.76) in [17]) Let η0, ε0 > 0 and E0 satisfy the +assumptions of Theorem 2.2, let f be a scaled test function defined in (2.3), and let φ(λ) be the +characteristic function of Tr f(H) − E [Tr f(H)], +φ(λ) := E [exp{iλ (Tr f(H) − E [Tr f(H)])}] , +λ ∈ R. +(5.1) +Then its derivative φ′(λ) satisfies the following equation, +φ′(λ) = −λφ(λ)V (f) + O≺ +� +N −1/2η−1/2 +0 +(1 + |λ|4) + (1 + |λ|)N −ε0/2� +, +λ ∈ R, +(5.2) +provided c ≤ V (f) ≤ C for some positive N-independent constants c and C. +Here the variance V (f) for a scaled test function f is defined by +V (f) := 1 +π2 +� +Ω0 +� +Ω′ +0 +∂ �f(ζ) +∂¯ζ +∂ �f(z) +∂¯z +K(z, ζ)d¯ζdζd¯zdz, +(5.3) +where for z, ζ ∈ C/R the kernel K(z, ζ) is defined by +K(z, ζ) := 2 +β +∂ +∂ζ Tr +�m′(z) +m(z) +� +1 − Sm(z)m(ζ) +�−1 +� ++ +� +1 − 2 +β +� +Tr [Sm′(z)m′(ζ)] + 1 +2 +∂2 +∂z∂ζ +� +m(z)m(ζ), C(4)m(z)m(ζ) +� +, +(5.4) +with C(4) denoting the matrix of fourth cumulants C(4) +jk . The integration domains Ω0, Ω′ +0 in (5.3) are +defined as +Ω0 := {z ∈ C : | Im z| > N −ε0/2η0}, +Ω′ +0 := {z ∈ C : | Im z| > 2N −ε0/2η0}, +(5.5) +and �f is the quasi-analytic extension of f, defined by +�f(x + iη) = χ(η) (f(x) + iηf ′(x)) , +(5.6) +where χ : R → [0, 1] is an even C∞ +c (R) function supported on [−1, 1], satisfying χ(η) = 1 for |η| < 1/2. +Proposition 5.2. (c.f. Lemma 6.7 in [17]) Let E0, η0 satisfy the conditions of Theorem 2.2. Let f be +the scaled test function with g ∈ C2 +c (R) given in (2.3), and let V (f) be the variance defined in (5.3), +then +V (f) = +1 +2βπ2 ∥g∥2 +˙H1/2 + O +� +η0 log N + N −ε0� +. +(5.7) +8 + +Proposition 5.2 implies that V (f) satisfies the condition of Proposition 5.1, hence +φ′(λ) = −λφ(λ)V (f) + o (1) , +(5.8) +as N → ∞, for any fixed λ ∈ R. It then follows by L´evy’s continuity theorem that Tr f(H)−E [Tr f(H)] +converges in distribution to a centered Gaussian with variance (2βπ2)−1 ∥g∥2 +˙H1/2. Therefore, to estab- +lish Theorem 2.2, it suffices to show that Propositions 5.1 and 5.2 hold, which is done in Sections 5.1 +and 7, respectively. +Remark 5.3. We restrict the proof to the real symmetric (β = 1) matrices for the sake of presentation. +The complex Hermitian (β = 2) case differs solely in replacing the cumulant expansion formula (Lemma +4.6) with its complex analog. The obvious modifications are left to the reader. +5.1 +Characteristic Function of Linear Eigenvalue Statistics +Proof of Proposition 5.1. Using standard techniques of the characteristic function method imported +from, e.g., Section 5.2 of [17] (see also Section 4.2 of [19] and references therein), we can obtain the +following series of estimates on the characteristic function of the linear eigenvalue statistics φ(λ) and +its derivative φ′(λ). The proof is a relatively straightforward modification of similar arguments in [17], +so we defer it to Appendix A. +Lemma 5.4. Let φ(λ) be the characteristic function defined in (5.1), then, under the conditions of +Theorem 2.2, the following estimates hold +φ(λ) = E [�e(λ)] + O≺ +� +N −ε0/2� +, +φ′(λ) = i +π +� +Ω0 +∂ �f +∂¯z E [�e(λ) {1 − E} [Tr G(z)]] d¯zdz + O≺ +� +|λ|N −ε0/2� +, +(5.9) +where +�e(λ) := exp +�iλ +π +� +Ω′ +0 +∂ �f +∂¯z {1 − E} [Tr G(z)] d¯zdz +� +. +(5.10) +Furthermore, for all z ∈ Dκ, we have +E [�e(λ) {1 − E} [Tr G(z)]] = E [�e(λ) {1 − E} T (z, z)] + 2iλ +π E +� +�e(λ) +� +Ω′ +0 +∂ �f +∂¯ζ +∂ +∂ζ T (z, ζ)d¯ζdζ +� ++ iλ +π E [�e(λ)] +� +Ω′ +0 +∂ �f +∂¯ζ Tr [Sm′(z)m′(ζ)] d¯ζdζ ++ iλ +2π E [�e(λ)] +� +Ω′ +0 +∂ �f +∂¯ζ +∂2 +∂z∂ζ +� +m(z)m(ζ), C(4)m(z)m(ζ) +� +d¯ζdζ ++ O≺ +� +(1 + |λ|4)(NΨ(z)Θ(z) + Ψ(z)η−1/2 +0 +) +� +, +(5.11) +where the random function T (z, ζ) is defined as +T (z, ζ) := Tr +�m′(z) +m(z) T (z, ζ) +� +. +(5.12) +We now proceed to estimate the first two terms on the right-hand side of (5.11) in such a way +that E [�e(λ)] factors out. By definition of the scaled test function (2.3), the support of �f is contained +inside a vertical strip centered at E0 of width ∼ η0, hence we limit the further analysis to the regime +| Re ζ − Re z| ≲ η0 ≪ ǫ, where ǫ is defined in the statement of Theorem 3.2. We estimate the function +T (z, ζ) using Corollary 3.3 with weight matrix A := m′(z) +m(z) . It follows from the bounds (4.2) and (4.14) +that ∥A∥ℓ∞→ℓ∞ ≲ 1, hence for all z, ζ ∈ Dκ with Re z, Re ζ ∈ supp(f), +T (z, ζ) = Tr +�m′(z) +m(z) +� +1 − Sm(z)m(ζ) +�−1� +Sm(z)m(ζ) +�2 +� ++ E(z, ζ), +(5.13) +9 + +where the error term E(z, ζ) is analytic in both variables and admits the bound +E(z, ζ) ≺ NΨ2(z)Ψ(ζ) + NΨ(z)Ψ2(ζ) + 1{Im z Im ζ<0}NΘ(z)Θ(ζ). +(5.14) +It follows from (5.13) and (5.14) for ζ = z that +E [�e(λ){1 − E} [T (z, z)]] ≺ NΨ(z)3, +(5.15) +yielding the desired bound on the first term on the right-hand side of (5.11). +We now estimate the second term in (5.11). Fix z ∈ Dκ, and consider ζ that lie in Ω′ +0 defined in +(5.5). Differentiating (5.13) with respect to ζ yields +∂ +∂ζ T (z, ζ) = ∂ +∂ζ Tr +�m′(z) +m(z) +� +1 − Sm(z)m(ζ) +�−1� +Sm(z)m(ζ) +�2 +� ++ ∂ +∂ζ E(z, ζ). +(5.16) +To bound the derivative of the error term E(z, ζ), we use the following technical lemma. +Lemma 5.5. (Lemma 5.5 in [17]) Let K(z) be a holomorphic function on C\R, then for all z ∈ C\R +and any p ∈ N, +���� +∂pK +∂zp (z) +���� ≤ Cp| Im z|−p +sup +|ζ−z|≤| Im z|/2 +|K(ζ)|, +(5.17) +where Cp > 0 is a constant depending only on p. +Lemma 5.5 applied to the estimate (5.14) implies that the error term ∂ζE(z, ζ) admits the bound +∂ +∂ζ E(z, ζ) ≺ N| Im ζ|−1� +Ψ(z)2Ψ(ζ) + Ψ(z)Ψ(ζ)2 + Θ(z)Θ(ζ) +� +. +(5.18) +To proceed we require another technical lemma. +Lemma 5.6. (c.f. Lemma 4.4 in [19]) Let f be the scaled test function defined in (2.3). Let Ω be a +domain of the form +Ω := {z ∈ C : cN −τ ′η0 < | Im z| < 1, a < Re z < b}, +(5.19) +such that supp(f) ⊂ (a, b) and τ ′, c are positive constants. Let K(z) be a holomorphic function on Ω +satisfying +|K(z)| ≤ C| Im z|−s, +z ∈ Ω, +(5.20) +for some 0 ≤ s ≤ 2. Then there exists a constant C′ > 0 depending only on g in (2.3), χ in (5.6), and +s, such that +���� +� +Ω +∂ �f +∂¯z (x + iy)K(x + iy)dxdy +���� ≤ CC′η1−s +0 +log N. +(5.21) +Proof of Lemma 5.6. It follows from (2.3) that ∥f∥1 ∼ η0, ∥f ′∥1 ∼ 1, ∥f ′′∥1 ∼ η−1 +0 . In case 1 ≤ s ≤ 2 +the inequality (5.21) follows from Lemma 4.4 in [19]. For 0 ≤ s < 1, the proof is conducted along the +same lines, except the integration by parts is performed twice in the regime η0 ≤ | Im z| ≤ 1. +Lemma 5.6 and the matrix identity (1−X)−1X2 = (1−X)−1−X −1 yield the following expression. +E +� +�e(λ) +� +Ω′ +0 +∂ �f +∂¯ζ +∂T +∂ζ d¯ζdζ +� += E [�e(λ)] +� +Ω′ +0 +∂ �f +∂¯ζ +∂ +∂ζ Tr +�m′(z) +m(z) +� +1 − Sm(z)m(ζ) +�−1 +� +d¯ζdζ +− E [�e(λ)] +� +Ω′ +0 +∂ �f +∂¯ζ Tr +� +Sm′(z)m′(ζ) +� +d¯ζdζ ++O≺ +� +N 1/2Ψ(z)2η−1/2 +0 ++ Ψ(z)η−1 +0 ++ Θ(z)η−1 +0 +� +, +(5.22) +Finally, from (5.11) and (5.22), combined with (5.9) we conclude that +φ′(λ) = −λV (f) E [�e(λ)] + �E(λ), +(5.23) +10 + +where V (f) is defined in (5.3), and �E(λ) is the total error term collected from previous derivations +and integrated over d¯zdz. Lemma 5.6 together with error estimates in (5.9), (5.11), (5.15) and (5.18) +provides the following bound on the error term +�E = O≺ +� +N −1/2η−1/2 +0 +(1 + |λ|4) + |λ|N −ε0/2� +. +(5.24) +Under the conditions of Proposition 5.1 V (f) is bounded, hence we conclude from the first estimate +in (5.9) and (5.23) that (5.2) holds. This concludes the proof of Proposition 5.1. +6 +Proof of the Local Laws for Two-point Functions +In this section, we derive all the tools necessary to prove Theorem 3.2 and its specification for the two- +point function T (z, ζ), Corollary 3.3. To make the notation more concise we introduce the convention +G ≡ G(z), �G ≡ G(ζ), m ≡ m(z), �m ≡ m(ζ), �Ψ ≡ Ψ(ζ), Ψ ≡ Ψ(z), Θ ≡ Θ(z), �Θ ≡ Θ(ζ). +For a deterministic matrix W with entries |Wab| ≲ N −1, the quantity � +a̸=y WaxGαa �Gaβ can be +readily estimated in two special cases. First, if each column of W is proportional to the vector of ones, +i.e., Wab = wb depends only on b, then the summation over a yields wx([G �G]αβ − Gαy �Gyβ), and the +estimate follows from the resolvent identity and the local laws in Theorem 4.2. Second, if the entries +of X := (1 − Sm �m)−1W are bounded by CN −1, then one can obtain the estimate from Lemma 6.1 +below. We show that these two special cases are exhaustive in the sense that any W can be represented +as their linear combination with controlled coefficients. +To this end, we prove that in the relevant regime, the operator (1 − Sm �m) has a very small +destabilizing eigenvalue and an order one spectral gap above it. Moreover, if Π is the eigenprojector +corresponding to the principal eigenvalue of (1 − Sm �m), then the ℓ∞ → ℓ∞-norm of the restriction +of (1 − Sm �m)−1 to the kernel of Π is also an order one quantity. Finally, we show that the vector of +ones 1 is sufficiently separated from the kernel of Π. +6.1 +Stable Direction Local Law +For any N × N deterministic matrix W, and any indices x, y, α, β, we define the quantities +Fxy +αβ(W) := +� +a̸=y +WaxGαa �Gaβ, +f xy +α (W) := mα �mα([(1 − Sm �m)−1W]αx − δαyWαx). +(6.1) +We prove the following estimate. +Lemma 6.1. For any z, ζ ∈ Dκ and any deterministic N × N matrix X, +Fxy +αβ((1 − Sm �m)X) = δαβf xy +α ((1 − Sm �m)X) + O≺ +� +N ∥X∥max Ψ�Ψ(Ψ + �Ψ) +� +. +(6.2) +provided ∥X∥max := max +j,k |Xjk| ≲ 1. +We use the following self-improving mechanism for stochastic domination bounds, borrowed, e.g., +from [14]. +Lemma 6.2. (Lemma 6.3 in [14]) Let X be a random variable such that 0 ≤ X ≺ N C for some C > 0, +and let Ξ ≥ 0 be a deterministic quantity. Suppose there exists a constant q ∈ [0, 1), such that for any +Φ satisfying Ξ ≤ Φ ≤ N C, and any d ∈ N, we have the implication +X ≺ Φ +=⇒ +E +� +|X|2d� +≺ +2d +� +k=1 +� +ΦqΞ1−q)k E +� +|X|2d−k� +, +(6.3) +then X ≺ Ξ. +Proof of Lemma 6.1. Let Y := (1 − Sm �m)X, then the quantity we need to estimate is [GY ]yx = +Fxy +yy (Y ). It follows from the local law in the form (4.4) that +Fxy +αβ(Y ) ≺ N ∥X∥max Ψ�Ψ =: Λ. +(6.4) +11 + +Let Φ be a deterministic control parameter admitting the bounds (Ψ + �Ψ)Λ ≤ Φ ≤ Λ, such that +Fxy +αβ(Y ) − δαβf xy +α (Y ) ≺ Φ. +(6.5) +It follows trivially from (6.4) and (6.5) that +Fxy +αβ(Y ) ≺ Φ + δαβΛ. +(6.6) +Let ∂jk denote the partial derivative with respect to the matrix element Hjk, then the partial derivatives +of Fxy +αβ are given by +∂abFxy +αβ(Y ) = −(1 + δab)−1(GαaFxy +bβ (Y ) + GαbFxy +aβ(Y ) + Fxy +αb (Y ) �Gaβ + Fxy +αa(Y ) �Gbβ). +(6.7) +We combine the vector Dyson equation (3.4) and the resolvent identity zG = HG − 1 to obtain +�Gaβ = − �ma +� +b +� +Hab �Gbβ + Sab �mb �Gaβ +� ++ �maδaβ. +(6.8) +Let d ∈ N, define P ≡ P(d − 1, d) := (Fxy +αβ(Y ) − δαβf xy +α (Y ))d−1(Fxy +αβ(Y ) − δαβf xy +α (Y ))d. For any +p ∈ N, define Mp := E +� +|Fxy +αβ(Y ) − δαβf xy +α (Y )|p� +. Plugging (6.8) into the definition (6.1) and applying +the cumulant expansion formula of Lemma 4.6, we obtain +E +� +Fxy +αβ(X)P +� += +� +a̸=y +ma �maXax E +� +Fay +αβ(S)P +� ++ δαβf xy +α (Y ) E[P] + δαβδβySyym2 +y �m2 +yXyx E[P] +(6.9a) ++ E +�� +a̸=y +� +b +�maXaxSab +� +Gαa( �Gbb − �mb) �Gaβ + Gαb(Gaa − ma) �Gbβ +�P +� +(6.9b) ++ E +�� +a̸=y +� +b̸=a +�maXaxSabGαa +� +Gba + �Gba +� �GbβP +� ++ R2 +(6.9c) ++ +� +a̸=y +Xaxma �maSay E +�� +Gαy �Gyβ − δαyδyβmy �my +�P +� +(6.9d) ++ δβ̸=y �mβXβx E +� +(Gαβ − δαβmβ)P +� +− E +�� +a̸=y +�maXaxGαa +� +b +Sab �Gbβ∂abP +� +, +(6.9e) +where R2 is the total error coming from the higher order cumulants, and all unrestricted summations +are from 1 to N. We successively bound the terms (6.9b)-(6.9e) appearing on the right-hand side of +(6.9). By condition (A), local law (4.4), upper bound (4.2), and (6.5), it follows that the terms (6.9b) +and the first term in (6.9c) are bounded by O≺((Ψ + �Ψ)ΛM2d−1). Similarly, the term (6.9d) and the +first term in (6.9e) are bounded by O≺(∥X∥max (Ψ + �Ψ)M2d−1). +We bound the second term in (6.9e). It follows by (A), (4.4), bounds (4.2), (6.6), and (6.7) that +� +b +Sab �Gbβ∂abP ≺ (Ψ + �Ψ + δαa + δaβ)�ΨΦM2d−2. +(6.10) +Hence, the second term in (6.9e) is bounded by O≺ +� +(Ψ + �Ψ)ΛΦM2d−2 +� +. Finally, it is easy to check +using estimates (4.16), (6.6) and identity (6.7), together with condition (A) and (4.2), that the error +term R2 ≺ (Ψ + �Ψ)ΛM2d−1 + (Ψ + �Ψ)ΛΦM2d−2 + (Ψ + �Ψ)ΛΦ2M2d−3. +Observe that the first term on the right-hand side of (6.9a) can be expressed as +� +a̸=y +ma �maXax E +� +Fay +αβ(S)P +� += E +� +Fay +αβ(X)P +� +− E +� +Fay +αβ(Y )P +� +− my �myXyx E +� +Fyy +αβ(S)P +� +, +(6.11) +where the last term is bounded by O≺(N −1ΛM2d−1). Combining (6.9) and (6.11) yields +E +� +|Fxy +αβ(Y ) − δαβf xy +α (Y )|2d� +≺ +� +Ψ + �Ψ +� +ΛΦ2M2d−3, +(6.12) +for any control parameter Φαβ,y satisfying (6.5). Hence, by Lemma 6.2, +Fxy +αβ(Y ) = δαβf xy +α (Y ) + O≺ +� +Λ(Ψ + �Ψ) +� +, +(6.13) +which concludes the proof of Lemma 6.1. +12 + +Remark 6.3. If z and ζ are in the same (upper or lower) half-plane, Lemma 6.1 implies Theorem 3.2. +Indeed, the bound (4.12) in Proposition 4.4 shows that provided η�η > 0, X := (1−Sm �m)−1W satisfies +|Xjk| ≲ N −1. Applying Lemma 6.1 to X = (1 − Sm �m)−1W then yields (3.6), and (3.7) follows by +summing (3.6). We turn to the case of z and ζ lying in different (upper and lower) half-planes. +6.2 +Stability Operator Analysis +In this subsection we obtain all the properties of the stability operator (1 − Sm(z)m(ζ)) that we use +in combination with Lemma 6.1 to finish the proof of Theorem 3.2 for z, ζ lying in opposite half-planes, +as outlined in the beginning of Section 6. +For two spectral parameters z, ζ, let η := Im z, and �η := Im ζ. Without loss of generality, we assume +in the following that Re z ∈ Iκ, η > 0 and Re ζ ∈ Iκ, �η < 0. For the remainder of this subsection, we +use the following notation +F ≡ F(z) := |m(z)|S|m(z)|, +B ≡ B(z, ζ) := 1 − Sm(z)m(ζ), +B0 ≡ B0(z) := 1 − S|m(z)|2 = |m(z)|−1(1 − F)|m(z)|. +(6.14) +We view the operator B as a perturbation of B0 = B(z, ¯z), since |ζ − ¯z| is small. We deduce the +desired properties of B from those of B0, which, in turn, follow from the lower bound on the spectral +gap of F found in (4.9). +Let {ψj}N +j=1 denote the eigenvalues of F (with multiplicity) in descending order. Then, by Per- +ron–Frobenius theorem, the principal eigenvalue ψ1 is real, and it coincides with the spectral radius +∥F∥ℓ2→ℓ2. Furthermore, by taking the imaginary part of the vector Dyson equation (3.4) and multi- +plying both sides by |m| coordinate-wise, we obtain +� +1 − F +�Im m +|m| = η|m|. +(6.15) +Furthermore, by condition (A), for every j we have (S Im m)j ∼ ⟨Im m(z)⟩ ∼ ρ(z), where ρ(z) is +the harmonic extension of the self-consistent density of states ρ(x) defined in (3.3) into C. Hence by +taking the imaginary part of (3.4), we get +Im mj +|mj| +∼ |mj|(ρ(z) + η), , +j ∈ {1, . . . , N}. +(6.16) +Therefore, by (6.15) and (6.16), 1 − ψ1 ≲ η. Together with an upper bound (4.10) on ∥F∥ℓ2→ℓ2, this +implies that 1 − ψ1 ∼ η. It follows from (4.9) that the principal eigenvalue of F is separated from the +rest of the spectrum by an annulus, i.e., there exist r > 0 and δ > 0 independent of z and N such that +|1 − ψ1| < r − δ, +and +|1 − ψj| > r + δ, +j ∈ {2, . . . , N}. +(6.17) +In the remainder of this subsection, we show that for all ζ sufficiently close to ¯z, the eigenvalue of +B with the smallest modulus is also separated from the rest of the spectrum by an annulus of order +one width. +Using the argument principle and Jacobi’s formula, one can express the number of eigenvalues +(with multiplicity) of a matrix X inside a domain Ω by a contour integral +NX(Ω) = +1 +2πi +� +∂Ω +Tr(w − X)−1dw. +(6.18) +To show the eigenvalue separation for B, we begin by estimating the norm of the resolvent of B inside +the annulus +Ar,δ := {w ∈ C : r − 3δ/4 ≤ |w| ≤ r + 3δ/4}, +(6.19) +with r and δ as in (6.17). +13 + +Claim 6.4. There exists ε1 > 0 and �C > 0 independent of N and z such that +���(w − B(z, ζ))−1��� ≤ �C +(6.20) +holds for all w ∈ Ar,δ and all ζ such that Re ζ ∈ Iκ, Im ζ < 0 and |ζ − ¯z| ≤ ε1. (The norm ∥·∥ is +induced by either ℓ2 or ℓ∞.) +Proof. Observe that +��(w − B)−1�� ≤ +��� +� +1 − (w − B0)−1(B − B0) +�−1��� +��(w − B0)−1��. +Since (w − B0)−1 = −|m|−1(1 − w − F)−1|m| and |m| ∼ 1, (6.17) implies that +��(w − B0)−1�� ≤ +C +min +j +|ψj − w| ≤ 4C +δ , +w ∈ Ar,δ. +(6.21) +From the uniform bounds (4.2), (4.14) on |m| and |m′| we have ∥B − B0∥ ≲ |ζ − ¯z|, which implies +that there exists ε1 > 0 such that +∀ζ : |ζ − ¯z| ≤ ε1, ∥B − B0∥ ≤ +δ +8C , +(6.22) +where C is the constant in (6.21). +It follows immediately that +��� +� +1 − (w − B0)−1(B − B0) +�−1��� ≤ 2 and hence +��(w − B)−1�� ≤ 8C +δ . +(6.23) +Claim 6.4 implies that for any sufficiently large fixed N the integrand in (6.18) with X := B is +uniformly bounded in Ω := Ar,δ for all ζ such that |ζ − ¯z| ≤ ε1, hence by analyticity +NB(z,ζ)(Ar,δ) = 0, +|ζ − ¯z| ≤ ε1. +(6.24) +Since the eigenvalues of B(z, ζ) are continuous in ζ, (6.24) implies that no eigenvalue can move between +the two connected components of C\Ar,δ, which together with (6.17) yields the following claim. +Claim 6.5. For any sufficiently large N, the equalities +NB({|w| < r − 3δ/4}) = NB0({|w| < r − 3δ/4}) = 1, +NB({|w| > r + 3δ/4}) = NB0({|w| > r + 3δ/4}) = N − 1, +(6.25) +hold for any ζ such that Re ζ ∈ Iκ, Im ζ < 0 and |ζ − ¯z| ≤ ε1. +Claim 6.5 now allows us to define the principal eigenprojector Π of B as a contour integral +Π ≡ Π(z, ζ) := +1 +2πi +� +|ξ|=r +(ξ − B(z, ζ))−1dξ. +(6.26) +Claim 6.5 asserts that the contour {|ξ| = r} encircles exactly one eigenvalue of B with multiplicity, +hence Π is a rank one eigenprojector. +We now prove that the restriction of B−1 to the range of (1 − Π) is bounded by a constant. +Claim 6.6. For all z, ζ such that Re z, Re ζ ∈ Iκ, Im z Im ζ < 0 and |ζ − ¯z| ≤ ε1, +��B−1(1 − Π) +�� +ℓ∞→ℓ∞ ≤ �c, +(6.27) +where �c depends only on the constants in conditions (A), (B) and κ. +Proof. By expression (6.26) for Π we have +B−1(1 − Π) = − 1 +2πi +� +|ξ|=r +1 +ξ (ξ − B)−1dξ +(6.28) +14 + +Hence the norm of B−1(1 − Π) is bounded by +��B−1(1 − Π) +�� +ℓ∞→ℓ∞ ≤ 1 +2π +2π +� +0 +��� +� +reiθ − B +�−1��� +ℓ∞→ℓ∞ dθ ≤ 8C +δ , +(6.29) +using the bound in Claim 6.4 on the circle {|ξ| = r} which lies inside Ar,δ. +Finally, we show that the vector of ones is sufficiently separated from the kernel of Π. This ensures +a stable decomposition of the space into the direct sum of the range of (1 − Π) and the span of 1, so +we can apply the local laws to each of the components separately. +Claim 6.7. There exists ε > 0 independent of N and z such that for all ζ with Re ζ ∈ Iκ, Im ζ < 0 +and |ζ − ¯z| ≤ ε, +∥Π1∥∞ +∥Π∥ℓ∞→ℓ∞ ≥ c, +(6.30) +where c > 0 is a constant independent of N and z. +Proof. Define the projector Π0 corresponding to B0 via (6.26). Then Π0 = |m|−1�Π0|m|, where �Π0 is +the orthoprojector corresponding to the principal eigenvalue of the Hermitian operator F. +Since |m| ∼ 1 we have ∥Π0∥ℓ∞→ℓ∞ ≤ C0. Moreover, by Proposition 4.3, the ℓ2-normalized eigenvector +v corresponding to the principal eigenvalue of F has entries vj ≥ 0 with vj ∼ N −1/2, hence +∥Π01∥∞ = +���|m|−1�Π0|m|1 +��� +∞ = +��|m|−1v +�� +∞ ⟨v, |m|⟩ ≥ c0, +(6.31) +where c0 > 0 is a constant independent of N and z. +Similarly to the proof of (6.22), for any γ ∈ (0, 1] there exists εγ > 0, such that the bound +∥B − B0∥ℓ∞→ℓ∞ ≤ γ δ +8C +(6.32) +holds for all ζ ∈ D− +κ with |ζ − ¯z| ≤ εγ. Here δ is defined in (6.17) and C > 0 is the constant in (6.21). +We choose εγ to be smaller than ε1 of Claim 6.4, then for all ζ with Re ζ ∈ Iκ, Im ζ < 0 such that +|ζ − ¯z| ≤ εγ we have +∥Π − Π0∥ℓ∞→ℓ∞ ≤ r +2π +2π +� +0 +��(reiθ − B)−1 − (reiθ − B0)−1�� +ℓ∞→ℓ∞ dθ +≤ r +2π +2π +� +0 +��(reiθ − B)−1(B − B0)(reiθ − B0)−1�� +ℓ∞→ℓ∞ dθ +≤ r · 8C +δ · γ δ +8C · 4C +δ += γ 4Cr +δ +. +(6.33) +Here we used inequalities (6.21) and (6.23) in the second to last step. We set the value of γ to be +γ0 := min +� +1, c0δ +8Cr +� +, which guarantees that +∥Π1∥∞ ≥ +��∥Π01∥∞ − ∥Π − Π0∥ℓ∞→ℓ∞ ∥1∥∞ +�� ≥ c0 − γ0 +4Cr +δ +≥ c0 +2 . +(6.34) +Finally, observe that +∥Π∥ℓ∞→ℓ∞ ≤ ∥Π0∥ℓ∞→ℓ∞ + ∥Π − Π0∥ℓ∞→ℓ∞ ≤ C0 + c0/2. +(6.35) +This proves the claim with c := c0/(2C0 + c0). +15 + +6.3 +Finishing the Proof of Theorem 3.2 +Proof of Theorem 3.2. Recall that the objective is to estimate the quantities defined in (3.1). +In- +stead of estimating � +a̸=y waGαa �Gaβ directly, it is more convenient to work with objects of the type +� +a̸=y WaxGαa �Gaβ, since they generalize quantities appearing in both (3.6) and (3.7). The redundant +index x can be eliminated by setting Wax := wa. +In the case Im z Im ζ > 0, (3.6) and (3.7) follow immediately from (4.12) and Lemma 6.1 (see +Remark 6.3). Therefore, we focus on the case Im z Im ζ < 0. +Since Π has rank one and Claim 6.7 asserts that Π1 ̸= 0, the kernel of Π together with 1 span +CN. Therefore we can decompose each column of the matrix W into a linear combination of 1 and an +element of ker Π, that is, there exists an N × N matrix Y and a vector s ∈ CN such that +W = Y + 1s∗, +ΠY = 0. +(6.36) +We multiply the first equality in (6.36) by Π from the left, apply both sides to the a-th standard basis +vector ea of CN and take the ℓ∞-norm to deduce +∥ΠWea∥∞ = |sa| ∥Π1∥∞ , +a ∈ {1, . . ., N}. +(6.37) +By assumption, ∥W∥max ≲ N −1, hence ∥Wea∥∞ ≲ N −1. Using Claim 6.7 we get +|sa| ≲ N −1 ∥Π∥ℓ∞→ℓ∞ +∥Π1∥∞ +≲ N −1, +a ∈ {1, . . . , N}. +(6.38) +We combine (6.36) and the resolvent identity in the form (z − ζ)G �G = G − �G to obtain +� +a̸=y +WaxGαa �Gaβ = +� +a̸=y +YaxGαa �Gaβ + gy +αβ¯sx, +gy +αβ := Gαβ − �Gαβ +z − ζ +− Gαy �Gyβ. +(6.39) +Define the N × N matrix X := (1 − Sm �m)−1 Y . It follows from (6.36) that Y = (1 − Π)Y , hence +X = (1 − Sm �m)−1(1 − Π)Y . Furthermore, estimates ∥W∥max ≲ N −1, (6.36), and (6.38) imply that +|Yab| ≲ N −1 for all a and b. Since by Claim 6.6 +��(1 − Sm �m)−1(1 − Π) +�� +ℓ∞→ℓ∞ ≲ 1, we conclude that +∥X∥max = max +a,b |Xab| ≲ N −1. +(6.40) +First, using (6.40), we can apply Lemma 6.1 to the first term in (6.39) to obtain +� +a̸=y +YaxGαa �Gaβ = δαβmα �mα([(1 − Sm �m)−1Y ]αx − δαyYαx) + O≺ +� +Ψ2 �Ψ + Ψ�Ψ2� +. +(6.41) +Using (6.36), we proceed by computing +mα �mα[(1 − Sm �m)−1Y ]αx = +� +m �m +� +1 − Sm �m +�−1 (W − 1s∗) +� +αx += +� +m �m +� +1 − Sm �m +�−1W +� +αx − +� +m �m +� +1 − Sm �m +�−11 +� +α¯sx. +(6.42) +Finally, it follows from subtracting the vector Dyson equations (3.4) for z and ζ that +m �m +� +1 − Sm �m +�−11 = m − �m +z − ζ . +(6.43) +Next, we estimate the second term in (6.39). Applying the local law in the form (4.4), we obtain +gy +αβ = δαβ +mα − �mα +z − ζ +− δαβδαymα �mα + O≺ +� +(|η| + |�η|)−1(Ψ + �Ψ) +� +, +(6.44) +where we used that |z − ζ| ≥ |η| + |�η|, since η�η < 0. Combining (6.38), (6.39), and (6.41)-(6.44) yields +� +a̸=y +WaxGαa �Gaβ = δαβ +� +m �m +� +1 − Sm �m +�−1W +� +αx − δαβδαy[m �mW]αx ++ O≺ +� +(Ψ + �Ψ)(Ψ�Ψ + min{Θ, �Θ}) +� +, +(6.45) +16 + +which proves (3.6) by setting Wax := wa. +To prove (3.7), we observe that by setting x = y = α = β = b in (6.39) and summing over b yields +� +b +� +a̸=b +WabGba �Gab = +� +b +� +a̸=b +YabGaa �Gab+⟨s, g⟩, +gb := Gbb − �Gbb +z − ζ +−Gbb �Gbb, b ∈ {1, . . ., N}. (6.46) +To estimate ⟨s, g⟩, we use (6.38) and the averaged local law (4.3) to obtain +� +s, g +� += +� +s, m − �m +z − ζ +− m �m +� ++ O≺ +� +(|η| + |�η|)−1(Θ + �Θ) +� +, +(6.47) +where we used that |z − ζ| ≥ |η| + |�η|, since η�η < 0. +Setting x = y = α = β = b in (6.41), summing over b, using the identities (6.42) and (6.43), and +combining the result with (6.47), we deduce that +� +b +� +a̸=b +WabGba �Gab = Tr +� +m �mSm �m +� +1 − Sm �m +�−1W +� ++ NO≺ +� +Ψ�Ψ(Ψ + �Ψ) + Θ�Θ +� +, +(6.48) +where we used that (|η| + |�η|)−1(Θ + �Θ) = NΘ�Θ. This establishes (3.7) and concludes the proof of +Theorem 3.2. +Remark 6.8. We outline the steps needed to achieve the optimal error estimate (3.12). First, one +needs to adapt the proof of Theorem 3.2. More specifically, replace the decomposition (6.36) with +W = Y + 1s∗ + q1∗, such that Π(z, ζ)Y = Y Πt(ζ, z) = 0, +(6.49) +where Π(z, ζ) is the destabilizing eigenprojector defined in (6.26). The terms involving s and q are +handled using the averaged local law (4.3), similarly to (6.47). +For the remaining term, R := � +y Fyy +yy , we adapt the mechanism of Lemma 6.1 by using the +following iterative scheme. +In the first step, we apply an expansion similar to (6.9) to the partial +derivative ∂jkR. This improves the error in the estimate on R by a factor of (Ψ + �Ψ)1/2. If we expand +∂lp∂jkR in a similar manner, we gain another (Ψ + �Ψ)1/4. Iterating this approach we can estimate R +with an error stochastically dominated by NΨ�Ψ(Ψ + �Ψ)2−2−d for any given integer d (where d is the +maximal order of expanded partial derivatives). By Definition 3.1, this is sufficient to establish (3.12). +Similar arguments in the context of random band matrices can be found in [10]. +Proof of Corollary 3.3. Estimate (3.9) on Txy(ζ, z) follows from (3.6) by setting α = β = y and +wa := Sxa. Estimate (3.10) on Tr[AT (z, ζ)] follows from (3.7) by setting W := SAt, which satisfies +|Wab| ≲ N −1 ∥A∥ℓ∞→ℓ∞. This concludes the proof of Corollary 3.3. +Remark 6.9. Note that estimates (3.6) and (3.7) (also with the improved error term (3.12)) hold +without omission of indices in the a summation. Indeed, it follows from Theorems 3.2 and 4.2 that +� +a +waGαa �Gaβ = δαβ +� +m �m +� +1 − Sm �m +�−1w +� +α + O≺ +� +(Ψ + �Ψ)(Ψ�Ψ + 1{η�η<0} min{Θ, �Θ}) +� +, +� +a,b +WabGba �Gab = Tr +� +m �m +� +1 − Sm �m +�−1W +� ++ O≺ +� +N(Ψ + �Ψ)Ψ�Ψ + 1{η�η<0}NΘ�Θ +� +. +(6.50) +7 +Proof of Proposition 5.2 +In this section, we compute the variance V (f) defined in (5.3) for mesoscopic C2 +c test functions f. In +[17], the limiting variance was computed for several types of C∞ test functions, including compactly +supported ones; however, V (f) is computed with an O(1) error (see, e.g., Lemma 6.7 in [17]), which +is not negligible in the setting of the present paper. To obtain effective error bounds, we augment the +proof laid out in [17] by performing further integration by parts in the integral representation of V (f), +thus eliminating the f ′ terms, improving the error by a factor of O(η0). +Throughout this section, we adhere to the notation m ≡ m(z), �m ≡ m(ζ), η := Im z, �η := Im ζ. +17 + +The stability operator (1 − Sm �m) can be expressed in terms of the self-saturated energy operator +F, defined in (4.6), via the following identity +1 − Sm �m = |m �m|−1/2 (U∗ − F(z, ζ)) |m �m|1/2U, +U := m �m +|m �m|. +(7.1) +Furthermore, by (4.9), the operator F can be decomposed such that +F(z, ζ) = ψ1(z, ζ) v(z, ζ) +� +v(z, ζ) +�∗ + A(z, ζ), +A(z, ζ)v(z, ζ) = 0, +∥A(z, ζ)∥ℓ2→ℓ2 ≤ 1 − �δ, +(7.2) +where ψ1, v is the principal eigenvalue-eigenvector pair of F, and �δ is the constant in (4.9). +Let R ≡ R(z, ζ) denote (U∗(z, ζ) − A(z, ζ))−1. In the sequel, we drop the arguments and write +A ≡ A(z, ζ). Lower bound (4.8) and the inequality in (7.2) imply that +∥R∥ℓ2→ℓ2 + ∥R∥ℓ∞→ℓ∞ ≲ 1. +(7.3) +In the following lemma, we collect the perturbative estimates on the saturated self-energy operator F +and related quantities established in [17]. +Lemma 7.1. (Proposition 6.5, (6.52), (6.60), (6.71), and (6.67) in [17]) Let w, ζ1, ζ2 be spectral +parameters in Iκ + i[−1, 1], and let F be the operator defined in (4.6), then the principal eigenvalue- +eigenvector pair ψ1, v of F satisfies +∥v(w, ζ1) − v(w, ζ2)∥ℓ2→ℓ2 + |ψ1(w, ζ1) − ψ1(w, ζ2)| ≲ |ζ1 − ζ2|. +(7.4) +Furthermore, for operator A defined in (7.2), we have the estimate +∥F(w, ζ1) − F(w, ζ2)∥ℓ2→ℓ2 + ∥A(w, ζ1) − A(w, ζ2)∥ℓ2→ℓ2 ≲ |ζ1 − ζ2|. +(7.5) +Let z := x + iη, ζ := y − iη, with x, y ∈ Iκ, 0 ≤ η ≤ 1, then +ψ1 +� +v, Rm′ +m U∗Rv +� += ψ1(z, z) +� +v(z, z)m′ +m v(z, z) +� ++ O(|x − y|) +(7.6) +Let ω ≡ ω(z, ζ) := 1 − ψ1⟨v, Rv⟩, then +ω(z, ζ) = 1 − ψ1(z, z) + ψ1(z, z)(x − y) +� +v(z, z)m′ +m v(z, z) +� ++ O(|x − y|2), +(7.7) +Moreover, there exists ε > 0 independent of N, such that for all x, y ∈ Iκ satisfying |x − y| ≤ ε, +|ω(z, ζ)| ≳ η + |x − y|. +(7.8) +Finally, for z := x + iη with x ∈ Iκ, the following identity holds +lim +η→+0 +� +v(z, z)m′ +m v(z, z) +� += iπ +2 ρ(x) +���� +Im m(x + i0) +|m(x)| +���� +−2 +2 +(7.9) +By our choice of κ, E0 is in the interior of the bulk interval Iκ, defined in (3.5) , hence if we define +ˆε := min{ε/4, dist(E0, R\Iκ)}, then ˆε ∼ 1. Furthermore, since the function g is compactly supported, +we assume that supp(f) ⊂ [E0 − ˆε, E0 + ˆε] for large N. +Lemma 7.2. Let η∗ ≡ η∗(N) satisfy 0 < η∗ ≤ N −100, then V (f), defined in (5.3), admits the estimate +V (f) = +1 +4π2 +�� +[E0−ˆε,E0+ˆε]2 +(f(y) − f(x))2 �K(x + iη∗, y − iη∗)dxdy + O +� +η0 + N −ε0� +, +(7.10) +where +�K(z, ζ) := −2 Re Tr +�m′ +m (1 − Sm �m)−1Sm �m′(1 − Sm �m)−1 +� +. +(7.11) +18 + +In preparation for the proof of Lemma 7.2 we define an auxiliary function L(z, ζ) +L(z, ζ) := Llog(z, ζ) + L1(z, ζ), +Llog(z, ζ) := −2 log det {1 − Sm �m} , +L1(z, ζ) := − Tr [Sm �m] + 1 +2 +� +m �m, C(4)m �m +� +, +(7.12) +where log is the principal branch of the complex logarithm, and C(4) is the matrix of the fourth cumu- +lants of H. By Jacobi’s formula for the derivative of the determinant, it follows from the definitions +of L and K, that for all z, ζ ∈ C\R +∂2 +∂ζ∂z L(z, ζ) = K(z, ζ). +(7.13) +Furthermore, by condition (A) and the upper bound (4.2), it follows that +|Llog(z, ζ)| ≤π + log |det {1 − Sm �m}| ≲ 1 + Tr +� +(1 − Sm �m)∗ (1 − Sm �m) − I +� +≲ 1, +(7.14) +where in the last line we used +� +(1 − Sm �m)∗ (1 − Sm �m) − I +� +jj ≲ N −1. +The partial derivatives of L1 contribute only sub-leading terms to L. Indeed, we have the estimates +L1(z, ζ) ≲ 1, +∂ +∂zL1(z, ζ) ≲ 1, +∂2 +∂ζ∂z L1(z, ζ) ≲ 1, +(7.15) +where we used the moment condition (2.2) to bound Sjk and C(4) +jk , (4.2) to get the upper bound +m, �m ≲ 1, and (4.14) to obtain m′, �m′ ≲ 1, since [E0 + ˆε, E0 − ˆε] ⊂ Iκ. +The following claim collects the bounds on K and ∂zL that together with (7.14) enable integration +by parts in the definition (5.3) of the variance V (f), which is the essence of Lemma 7.2. +Claim 7.3. (Proposition 6.2 and Proposition 6.6 in [17]) Let K(z, ζ) and L(z, ζ) be as defined in (5.4) +(with β = 1) and (7.12) respectively, then for all z, ζ ∈ C\R with Re z, Re ζ ∈ [E0 − ˆε, E0 + ˆε] and +| Im z|, | Im ζ| ≤ 1 we have +K(z, ζ) ≲ 1 + 1{η�η<0}(|η| + |�η|)−2, +∂ +∂z L(z, ζ) ≲ 1 + (| Re z − Re ζ| + |η| + |�η|)−1, +(7.16) +where η := Im z, �η := Im ζ. +Proof of Lemma 7.2. Define Ω∗ := {z ∈ C : 1 > | Im z| > η∗}. Recall the definition of V (f) from (5.3). +First, we prove that +V (f) = 1 +π2 +� +Ω∗ +� +Ω∗ +∂ �f(ζ) +∂¯ζ +∂ �f(z) +∂¯z +K(z, ζ)d¯ζdζd¯zdz + O +� +N −ε0� +. +(7.17) +It follows from (5.6) that +∂ �f +∂¯z = 1 +2 +� +−ηχ′(η)f ′(x) + i +� +ηχ(η)f ′′(x) + χ′(η)f(x) +�� +. +(7.18) +Moreover, for all z with | Im z| < 1/2, (7.18) and the properties of χ in (5.6) imply +∂ �f +∂¯z = i Im z +2 +f ′′(Re z). +(7.19) +Let V∗(f) denote the integral on right hand side of (7.17), and define η1 := N −ε0/2η0. It follows +from the first inequality in (7.16), and (7.19) that +|V (f) − V∗(f)| ≲ +�� +R2 +|f ′′(x)f ′′(y)| dxdy +η1 +� +η∗ +2η1 +� +η∗ +η�η +(η + �η)2 d�ηdη. +(7.20) +19 + +Note that η�η ≤ (η + �η)2/4, hence the integral over d�ηdη is bounded by η2 +1/2, and since ∥f ′′∥1 ∼ η−1 +0 , +(7.17) is established. +We write z := x + iη, ζ := y + i�η and plug (7.13) into the expression (7.17) for V (f). Using the +fact that ∂zu = −i∂ηu for any holomorphic function u(z), and integrating by parts in η, we obtain +V (f) = i +π2 +�� +R2 +dxdy +� +|�η|>η∗ +∂ �f(ζ) +∂¯ζ +� +|η|>η∗ +∂2 �f(z) +∂η∂¯z +∂ +∂ζ L(z, ζ)d�ηdη +− i +π2 +�� +R2 +dxdy +� +|�η|>η∗ +∂ �f(ζ) +∂¯ζ +� +η=±η∗ +∂ �f +∂¯z (x + iη) ∂ +∂ζ L(z, ζ)d�η + O +� +N −ε0� +. +(7.21) +The second estimate in (7.16), expression (7.18) and the estimates ∥f ′′∥1 ∼ η−1 +0 , ∥f ′∥1 ∼ 1, ∥f∥1 ∼ η0 +imply that the boundary term in (7.21) is dominated by O≺(η∗η−2 +0 ), which is smaller than O (N −ε0). +Similarly, integrating the first term on the right hand side of (7.21) by parts in �η we get +V (f) = − 1 +π2 +� +Ω∗ +� +Ω∗ +∂2 �f(z) +∂¯z∂η +∂2 �f(ζ) +∂¯ζ∂�η L(z, ζ)d¯ζdζd¯zdz ++ 1 +π2 +�� +R2 +dxdy +� +|η|>η∗ +∂2 �f(z) +∂η∂¯z +� +�η=±η∗ +∂ �f +∂¯ζ (y + i�η)L(z, y + i�η)dη + O +� +N −ε0� +. +(7.22) +It follows from (7.14) and the expression (7.18) that the boundary term (the second line of (7.22)) is +again dominated by O≺(N −ε0). +We apply Stokes’ theorem to (7.22) twice: once in z and once in ζ. Considering that ∂η �f(z) vanishes +on the boundary of Ω∗ except for the lines {Im z = ±η∗}, this results in +V (f) = 1 +4π2 +�� +R2 +� +η,�η=±η∗ +sign (η�η) ∂ �f(x + iη) +∂η +∂ �f(y + i�η) +∂�η +L(x + iη, y + i�η)dxdy + O +� +N −ε0� += − +1 +2π2 +�� +R2 +f ′(x)f ′(y) �L(x, y)dxdy + O +� +N −ε0� +, +(7.23) +where +�L(x, y) := Re [L(x + iη∗, y + iη∗) − L(x + iη∗, y − iη∗)] +(7.24) +We restrict the integrations in (7.23) to [E0 − ˆε, E0 + ˆε], since this interval contains the support of f. +Furthermore, for all y ∈ supp(f), y − E0 ≲ η0, hence |y − E0 ± ˆε| ∼ 1. By symmetry of L(z, ζ), and +the second estimate in (7.16) it follows that +∂ +∂y +�L(E0 ± ˆε, y) ≲ 1, +y ∈ supp(f). +(7.25) +We write f ′(y) = ∂y (f(y) − f(x)), perform integration by parts in y and integrate the boundary term +by parts in x to obtain +V (f) = 1 +2π2 +E0+ˆε +� +E0−ˆε +E0+ˆε +� +E0−ˆε +f ′(x) (f(y) − f(x)) ∂ +∂y +�L(x, y)dxdy ++ +1 +4π2 +E0+ˆε +� +E0−ˆε +(f(x))2 ∂ +∂x +� +�L(x, E0 + ˆε) − �L(x, E0 − ˆε) +� +dx + O +� +N −ε0� +. +(7.26) +Since ∥f∥2 +2 ≲ η0, it follows from (7.25) that the second integral in (7.26) is O (η0). +Similarly, +integrating (7.26) by parts in x and using (7.26) to substitute one of the emerging itegrals for +−V (f) + O (N −ε0 + η0), we get +2V (f) = 1 +2π2 +E0+ˆε +� +E0−ˆε +E0+ˆε +� +E0−ˆε +(f(y) − f(x))2 +∂2 +∂x∂y +�L(x, y)dxdy + O +� +η0 + N −ε0� +, +(7.27) +20 + +where we again used (7.25) to estimate the boundary term. For any holomorphic function u(z) of +z = x + iη, we have ∂xu = Re[∂zu], hence ∂x∂y �L(x, y) = Re [K(x + iη∗, y + iη∗) − K(x + iη∗, y − iη∗)]. +Finally, in view of in view of the first estimate in (7.16), ∂z∂ζLlog(x + iη∗, y + iη∗) ≲ 1, so its +contribution is also bounded by O≺(η0 ∥g∥2 +2 + η2 +0 ∥g∥2 +1). Moreover, it follows from the last estimate in +(7.15) that we can replace K(x+iη∗, y −iη∗) by ∂z∂ζLlog(x+iη∗, y −iη∗), since the contribution of the +remaining terms is bounded by O≺(η0 ∥g∥2 +2 + η2 +0 ∥g∥2 +1). This concludes the proof of Lemma 7.2. +Once Lemma 7.2 is established, we can follow the method of Lemma 6.7 in [17] to finish the proof +of Proposition 5.2. +Fix x, y ∈ [E0 − ˆε, E0 + ˆε] and write z := x + iη∗, ζ := y − iη∗, as in (7.10). It follows from (7.1) +and (7.2) that the kernel �K(z, ζ) can be written as +�K(z, ζ) = −2 Re Tr +�m′ +m U∗� +R + ψ1 +ω Rvv∗R +� +F �m′ +�m +� +R + ψ1 +ω Rvv∗R +�� +, +(7.28) +where ω is defined in (7.7). Expanding the brackets in (7.28), collecting like terms according to the +powers of ω−1, and using the cyclic property of trace yields +�K(z, ζ) = −2 Re +�ψ2 +1 +ω2 +� +v, Rm′ +m U∗Rv +�� +v, RF �m′ +�m Rv +�� ++ O +� +1 + ω−1� +, +(7.29) +since Tr +� m′ +m U∗RF � +m′ +� +m R +� +, Tr +� m′ +m U∗RF � +m′ +� +m Rvv∗R +� +, and Tr +� m′ +m U∗Rvv∗RF � +m′ +� +m R +� +are all O(1). The first +scalar product in (7.29) can be estimated using (7.6). +We compute the second scalar product in (7.29). It follows from uniform bounds (4.2) and (4.14) +that ∥m(z)− m(¯ζ)∥∞ ≲ |x− y|, and hence ∥U(z, ζ) − 1∥ℓ2→ℓ2 ≲ |x− y|. Together with estimates (7.5) +and (7.4), this yields +ψ1 +� +v, RF �m′ +�m Rv +� += ⟨v(ζ, ζ), F(ζ, ζ) � +m′ +�m v(ζ, ζ) +� ++ O(|x − y|), +(7.30) +where we used the identity R(¯ζ, ζ)v(ζ, ζ) = (1 − A(ζ, ζ))−1v(ζ, ζ) = v(ζ, ζ). +It follows from the estimate on v in (7.4) that ∥v(ζ, ζ) − v(y, y)∥2 ≲ η∗. Vector v(y, y) is the ℓ2- +normalization of |m(y)|−1 Im m(y + i0), hence it satisfies F(y, y)v(y, y) = v(y, y) by (3.4). Therefore +using (4.14) and the lower bound in (7.5), we obtain +∥F(ζ, ζ)v(ζ, ζ) − v(ζ, ζ)∥2 ≲ η∗. +(7.31) +Substituting (7.31) into (7.30) yields +ψ1 +� +v, RF �m′ +�m Rv +� += ⟨v(ζ, ζ), �m′ +�m v(ζ, ζ) +� ++ O(|x − y| + η∗), +(7.32) +Combining (7.28) with estimates (7.4), (7.6), (7.8) and (7.32) yield +�K(z, ζ) = −2 Re +�ψ1(z, z)ψ1(ζ, ζ) +ω2 +� +v(z, z)m′ +m v(z, z) +� +⟨v(ζ, ζ), �m′ +�m v(ζ, ζ) +�� ++ O(1 + ω−1). +(7.33) +It follows by (7.9) and (7.7) that +lim +η∗→+0 +�K(x + iη∗, y − iη∗) = 2|x − y|−2 + O(|x − y|−1). +(7.34) +Since f ∈ C2 +c (R), (7.33) implies that the integrand in (7.10) is uniformly bounded in η∗ ∈ [0, N −100]. +Therefore, we can take the limit η∗ → 0 in (7.10), and apply the boundary estimate (7.34) to obtain. +V (f) = +1 +2π2 +�� +[E0−ˆε,E0+ˆε]2 +(f(x) − f(y))2 +(x − y)2 +dxdy + O +� +η0 log N + N −ε0� +, +(7.35) +because the contribution of O(|x − y|−1) to the integral (7.10) is bounded by O(η0 log N). +21 + +Finally, the contribution of the regime (x, y) /∈ [E0 − ˆε, E0 + ˆε]2 to the integral +�� +R2 +(f(x) − f(y))2 +(x − y)2 +dxdy = ∥f∥2 +˙H1/2 = ∥g∥2 +˙H1/2 , +(7.36) +is bounded by O≺(η0), therefore +V (f) = +1 +2π2 ∥g∥2 +˙H1/2 + O +� +η0 log N + N −ε0� +. +(7.37) +This concludes the proof of Proposition 5.2. +Appendix A +Proof of Lemma 5.4 +We use the Helffer–Sj¨ostrand representation to express the linear eigenvalue statistics in terms of the +resolvent of H (see Section 4.2 in [19] for references), +{1 − E} [Tr f(H)] = 1 +2π +� +C +∂ �f +∂¯z {1 − E} [Tr G(z)] d¯zdz. +(A.1) +The characteristic function φ then admits the form +φ(λ) = E [e(λ)] , +e(λ) := exp +� +iλ 1 +2π +� +C +∂ �f +∂¯z {1 − E} [Tr G(z)] d¯zdz +� +, +λ ∈ R, +(A.2) +and its derivative φ′ is given by +φ′(λ) = E +� +e(λ) i +2π +� +C +∂ �f +∂¯z {1 − E} [Tr G(z)] d¯zdz +� +, +λ ∈ R. +(A.3) +As observed in [19], the regime | Im z| ≤ N −ε0/2η0, referred to as the ultra-local scales, does not +contribute to the integrals in (A.2) and (A.3). This yields the estimates (5.9) (see equations (4.21) +and (4.22) in [19] for further detail). +It remains to show that (5.11) holds. +Applying the cumulant expansion formula (4.15) to the +quantity E [�e(λ) {1 − E} [Gjj(z)]] yields the following lemma. +Lemma A.1. (Lemma 5.7 in [17]) For all z ∈ D defined in (3.2) and j ∈ {1, . . ., N} we have +−1 +mj(z) E [�e(λ) {1 − E} [Gjj(z)]] = − mj(z) +N +� +k=1 +Sjk E [�e(λ) {1 − E} [Gkk(z)]] +− E [�e(λ) {1 − E} [Tjj(z, z)]] ++ E +� N +� +k=1 +SjkGkj(z)∂�e(λ) +∂Hjk +� +− 1 +2 +N +� +k=1 +C(4) +jk mj(z)mk(z) E +�∂2�e(λ) +∂H2 +jk +� ++ O≺ +� +(1 + |λ|4) +� +Ψ(z)Θ(z) + N −1Ψ(z)η−1/2 +0 +�� +, +(A.4) +where η0 is from (2.3), and for a, b ∈ {1, . . ., N}, z, ζ ∈ C\R, Txy(z, ζ) is defined in (1.1). +Let gj := E [�e(λ) {1 − E} [Gjj(z)]] and let rj denote the right-hand side of (A.4) without the first +term, then (A.4) reads +�� +1 − Sm2(z) +�g +� +j = −mj(z)rj. The operator +� +1 − Sm2(z) +� +can be inverted to +22 + +deduce that gj = − +�� +1 − Sm2(z) +�−1 m(z)r +� +j, where m(z) is interpreted as a multiplication operator +acting on the vector r. Summing over j, we obtain +E [�e(λ) {1 − E} [Tr G(z)]] = +N +� +j=1 +gj = − +N +� +j,k=1 +�� +1 − Sm2(z) +�−1� +jk mk(z)r k = − +N +� +j=1 +m′ +j(z) +mj(z)rj, +(A.5) +where in the last step we applied the identity m′(z)/m2(z) = (1 − Sm2(z))−11. The second term on +the right-hand side of (A.4) contributes the first term to the right hand side of (5.11), which, as we +show in Section 6, is negligible. Therefore, it suffices to estimate the contribution of the third and +fourth terms on the right-hand side. The necessary estimates on the partial derivatives of �e(λ) are +collected in the following lemma. +Lemma A.2. (Lemma 5.6 in [17]) For all j, k ∈ {1, . . . , N} we have +∂�e(λ) +∂Hjk += −iλ +π +2 +1 + δjk +�e(λ) +� +Ω′ +0 +∂ �f +∂¯ζ +∂Gkj(ζ) +∂ζ +d¯ζdζ. +(A.6) +Moreover, for all p ∈ N, the following bound holds +���� +∂p�e(λ) +∂Hp +jk +���� = O≺ +� +(1 + |λ|)p� +, +(A.7) +and for k ̸= j +���� +∂�e(λ) +∂Hjk +���� = O≺ +� +N −1/2(1 + |λ|)η−1/2 +0 +� +. +(A.8) +Second derivatives with k ̸= j are given by +∂2�e(λ) +∂H2 +jk += 2iλ +π �e(λ) +� +Ω′ +0 +∂ �f +∂¯ζ +∂ {mj(ζ)mk(ζ)} +∂ζ +d¯ζdζ + O≺ +� +N −1/2(1 + |λ|)2η−1/2 +0 +� +. +(A.9) +The form in which we write the error terms in Lemmas A.1 and A.2 slightly differs from their +original form in [17] because we have already applied the estimate ∥f ′′∥1 ∼ η−1 +0 . The leading term in +(A.9) results in the third line of (5.11). +Using Lemmas A.2 and 5.6 we proceed to estimate the third term on the right hand side of (A.4). +Lemma A.3. (c.f. Equation (5.65) of Lemma 5.8 in [17]) For all z ∈ D defined in (3.2) and all +j ∈ {1, . . . , N} we have +E +� N +� +k=1 +SjkGkj(z)∂�e(λ) +∂Hjk +� += − 2iλ +π E +� +�e(λ) +� +Ω′ +0 +∂ �f +∂¯ζ +∂Tjj(z, ζ) +∂ζ +d¯ζdζ +� +− iλ +π Sjj E [�e(λ)] +� +Ω′ +0 +∂ �f +∂¯ζ m′ +j(ζ)mj(z)d¯ζdζ + O≺ +�Ψ(z)(1 + |λ|) +Nη1/2 +0 +� +. +(A.10) +Proof of Lemma A.3. In view of (1.1), multiplying (A.6) by SjkGkj(z), summing over k ̸= j and +taking expectations gives the first term on the right hand side of (A.4). For the remaining k = j term, +observe that the function K(ζ) := Gjj(ζ) − mj(ζ) is analytic in C\R and is stochastically dominated +by Ψ(ζ) in D. Applying Lemma 5.5 with p = 1 to K(ζ), we obtain +∂Gjj(ζ) +∂ζ += m′ +j(ζ) + O≺ +� +| Im ζ|−1Ψ(ζ) +� +. +(A.11) +Plugging (A.11) into (A.6) with k = j and applying Lemma 5.6 with K(ζ) := ∂ζGjj(ζ) − m′ +j(ζ) with +s = 3/2, we get +∂�e(λ) +∂Hjj += −iλ +π �e(λ) +� +Ω′ +0 +∂ �f +∂¯ζ m′ +j(ζ)d¯ζdζ + O≺ +� +1 + |λ|)N −1/2η−1/2 +0 +� +. +(A.12) +23 + +where we used the the fact that |e(λ)| = 1 and the first line of (5.9) to bound |�e(λ)| by O≺(1). +Multiplying (A.12) by SjjGjj(z) and using the local law (4.4) to estimate Gjj(z) gives the second +term on the right hand side of (A.4). Application of the local law (4.4) is justified by (A.7) with p = 1. +This concludes the proof of Lemma A.3. +Summing up the leading terms in (A.10) results in the second and third terms on the right-hand +side of (5.11). Collecting all the error terms, the estimate in (5.11) now follows from (4.13), (A.5), +(A.7) (A.9) and Lemma A.3. This concludes the proof of Lemma 5.4. +References +[1] +Oskari Ajanki, L´aszl´o Erd˝os, and Torben Kr¨uger. Quadratic Vector Equations On Complex +Upper Half-Plane. Memoirs of the American Mathematical Society 261.1261 (2019). +[2] +Oskari Ajanki, L´aszl´o Erd˝os, and Torben Kr¨uger. Universality for general Wigner-type matrices. +Probability Theory and Related Fields 169 (2015), pp. 667–727. +[3] +Oskari Ajanki, Torben Kr¨uger, and L´aszl´o Erd˝os. Singularities of Solutions to Quadratic Vector +Equations on the Complex Upper Half-Plane. Communications on Pure and Applied Mathematics +70 (2017), pp. 1672–1705. +[4] +Zhidong Bai and Jian-Feng Yao. On the convergence of the spectral empirical process of Wigner +matrices. Bernoulli 11 (2005), pp. 1059–1092. +[5] +Zhigang Bao, Kevin Schnelli, and Yuanyuan Xu. Central Limit Theorem for Mesoscopic Eigen- +value Statistics of the Free Sum of Matrices. 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Annals +of Mathematics 65.2 (1957), pp. 203–207. +25 + diff --git a/ANAzT4oBgHgl3EQfvv6Q/content/tmp_files/load_file.txt b/ANAzT4oBgHgl3EQfvv6Q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6029ce41558bb56d1949362d177cf251281e630d --- /dev/null +++ b/ANAzT4oBgHgl3EQfvv6Q/content/tmp_files/load_file.txt @@ -0,0 +1,1275 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf,len=1274 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='01712v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='PR] 4 Jan 2023 1 Mesoscopic eigenvalue statistics for Wigner-type matrices Volodymyr Riabov∗ Institute of Science and Technology Austria volodymyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='riabov@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='at Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We prove a universal mesoscopic central limit theorem for linear eigenvalue statistics of a Wigner- type matrix inside the bulk of the spectrum with compactly supported twice continuously differentiable test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The main novel ingredient is an optimal local law for the two-point function T(z, ζ) and a general class of related quantities involving two resolvents at nearby spectral parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Date: January 5, 2023 Keywords and phrases: Wigner-type matrix, mesoscopic eigenvalue statistics, central limit theorem 2010 Mathematics Subject Classification: 60B20, 15B52 1 Introduction In the study of the eigenvalue distribution of large random matrices, the most celebrated analog of the Law of Large Numbers is the Wigner semicircle law [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It states that the empirical density of eigenvalues converges to a deterministic limit known as the semicircle distribution ρsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' More explicitly, if H is an N ×N Wigner matrix and f is a sufficiently smooth test function, then the linear eigenvalue statistics N −1 Tr f(H) converge in probability to � R f(x)ρsc(x)dx in the large N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The corresponding Central Limit Theorem (CLT) asserts that the asymptotic fluctuations of the linear eigenvalue statistics Tr f(H)−E [Tr f(H)] are Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The absence of the N −1/2 normalization factor, appearing in the classical CLT, can be viewed as a manifestation of the strongly-correlated nature of the eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For the special case of f(x) = (x − z)−1 with Im z ̸= 0, this result was obtained by Khorunzhy, Khoruzhenko and Pastur [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Johansson obtained the CLT for invariant ensembles with arbitrary polynomial potentials in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In [4], Bai and Yao used martingale CLT to establish the result for Wigner matrices with analytic test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The proof for bounded test functions f with bounded derivatives appeared in the work of Lytova and Pastur [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In subsequent works, different moment conditions on the matrix and regularity conditions on the test function were studied extensively by many authors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', [6, 18, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' While fixed test functions represent macroscopic averaging in the spectrum, one can introduce N- dependent scaling and consider scaled test functions of the form f(x) = g(η−1 0 (x − E0)), where E0 is a fixed reference energy in the bulk, η0 ≡ η0(N) ≪ 1 is a scaling parameter, and g is compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Then Tr f(H) involves only about Nη0 eigenvalues of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In particular, on mesoscopic scales, corresponding to N −1 ≪ η0 ≪ 1, the limiting variance is given by the square of the ˙H1/2 norm of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Mesoscopic test functions were first studied by Boutet de Monvel and Khorunzhy in [7] for the Gaussian Orthogonal Ensemble, with subsequent extension to real Wigner matrices in [8] with N −1/8 ≪ η0 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In [13], He and Knowles proved the CLT for Wigner matrices with general mesoscopic test functions for all scaling parameters N −1 ≪ η0 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' ∗Supported by the ERC Advanced Grant ”RMTBeyond” No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 101020331 The result was extended to ensembles of greater generality in the more recent works, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', [5] and [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In particular, Li and Xu obtained mesoscopic CLT for generalized Wigner matrices 1 in the bulk and at the spectral edge with C2 c test functions in the full range of scales [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Finally, Landon, Lopatto, and Sosoe proved the bulk CLT for the much more general ensemble of Wigner-type matrices in [17] for two classes of C∞ test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For a special class of globally supported regularized bump functions , the proof is performed via resolvent techniques for large scales and extended to the entire mesoscopic range using Dyson Brownian motion (DBM) dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For the more conventional compactly supported scaled test functions, the bulk CLT is established on all mesoscopic scales N −1 ≪ η0 ≪ 1 using a combination of DBM and Green’s function comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Wigner-type matrices were first introduced in [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' they have centered entries Hjk independent up to the symmetry constraint H = H∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The matrix of variances S, defined by Sjk := E � |Hjk|2� , is assumed to be flat, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', Sjk ∼ N −1 and satisfy a piece-wise H¨older regularity condition (see (B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' As the main step towards CLT in the present paper, we prove the optimal averaged and entry-wise local laws (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) for the two-point function T , defined by Txy(z, ζ) := � a̸=y SxaGay(z)Gya(ζ), x, y ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) where G(z) is the resolvent of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The corresponding result in the simpler setting of generalized Wigner matrices was obtained in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Using the optimal local law for T (z, ζ), we prove the bulk mesoscopic CLT for Wigner-type matrices in the full range of scales N −1 ≪ η0 ≪ 1 for compactly supported C2 scaled test functions (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Our proof relies entirely on resolvent methods, circumventing the DBM dynamics used in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Understanding T (z, ζ) is the crucial ingredient for the CLT as it was realized in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In fact, a suboptimal entry-wise local law for Txy(z, ζ) was proved in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 of [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' If one relies solely on resolvent methods, this local law provides sufficient control for mesoscopic CLT only on scales η0 ≫ N −1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The main reason for this limitation is that the error term in [17] contains the norm of the inverted stability operator (defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In the present paper, we show that this factor can be removed by separating the destabilizing eigendirection corresponding to the smallest eigenvalue of the stability operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Using this method, we prove a local law for a general class of quantities involving two resolvents (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) and deduce the optimal averaged and entry-wise local laws for T (z, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In particular, this allows us to obtain the CLT on all mesoscopic scales without relying on DBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The main difficulty lies in the fact that the deterministic approximation of the resolvent for Wigner- type matrices is not a multiple of the identity matrix, contrary to the generalized Wigner case [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Consequently, the destabilizing direction is no longer parallel to the vector of ones, and generally, no closed-form expression is known for the corresponding eigenprojector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It is important to note that for the deformed Wigner matrices studied in [20], the deterministic approximation is also not a multiple of the identity, but Sjk = N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Therefore, the two-point function can be expressed as the square of the resolvent and can be studied using the local law, similarly to the standard Wigner case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Instead of approximating the destabilizing direction to circumvent this difficulty, we use a contour integral representation for the eigenprojector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It allows us to extend the decomposition approach of [21] to the Wigner-type ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This method benefits from yielding an integral representation for the variance on all mesoscopic scales, under weaker regularity conditions on the test function than in [17], and relying only on resolvent methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The paper is organized in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Section 2 contains the precise definition of the model and the statement of our main mesoscopic CLT result, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In Section 3, we present our main technical result, the optimal local law for two-point functions in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In Section 4, we collect notations and preliminary results to which we refer throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In Section 5, we deduce Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 from Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2, and prove Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 using a local law for T (z, ζ) (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The proofs of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3 are presented in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In Section 7, we prove Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2, which relates the variance of the linear eigenvalue statistics to the ˙H1/2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' I would like to express my gratitude to L´aszl´o Erd˝os for suggesting the project and supervising my work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' I am also thankful to Yuanyuan Xu and Oleksii Kolupaiev for many helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 1Generalized Wigner matrices are characterized by a flat doubly-stochastic matrix of variances S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Unlike the Wigner case, the entries Sjk are not assumed to be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The limiting eigenvalue distribution remains semicircular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 2 2 Model and Main Result We begin with the definition of Wigner-type matrices originally introduced in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 of [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 (Wigner-type matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let H = (Hjk)N j,k=1 be an N × N matrix with independent entries up to the Hermitian symmetry condition H = H∗ satisfying E [Hjk] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) We consider both real and complex Wigner-type matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In case the matrix H is complex we assume additionally that Re Hjk and Im Hjk are independent and E[H2 jk] = 0 for k ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Denote by S the matrix of variances Sjk := E[|Hjk|2], and assume it satisfies cinf N ≤ Sjk ≤ Csup N , (A) for all j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N} and some strictly positive constants Csup, cinf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We assume a uniform bound on all other moments of √ NHjk, that is, for any p ∈ N there exists a positive constant Cp such that E � | √ NHjk|p� ≤ Cp (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) holds for all j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Additionally, we assume that S satisfies a H¨older regularity condition1, that is, |Sjk − Sj′k′| ≤ L N �|j − j′| + |k − k′| N �1/2 , (B) for all j, j′, k, k′ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N} and some positive constant L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The constants cinf, Csup, Cp and L are independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 Central Limit Theorem for Mesoscopic Linear Eigenvalue Statistics Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5 in [17]) Let g be a C2 c (R) test function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let ε0 be a small fixed constant and let N −1+ε0 ≤ η0 ≤ N −ε0, and let E0 be a fixed reference energy in the bulk of the spectrum, that is, ρ(E0) ≥ ε0 (here ρ is the density of states to be defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) below ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Define the scaled test function f to be f(x) := g �x − E0 η0 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) then Tr f(H) − E [Tr f(H)] d−→ N � 0, 1 2βπ2 ∥g∥2 ˙H1/2 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) where β = 1 and β = 2 corresponds to real symmetric and complex Hermitian H, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We remark that the universal limiting variance in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) coincides with the corre- sponding formulas for standard Wigner matrices [13], where Sjk = N −1, mj(z) = msc(z) for all j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' , N}, and msc(z) is the Stieltjes transform of the semicircle law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 1As stated in [2], assumption (B) can be weakened to piece-wise 1/2-H¨older regularity condition for some positive constant L on finitely many intervals, in the sense that max a,b max j,j′∈(NIb) max k,k′∈(NIa) N3/2 |Sjk − Sj′k′| |j − j′|1/2 + |k − k′|1/2 ≤ L, where {Ia}n a=1 is a fixed finite partition of [0, 1] into smaller intervals, and (NIa) denotes the set of positive integers j such that j/N lies in Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 3 3 Local Laws for the Two-point Functions In this section, we introduce our main technical result, local laws for quantities that involve two resolvents of a Wigner-type matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Our prime motivation is to study the function T (z, ζ) defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1), but our methods allow us to estimate a more general class of quantities, namely � a̸=y waGαa(z)Gaβ(ζ), � b � a̸=b WabGba(z)Gab(ζ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) for fixed indices α, β, y, and deterministic weights wa, Wab satisfying |wa|, |Wab| ≤ cN −1 for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Here G(z) := (H − z)−1 denotes the resolvent of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Objects of this type were first studied in [11] in the setting of random band matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We obtain the estimates in the sense of stochastic domination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 in [12]) Let X = X (N)(u) and Y = Y(N)(u) be two families of random variables possibly depending on a parameter u ∈ U (N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We say that Y stochastically dominates X uniformly in u if for any ε > 0 and D > 0 there exists N0(ε, D) such that for any N ≥ N0(ε, D), sup u∈U(N) P � X (N)(u) > N εY(N)(u) � < N −D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We denote this relation by X ≺ Y or X = O≺(Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We consider spectral parameters z lying in the domain D, defined by D := {z ∈ C : N −1+τ ≤ | Im z| ≤ τ −1, | Re z| ≤ τ −1}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) for a fixed τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' As in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2, our analysis is limited to the bulk of the spectrum, which we define via the self-consistent density of states ρ(E) ≡ ρN(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The density ρ(E) is recovered by the Stieltjes inversion formula, ρ(E) := π−1 lim η→+0 Im m(E + iη), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) where m(z) := N −1 �N j=1 mj(z), and m(z) = (mj(z))N j=1 is the unique (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 in [2]) solution to the vector Dyson equation −1 m(z) = z + Sm(z), Im m(z) Im z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) Let I be the set on which ρ(E) is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 of [2] guarantees that I consists of a finite union of open intervals (a(j), b(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Then for κ > 0, we define the bulk domain by Dκ := {z ∈ D : Re z ∈ Iκ}, Iκ := � j [a(j) + κ, b(j) − κ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5) In particular, for all z ∈ Dκ, ρ(z) ≥ C(κ) for some constant C(κ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Given E0 as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2, we choose κ so that E0 ∈ I2κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' There exists a positive constant ǫ = ǫκ which is independent of N, such that for all z, ζ in Dκ with | Re ζ − Re z| ≤ ǫ, and deterministic vectors w ∈ CN satisfying ∥w∥∞ ≤ cN −1, the following estimate holds, � a̸=y waGαa(z)Gaβ(ζ) = δαβ � m(z)m(ζ) � 1 − Sm(z)m(ζ) �−1w � α − δαβδαy[m(z)m(ζ)w]α + O≺ � (Ψ(z) + Ψ(ζ))(Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0} min{Θ(z), Θ(ζ)}) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) where the vector m is identified with the diagonal operator diag (m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Under the same conditions on z, ζ, for any deterministic N ×N matrix W satisfying |Wab| ≤ cN −1 for all a, b, the following estimate holds, � b � a̸=b WabGba(z)Gab(ζ) = Tr � m(z)m(ζ)Sm(z)m(ζ) � 1 − Sm(z)m(ζ) �−1W � + NO≺ � (Ψ(z) + Ψ(ζ))Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0}Θ(z)Θ(ζ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) 4 Here Ψ(z) and Θ(z) denote control parameters defined as Ψ(z) := � | Im m(z)| N|η| + 1 N|η|, Θ(z) := 1 N|η|, z = E + iη ∈ C\\R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 implies the following averaged and entry-wise local laws for T (z, ζ) from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let z, ζ satisfy the assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The entries Txy(z, ζ) admit the estimate Txy(z, ζ) = � (Sm(z)m(ζ))2 � 1 − Sm(z)m(ζ) �−1� xy + O≺ � (Ψ(z) + Ψ(ζ)) � Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0} min{Θ(z), Θ(ζ)} �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) Furthermore, for all deterministic N × N matrices A, the following equality holds Tr[A T (z, ζ)] = Tr[A � 1 − Sm(z)m(ζ) �−1� Sm(z)m(ζ) �2] + N ∥A∥ℓ∞→ℓ∞ O≺ � (Ψ(z) + Ψ(ζ))Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0}Θ(z)Θ(ζ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The error estimates in the entry-wise local law (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6), and hence in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) are optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Indeed, for Sjk := N −1, which corresponds to the standard Wigner matrices, and ζ = ¯z, a simple calculation using the Ward identity shows that Txy(z, ¯z) = N −1| Im z|−1 Im msc(z) − N −1|msc(z)|2 + O≺ � Θ(z)Ψ(z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) The error estimate in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) is not optimal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' it can be improved to O≺ � N(Ψ(z) + Ψ(ζ))2� Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0}NΘ(z)Θ(ζ) �� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) However, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) is sufficient for establishing the CLT, so for the sake of brevity, we do not present the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) in full detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We only indicate the necessary ingredients in Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 4 Notations and Preliminaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 Notations For a vector x = (xj)N j=1 ∈ CN we use the standard definitions of ℓ2 and ℓ∞ norms, namely, ∥x∥2 = � N � j=1 |xj|2 �1/2 , ∥x∥∞ = max j |xj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For a linear operator T : CN → CN, we denote its matrix norms induced by ℓ2 and ℓ∞ norms, respectively, by ∥T ∥ℓ2→ℓ2 = sup ∥x∥2=1 ∥T x∥2 , ∥T ∥ℓ∞→ℓ∞ = sup ∥x∥∞=1 ∥T x∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For two vectors x, y ∈ CN we use angle brackets to denote the ℓ2 scalar product, while for a single vector x ∈ CN angle brackets denote the average of its coordinates ⟨x, y⟩ = N � j=1 ¯xjyj, ⟨x⟩ = 1 N N � j=1 xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We use xy to denote a coordinate-wise product of vectors x and y, (xy)j = xjyj, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Similarly, for a given vector x with non-zero entries, 1x denotes a coordinate-wise multiplicative inverse � 1 x � j = 1 xj , j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 5 We use 1 to denote the vector of ones (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' , 1)t in CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For a measurable function f : R → R we use the standard definition of the Lp norms for p ≥ 1, and the following definition of the ˙H1/2 norm ∥f∥ ˙H1/2 = \uf8eb \uf8ed �� R2 |f(x) − f(y)|2 |x − y|2 dxdy \uf8f6 \uf8f8 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For two deterministic quantities X, Y ∈ R depending on N, we write X ≪ Y if there exists ε, N0 > 0 such that |X| ≤ N −ε|Y | for all N ≥ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Similarly, we write X ≲ Y if there exists a constant C, N0 > 0 such that |X| ≤ C|Y | for all N ≥ N0, and X ∼ Y if both X ≲ Y and Y ≲ X hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We use C and c to denote constants, the precise value of which is irrelevant and may change from line to line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 Local Law for the Resolvent In this subsection, we summarize the facts on Wigner-type matrices that we use throughout our proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Majority of these results were obtained in [1] (see also [3]), but we refer to their concise versions from [2] adapted for the Wigner-type setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 in [2]) The solution m(z) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) satisfies the following properties: (1) For every j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N} there exists a generating probability measure νj(dx) such that mj(z) = � R νj(dx) x − z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) (2) If the matrix of variances S satisfies conditions (A) and (B), then for all z ∈ C\\R, the solution admits the following bounds ∥m(z)∥∞ ≤ c 1 + |z|, ���� 1 m(z) ���� ∞ ≤ C(1 + |z|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) We now state the optimal averaged and isotropic local laws for Wigner-type matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8 in [2]) Let w, x, y be deterministic vectors in CN satisfying ∥w∥∞ = 1 and ∥x∥2 = ∥y∥2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Then the following estimates hold uniformly in z ∈ D: N −1��Tr � w(G(z) − m(z)) ��� ≺ Θ(z), ��⟨x, (G(z) − m(z))y⟩ �� ≺ Ψ(z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) where vectors m and w are associated with corresponding diagonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In particular, it follows from the isotropic local law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) that for any j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}, |Gjk(z) − δjkmj(z)| ≺ Ψ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3 Preliminary Bounds on the Stability Operator A significant part of our proof revolves around the stability operator, originally introduced in [1], that emerges when studying the two-point function T (z, ζ) defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In this subsection, we collect the known bounds on the stability and related operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The stability operator (1 − Sm(z)m(ζ)) is defined by the matrix with entries (1 − Sm(z)m(ζ))jk := δjk − Sjkmk(z)mk(ζ), j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}, z, ζ ∈ C\\R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5) Throughout this paper we use m (and various functions of m, such as Im m, |m|, m−1, m′) to denote both a vector (mj)N j=1 and the corresponding multiplication operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', diag � (mj)N j=1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Note that this notation agrees with the point-wise multiplication of two vectors if the first multiplicand is interpreted as an operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We stress which interpretation is used whenever ambiguity may arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 6 The analysis of the stability operator relies on the corresponding saturated self-energy operator F, studied in [17], that depends on two spectral parameters z, ζ, and is defined as Fjk(z, ζ) := |mj(z)mj(ζ)|1/2Sjk|mk(z)mk(ζ)|1/2, j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}, z, ζ ∈ C\\R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) The following statements encompass the main properties of F and preliminary bounds on the stability operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3 in [17], c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9 and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4 in [9]) For any z, ζ ∈ C, the principal eigenvalue of F defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) is positive and simple, the corresponding ℓ2- normalized eigenvector v(z, ζ) has strictly positive entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The norm of F admits the following upper bound ∥F(z, ζ)∥ℓ2→ℓ2 ≤ 1 − 1 2 � | Im z| ⟨v(z, z), |m(z)|⟩ ⟨v(z, z), | Im m(z)| |m(z)| ⟩ + | Im ζ| ⟨v(ζ, ζ), |m(ζ)|⟩ ⟨v(ζ, ζ), | Im m(ζ)| |m(ζ)| ⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) If |z|, |ζ| ≲ 1, then the entries of v(z, ζ) are comparable in size, that is cκ ≤ √ Nvj(z, ζ) ≤ Cκ, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8) and moreover, let Gap (F) denote the difference between the two largest eigenvalues of |F| = √ FF ∗, then Gap (F) admits the bound Gap (F) ≥ �δ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) where �δ is a constant that depends only on the constants in conditions (A), (B) and κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Furthermore, for a fixed κ > 0 and z, ζ ∈ Dκ there exists a positive constant �cκ such that ∥F(z, ζ)∥ℓ2→ℓ2 ≤ 1 − �cκ (| Im z| + | Im ζ|) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7 in [17]) Let z, ζ ∈ C, such that |z|, |ζ| ≲ 1 and Re z, Re ζ ∈ Iκ, then ��(1 − Sm(z)m(ζ))−1�� ℓ2→ℓ2 + ��(1 − Sm(z)m(ζ))−1�� ℓ∞→ℓ∞ ≲ 1 | Im z| + | Im ζ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) If additionally Im z Im ζ > 0, the estimate is improved to ��(1 − Sm �m)−1�� ℓ∞→ℓ∞ ≤ Cκ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) where Cκ > 0 is a positive constants dependent on κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Finally, we state the bounds on the stability operator in the special case of ζ = z, which is related to the derivative of m via the (vector) identity m′(z) = (1 − m2(z)S)−1m2(z), obtained by taking the derivative of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9 in [1], Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 in [9]) Let C > 0 be a positive constant, then for z ∈ C\\R with |z| ≤ C we have ��(1 − m2(z)S)−1�� ℓ2→ℓ2 + ��(1 − m2(z)S)−1�� ℓ∞→ℓ∞ ≲ |ρ(z)|−2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='13) where ρ(z) = π−1⟨Im m(z)⟩ is the harmonic extension of ρ(E) defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Therefore for all z ∈ C\\R with Re z ∈ Iκ we have ∥m′(z)∥∞ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4 Cumulant Expansion Formula Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Section II in [7], Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 in [13]) Let h be a real-valued random variable with finite moments, let f be a C∞(R) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Then for any ℓ ∈ N the following expansion holds, E [h · f(h)] = ℓ � j=0 1 j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='c(j+1)(h) E � dj dhj f(h) � + Rℓ+1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='15) 7 where c(j) is the j-th cumulant of h defined by c(j)(h) = (−i)j dj dtj � log E � eith������ t=0 , and the remainder term Rℓ+1 satisfies |Rℓ+1| ≤ Cl E � |h|ℓ+2� sup |x|≤M |f (ℓ+1)(x)| + Cl E � |h|ℓ+2 · 1|h|>M � ���f (ℓ+1)(x) ��� ∞ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16) for any M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We apply formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='15) with h equal to the matrix element Hjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Correspondingly, in the real case (β = 1), C(p) denotes the matrix of p-th cumulants of H, C(p) jk := C(p)(Hjk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In the complex case (β = 2), C(p) is used as a notational shortcut and denotes the sum of matrices of p-th cumulants of real and imaginary parts of H, that is C(p) jk := C(p)(Re Hjk) + C(p)(Im Hjk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 5 Proof of the Main Result Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We divide the proof into two parts contained in the following propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We indicate their analogs in the settings of [21] and [17] in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 in [21] and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='76) in [17]) Let η0, ε0 > 0 and E0 satisfy the assumptions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2, let f be a scaled test function defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3), and let φ(λ) be the characteristic function of Tr f(H) − E [Tr f(H)], φ(λ) := E [exp{iλ (Tr f(H) − E [Tr f(H)])}] , λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) Then its derivative φ′(λ) satisfies the following equation, φ′(λ) = −λφ(λ)V (f) + O≺ � N −1/2η−1/2 0 (1 + |λ|4) + (1 + |λ|)N −ε0/2� , λ ∈ R, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) provided c ≤ V (f) ≤ C for some positive N-independent constants c and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Here the variance V (f) for a scaled test function f is defined by V (f) := 1 π2 � Ω0 � Ω′ 0 ∂ �f(ζ) ∂¯ζ ∂ �f(z) ∂¯z K(z, ζ)d¯ζdζd¯zdz, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) where for z, ζ ∈ C/R the kernel K(z, ζ) is defined by K(z, ζ) := 2 β ∂ ∂ζ Tr �m′(z) m(z) � 1 − Sm(z)m(ζ) �−1 � + � 1 − 2 β � Tr [Sm′(z)m′(ζ)] + 1 2 ∂2 ∂z∂ζ � m(z)m(ζ), C(4)m(z)m(ζ) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) with C(4) denoting the matrix of fourth cumulants C(4) jk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The integration domains Ω0, Ω′ 0 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) are defined as Ω0 := {z ∈ C : | Im z| > N −ε0/2η0}, Ω′ 0 := {z ∈ C : | Im z| > 2N −ε0/2η0}, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5) and �f is the quasi-analytic extension of f, defined by �f(x + iη) = χ(η) (f(x) + iηf ′(x)) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) where χ : R → [0, 1] is an even C∞ c (R) function supported on [−1, 1], satisfying χ(η) = 1 for |η| < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7 in [17]) Let E0, η0 satisfy the conditions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let f be the scaled test function with g ∈ C2 c (R) given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3), and let V (f) be the variance defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3), then V (f) = 1 2βπ2 ∥g∥2 ˙H1/2 + O � η0 log N + N −ε0� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) 8 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 implies that V (f) satisfies the condition of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1, hence φ′(λ) = −λφ(λ)V (f) + o (1) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8) as N → ∞, for any fixed λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It then follows by L´evy’s continuity theorem that Tr f(H)−E [Tr f(H)] converges in distribution to a centered Gaussian with variance (2βπ2)−1 ∥g∥2 ˙H1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Therefore, to estab- lish Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2, it suffices to show that Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 hold, which is done in Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 and 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We restrict the proof to the real symmetric (β = 1) matrices for the sake of presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The complex Hermitian (β = 2) case differs solely in replacing the cumulant expansion formula (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) with its complex analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The obvious modifications are left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 Characteristic Function of Linear Eigenvalue Statistics Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Using standard techniques of the characteristic function method imported from, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 of [17] (see also Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 of [19] and references therein), we can obtain the following series of estimates on the characteristic function of the linear eigenvalue statistics φ(λ) and its derivative φ′(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The proof is a relatively straightforward modification of similar arguments in [17], so we defer it to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let φ(λ) be the characteristic function defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1), then, under the conditions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2, the following estimates hold φ(λ) = E [�e(λ)] + O≺ � N −ε0/2� , φ′(λ) = i π � Ω0 ∂ �f ∂¯z E [�e(λ) {1 − E} [Tr G(z)]] d¯zdz + O≺ � |λ|N −ε0/2� , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) where �e(λ) := exp �iλ π � Ω′ 0 ∂ �f ∂¯z {1 − E} [Tr G(z)] d¯zdz � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) Furthermore, for all z ∈ Dκ, we have E [�e(λ) {1 − E} [Tr G(z)]] = E [�e(λ) {1 − E} T (z, z)] + 2iλ π E � �e(λ) � Ω′ 0 ∂ �f ∂¯ζ ∂ ∂ζ T (z, ζ)d¯ζdζ � + iλ π E [�e(λ)] � Ω′ 0 ∂ �f ∂¯ζ Tr [Sm′(z)m′(ζ)] d¯ζdζ + iλ 2π E [�e(λ)] � Ω′ 0 ∂ �f ∂¯ζ ∂2 ∂z∂ζ � m(z)m(ζ), C(4)m(z)m(ζ) � d¯ζdζ + O≺ � (1 + |λ|4)(NΨ(z)Θ(z) + Ψ(z)η−1/2 0 ) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) where the random function T (z, ζ) is defined as T (z, ζ) := Tr �m′(z) m(z) T (z, ζ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) We now proceed to estimate the first two terms on the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) in such a way that E [�e(λ)] factors out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' By definition of the scaled test function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3), the support of �f is contained inside a vertical strip centered at E0 of width ∼ η0, hence we limit the further analysis to the regime | Re ζ − Re z| ≲ η0 ≪ ǫ, where ǫ is defined in the statement of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We estimate the function T (z, ζ) using Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3 with weight matrix A := m′(z) m(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows from the bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) that ∥A∥ℓ∞→ℓ∞ ≲ 1, hence for all z, ζ ∈ Dκ with Re z, Re ζ ∈ supp(f), T (z, ζ) = Tr �m′(z) m(z) � 1 − Sm(z)m(ζ) �−1� Sm(z)m(ζ) �2 � + E(z, ζ), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='13) 9 where the error term E(z, ζ) is analytic in both variables and admits the bound E(z, ζ) ≺ NΨ2(z)Ψ(ζ) + NΨ(z)Ψ2(ζ) + 1{Im z Im ζ<0}NΘ(z)Θ(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) It follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='13) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) for ζ = z that E [�e(λ){1 − E} [T (z, z)]] ≺ NΨ(z)3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='15) yielding the desired bound on the first term on the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We now estimate the second term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Fix z ∈ Dκ, and consider ζ that lie in Ω′ 0 defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Differentiating (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='13) with respect to ζ yields ∂ ∂ζ T (z, ζ) = ∂ ∂ζ Tr �m′(z) m(z) � 1 − Sm(z)m(ζ) �−1� Sm(z)m(ζ) �2 � + ∂ ∂ζ E(z, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16) To bound the derivative of the error term E(z, ζ), we use the following technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5 in [17]) Let K(z) be a holomorphic function on C\\R, then for all z ∈ C\\R and any p ∈ N, ���� ∂pK ∂zp (z) ���� ≤ Cp| Im z|−p sup |ζ−z|≤| Im z|/2 |K(ζ)|, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17) where Cp > 0 is a constant depending only on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5 applied to the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) implies that the error term ∂ζE(z, ζ) admits the bound ∂ ∂ζ E(z, ζ) ≺ N| Im ζ|−1� Ψ(z)2Ψ(ζ) + Ψ(z)Ψ(ζ)2 + Θ(z)Θ(ζ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='18) To proceed we require another technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4 in [19]) Let f be the scaled test function defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let Ω be a domain of the form Ω := {z ∈ C : cN −τ ′η0 < | Im z| < 1, a < Re z < b}, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='19) such that supp(f) ⊂ (a, b) and τ ′, c are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let K(z) be a holomorphic function on Ω satisfying |K(z)| ≤ C| Im z|−s, z ∈ Ω, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='20) for some 0 ≤ s ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Then there exists a constant C′ > 0 depending only on g in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3), χ in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6), and s, such that ���� � Ω ∂ �f ∂¯z (x + iy)K(x + iy)dxdy ���� ≤ CC′η1−s 0 log N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21) Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) that ∥f∥1 ∼ η0, ∥f ′∥1 ∼ 1, ∥f ′′∥1 ∼ η−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In case 1 ≤ s ≤ 2 the inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21) follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4 in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For 0 ≤ s < 1, the proof is conducted along the same lines, except the integration by parts is performed twice in the regime η0 ≤ | Im z| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6 and the matrix identity (1−X)−1X2 = (1−X)−1−X −1 yield the following expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' E � �e(λ) � Ω′ 0 ∂ �f ∂¯ζ ∂T ∂ζ d¯ζdζ � = E [�e(λ)] � Ω′ 0 ∂ �f ∂¯ζ ∂ ∂ζ Tr �m′(z) m(z) � 1 − Sm(z)m(ζ) �−1 � d¯ζdζ − E [�e(λ)] � Ω′ 0 ∂ �f ∂¯ζ Tr � Sm′(z)m′(ζ) � d¯ζdζ +O≺ � N 1/2Ψ(z)2η−1/2 0 + Ψ(z)η−1 0 + Θ(z)η−1 0 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='22) Finally, from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='22), combined with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) we conclude that φ′(λ) = −λV (f) E [�e(λ)] + �E(λ), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='23) 10 where V (f) is defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3), and �E(λ) is the total error term collected from previous derivations and integrated over d¯zdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6 together with error estimates in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='15) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='18) provides the following bound on the error term �E = O≺ � N −1/2η−1/2 0 (1 + |λ|4) + |λ|N −ε0/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='24) Under the conditions of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 V (f) is bounded, hence we conclude from the first estimate in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='23) that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This concludes the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 6 Proof of the Local Laws for Two-point Functions In this section, we derive all the tools necessary to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 and its specification for the two- point function T (z, ζ), Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' To make the notation more concise we introduce the convention G ≡ G(z), �G ≡ G(ζ), m ≡ m(z), �m ≡ m(ζ), �Ψ ≡ Ψ(ζ), Ψ ≡ Ψ(z), Θ ≡ Θ(z), �Θ ≡ Θ(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For a deterministic matrix W with entries |Wab| ≲ N −1, the quantity � a̸=y WaxGαa �Gaβ can be readily estimated in two special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' First, if each column of W is proportional to the vector of ones, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', Wab = wb depends only on b, then the summation over a yields wx([G �G]αβ − Gαy �Gyβ), and the estimate follows from the resolvent identity and the local laws in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Second, if the entries of X := (1 − Sm �m)−1W are bounded by CN −1, then one can obtain the estimate from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We show that these two special cases are exhaustive in the sense that any W can be represented as their linear combination with controlled coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' To this end, we prove that in the relevant regime, the operator (1 − Sm �m) has a very small destabilizing eigenvalue and an order one spectral gap above it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Moreover, if Π is the eigenprojector corresponding to the principal eigenvalue of (1 − Sm �m), then the ℓ∞ → ℓ∞-norm of the restriction of (1 − Sm �m)−1 to the kernel of Π is also an order one quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Finally, we show that the vector of ones 1 is sufficiently separated from the kernel of Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 Stable Direction Local Law For any N × N deterministic matrix W, and any indices x, y, α, β, we define the quantities Fxy αβ(W) := � a̸=y WaxGαa �Gaβ, f xy α (W) := mα �mα([(1 − Sm �m)−1W]αx − δαyWαx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) We prove the following estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For any z, ζ ∈ Dκ and any deterministic N × N matrix X, Fxy αβ((1 − Sm �m)X) = δαβf xy α ((1 − Sm �m)X) + O≺ � N ∥X∥max Ψ�Ψ(Ψ + �Ψ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) provided ∥X∥max := max j,k |Xjk| ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We use the following self-improving mechanism for stochastic domination bounds, borrowed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', from [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3 in [14]) Let X be a random variable such that 0 ≤ X ≺ N C for some C > 0, and let Ξ ≥ 0 be a deterministic quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Suppose there exists a constant q ∈ [0, 1), such that for any Φ satisfying Ξ ≤ Φ ≤ N C, and any d ∈ N, we have the implication X ≺ Φ =⇒ E � |X|2d� ≺ 2d � k=1 � ΦqΞ1−q)k E � |X|2d−k� , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) then X ≺ Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let Y := (1 − Sm �m)X, then the quantity we need to estimate is [GY ]yx = Fxy yy (Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows from the local law in the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) that Fxy αβ(Y ) ≺ N ∥X∥max Ψ�Ψ =: Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) 11 Let Φ be a deterministic control parameter admitting the bounds (Ψ + �Ψ)Λ ≤ Φ ≤ Λ, such that Fxy αβ(Y ) − δαβf xy α (Y ) ≺ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5) It follows trivially from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5) that Fxy αβ(Y ) ≺ Φ + δαβΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) Let ∂jk denote the partial derivative with respect to the matrix element Hjk, then the partial derivatives of Fxy αβ are given by ∂abFxy αβ(Y ) = −(1 + δab)−1(GαaFxy bβ (Y ) + GαbFxy aβ(Y ) + Fxy αb (Y ) �Gaβ + Fxy αa(Y ) �Gbβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) We combine the vector Dyson equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) and the resolvent identity zG = HG − 1 to obtain �Gaβ = − �ma � b � Hab �Gbβ + Sab �mb �Gaβ � + �maδaβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8) Let d ∈ N, define P ≡ P(d − 1, d) := (Fxy αβ(Y ) − δαβf xy α (Y ))d−1(Fxy αβ(Y ) − δαβf xy α (Y ))d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For any p ∈ N, define Mp := E � |Fxy αβ(Y ) − δαβf xy α (Y )|p� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Plugging (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8) into the definition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) and applying the cumulant expansion formula of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6, we obtain E � Fxy αβ(X)P � = � a̸=y ma �maXax E � Fay αβ(S)P � + δαβf xy α (Y ) E[P] + δαβδβySyym2 y �m2 yXyx E[P] (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9a) + E �� a̸=y � b �maXaxSab � Gαa( �Gbb − �mb) �Gaβ + Gαb(Gaa − ma) �Gbβ �P � (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9b) + E �� a̸=y � b̸=a �maXaxSabGαa � Gba + �Gba � �GbβP � + R2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9c) + � a̸=y Xaxma �maSay E �� Gαy �Gyβ − δαyδyβmy �my �P � (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9d) + δβ̸=y �mβXβx E � (Gαβ − δαβmβ)P � − E �� a̸=y �maXaxGαa � b Sab �Gbβ∂abP � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9e) where R2 is the total error coming from the higher order cumulants, and all unrestricted summations are from 1 to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We successively bound the terms (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9b)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9e) appearing on the right-hand side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' By condition (A), local law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4), upper bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5), it follows that the terms (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9b) and the first term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9c) are bounded by O≺((Ψ + �Ψ)ΛM2d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Similarly, the term (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9d) and the first term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9e) are bounded by O≺(∥X∥max (Ψ + �Ψ)M2d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We bound the second term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows by (A), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4), bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) that � b Sab �Gbβ∂abP ≺ (Ψ + �Ψ + δαa + δaβ)�ΨΦM2d−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) Hence, the second term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9e) is bounded by O≺ � (Ψ + �Ψ)ΛΦM2d−2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Finally, it is easy to check using estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) and identity (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7), together with condition (A) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2), that the error term R2 ≺ (Ψ + �Ψ)ΛM2d−1 + (Ψ + �Ψ)ΛΦM2d−2 + (Ψ + �Ψ)ΛΦ2M2d−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Observe that the first term on the right-hand side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9a) can be expressed as � a̸=y ma �maXax E � Fay αβ(S)P � = E � Fay αβ(X)P � − E � Fay αβ(Y )P � − my �myXyx E � Fyy αβ(S)P � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) where the last term is bounded by O≺(N −1ΛM2d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Combining (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) yields E � |Fxy αβ(Y ) − δαβf xy α (Y )|2d� ≺ � Ψ + �Ψ � ΛΦ2M2d−3, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) for any control parameter Φαβ,y satisfying (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Hence, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2, Fxy αβ(Y ) = δαβf xy α (Y ) + O≺ � Λ(Ψ + �Ψ) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='13) which concludes the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 12 Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' If z and ζ are in the same (upper or lower) half-plane, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 implies Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Indeed, the bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4 shows that provided η�η > 0, X := (1−Sm �m)−1W satisfies |Xjk| ≲ N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Applying Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 to X = (1 − Sm �m)−1W then yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) follows by summing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We turn to the case of z and ζ lying in different (upper and lower) half-planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 Stability Operator Analysis In this subsection we obtain all the properties of the stability operator (1 − Sm(z)m(ζ)) that we use in combination with Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 to finish the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 for z, ζ lying in opposite half-planes, as outlined in the beginning of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For two spectral parameters z, ζ, let η := Im z, and �η := Im ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Without loss of generality, we assume in the following that Re z ∈ Iκ, η > 0 and Re ζ ∈ Iκ, �η < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For the remainder of this subsection, we use the following notation F ≡ F(z) := |m(z)|S|m(z)|, B ≡ B(z, ζ) := 1 − Sm(z)m(ζ), B0 ≡ B0(z) := 1 − S|m(z)|2 = |m(z)|−1(1 − F)|m(z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) We view the operator B as a perturbation of B0 = B(z, ¯z), since |ζ − ¯z| is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We deduce the desired properties of B from those of B0, which, in turn, follow from the lower bound on the spectral gap of F found in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let {ψj}N j=1 denote the eigenvalues of F (with multiplicity) in descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Then, by Per- ron–Frobenius theorem, the principal eigenvalue ψ1 is real, and it coincides with the spectral radius ∥F∥ℓ2→ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Furthermore, by taking the imaginary part of the vector Dyson equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) and multi- plying both sides by |m| coordinate-wise, we obtain � 1 − F �Im m |m| = η|m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='15) Furthermore, by condition (A), for every j we have (S Im m)j ∼ ⟨Im m(z)⟩ ∼ ρ(z), where ρ(z) is the harmonic extension of the self-consistent density of states ρ(x) defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) into C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Hence by taking the imaginary part of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4), we get Im mj |mj| ∼ |mj|(ρ(z) + η), , j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16) Therefore, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='15) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16), 1 − ψ1 ≲ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Together with an upper bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) on ∥F∥ℓ2→ℓ2, this implies that 1 − ψ1 ∼ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) that the principal eigenvalue of F is separated from the rest of the spectrum by an annulus, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', there exist r > 0 and δ > 0 independent of z and N such that |1 − ψ1| < r − δ, and |1 − ψj| > r + δ, j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17) In the remainder of this subsection, we show that for all ζ sufficiently close to ¯z, the eigenvalue of B with the smallest modulus is also separated from the rest of the spectrum by an annulus of order one width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Using the argument principle and Jacobi’s formula, one can express the number of eigenvalues (with multiplicity) of a matrix X inside a domain Ω by a contour integral NX(Ω) = 1 2πi � ∂Ω Tr(w − X)−1dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='18) To show the eigenvalue separation for B, we begin by estimating the norm of the resolvent of B inside the annulus Ar,δ := {w ∈ C : r − 3δ/4 ≤ |w| ≤ r + 3δ/4}, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='19) with r and δ as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 13 Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' There exists ε1 > 0 and �C > 0 independent of N and z such that ���(w − B(z, ζ))−1��� ≤ �C (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='20) holds for all w ∈ Ar,δ and all ζ such that Re ζ ∈ Iκ, Im ζ < 0 and |ζ − ¯z| ≤ ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (The norm ∥·∥ is induced by either ℓ2 or ℓ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Observe that ��(w − B)−1�� ≤ ��� � 1 − (w − B0)−1(B − B0) �−1��� ��(w − B0)−1��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Since (w − B0)−1 = −|m|−1(1 − w − F)−1|m| and |m| ∼ 1, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17) implies that ��(w − B0)−1�� ≤ C min j |ψj − w| ≤ 4C δ , w ∈ Ar,δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21) From the uniform bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) on |m| and |m′| we have ∥B − B0∥ ≲ |ζ − ¯z|, which implies that there exists ε1 > 0 such that ∀ζ : |ζ − ¯z| ≤ ε1, ∥B − B0∥ ≤ δ 8C , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='22) where C is the constant in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows immediately that ��� � 1 − (w − B0)−1(B − B0) �−1��� ≤ 2 and hence ��(w − B)−1�� ≤ 8C δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='23) Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4 implies that for any sufficiently large fixed N the integrand in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='18) with X := B is uniformly bounded in Ω := Ar,δ for all ζ such that |ζ − ¯z| ≤ ε1, hence by analyticity NB(z,ζ)(Ar,δ) = 0, |ζ − ¯z| ≤ ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='24) Since the eigenvalues of B(z, ζ) are continuous in ζ, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='24) implies that no eigenvalue can move between the two connected components of C\\Ar,δ, which together with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17) yields the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For any sufficiently large N, the equalities NB({|w| < r − 3δ/4}) = NB0({|w| < r − 3δ/4}) = 1, NB({|w| > r + 3δ/4}) = NB0({|w| > r + 3δ/4}) = N − 1, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='25) hold for any ζ such that Re ζ ∈ Iκ, Im ζ < 0 and |ζ − ¯z| ≤ ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5 now allows us to define the principal eigenprojector Π of B as a contour integral Π ≡ Π(z, ζ) := 1 2πi � |ξ|=r (ξ − B(z, ζ))−1dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='26) Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5 asserts that the contour {|ξ| = r} encircles exactly one eigenvalue of B with multiplicity, hence Π is a rank one eigenprojector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We now prove that the restriction of B−1 to the range of (1 − Π) is bounded by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For all z, ζ such that Re z, Re ζ ∈ Iκ, Im z Im ζ < 0 and |ζ − ¯z| ≤ ε1, ��B−1(1 − Π) �� ℓ∞→ℓ∞ ≤ �c, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='27) where �c depends only on the constants in conditions (A), (B) and κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' By expression (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='26) for Π we have B−1(1 − Π) = − 1 2πi � |ξ|=r 1 ξ (ξ − B)−1dξ (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='28) 14 Hence the norm of B−1(1 − Π) is bounded by ��B−1(1 − Π) �� ℓ∞→ℓ∞ ≤ 1 2π 2π � 0 ��� � reiθ − B �−1��� ℓ∞→ℓ∞ dθ ≤ 8C δ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='29) using the bound in Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4 on the circle {|ξ| = r} which lies inside Ar,δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Finally, we show that the vector of ones is sufficiently separated from the kernel of Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This ensures a stable decomposition of the space into the direct sum of the range of (1 − Π) and the span of 1, so we can apply the local laws to each of the components separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' There exists ε > 0 independent of N and z such that for all ζ with Re ζ ∈ Iκ, Im ζ < 0 and |ζ − ¯z| ≤ ε, ∥Π1∥∞ ∥Π∥ℓ∞→ℓ∞ ≥ c, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='30) where c > 0 is a constant independent of N and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Define the projector Π0 corresponding to B0 via (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Then Π0 = |m|−1�Π0|m|, where �Π0 is the orthoprojector corresponding to the principal eigenvalue of the Hermitian operator F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Since |m| ∼ 1 we have ∥Π0∥ℓ∞→ℓ∞ ≤ C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Moreover, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3, the ℓ2-normalized eigenvector v corresponding to the principal eigenvalue of F has entries vj ≥ 0 with vj ∼ N −1/2, hence ∥Π01∥∞ = ���|m|−1�Π0|m|1 ��� ∞ = ��|m|−1v �� ∞ ⟨v, |m|⟩ ≥ c0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='31) where c0 > 0 is a constant independent of N and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Similarly to the proof of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='22), for any γ ∈ (0, 1] there exists εγ > 0, such that the bound ∥B − B0∥ℓ∞→ℓ∞ ≤ γ δ 8C (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='32) holds for all ζ ∈ D− κ with |ζ − ¯z| ≤ εγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Here δ is defined in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17) and C > 0 is the constant in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We choose εγ to be smaller than ε1 of Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4, then for all ζ with Re ζ ∈ Iκ, Im ζ < 0 such that |ζ − ¯z| ≤ εγ we have ∥Π − Π0∥ℓ∞→ℓ∞ ≤ r 2π 2π � 0 ��(reiθ − B)−1 − (reiθ − B0)−1�� ℓ∞→ℓ∞ dθ ≤ r 2π 2π � 0 ��(reiθ − B)−1(B − B0)(reiθ − B0)−1�� ℓ∞→ℓ∞ dθ ≤ r · 8C δ · γ δ 8C · 4C δ = γ 4Cr δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='33) Here we used inequalities (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='23) in the second to last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We set the value of γ to be γ0 := min � 1, c0δ 8Cr � , which guarantees that ∥Π1∥∞ ≥ ��∥Π01∥∞ − ∥Π − Π0∥ℓ∞→ℓ∞ ∥1∥∞ �� ≥ c0 − γ0 4Cr δ ≥ c0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='34) Finally, observe that ∥Π∥ℓ∞→ℓ∞ ≤ ∥Π0∥ℓ∞→ℓ∞ + ∥Π − Π0∥ℓ∞→ℓ∞ ≤ C0 + c0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='35) This proves the claim with c := c0/(2C0 + c0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3 Finishing the Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Recall that the objective is to estimate the quantities defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In- stead of estimating � a̸=y waGαa �Gaβ directly, it is more convenient to work with objects of the type � a̸=y WaxGαa �Gaβ, since they generalize quantities appearing in both (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The redundant index x can be eliminated by setting Wax := wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In the case Im z Im ζ > 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) follow immediately from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 (see Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Therefore, we focus on the case Im z Im ζ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Since Π has rank one and Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7 asserts that Π1 ̸= 0, the kernel of Π together with 1 span CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Therefore we can decompose each column of the matrix W into a linear combination of 1 and an element of ker Π, that is, there exists an N × N matrix Y and a vector s ∈ CN such that W = Y + 1s∗, ΠY = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='36) We multiply the first equality in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='36) by Π from the left, apply both sides to the a-th standard basis vector ea of CN and take the ℓ∞-norm to deduce ∥ΠWea∥∞ = |sa| ∥Π1∥∞ , a ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='37) By assumption, ∥W∥max ≲ N −1, hence ∥Wea∥∞ ≲ N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Using Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7 we get |sa| ≲ N −1 ∥Π∥ℓ∞→ℓ∞ ∥Π1∥∞ ≲ N −1, a ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='38) We combine (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='36) and the resolvent identity in the form (z − ζ)G �G = G − �G to obtain � a̸=y WaxGαa �Gaβ = � a̸=y YaxGαa �Gaβ + gy αβ¯sx, gy αβ := Gαβ − �Gαβ z − ζ − Gαy �Gyβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='39) Define the N × N matrix X := (1 − Sm �m)−1 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='36) that Y = (1 − Π)Y , hence X = (1 − Sm �m)−1(1 − Π)Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Furthermore, estimates ∥W∥max ≲ N −1, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='36), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='38) imply that |Yab| ≲ N −1 for all a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Since by Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6 ��(1 − Sm �m)−1(1 − Π) �� ℓ∞→ℓ∞ ≲ 1, we conclude that ∥X∥max = max a,b |Xab| ≲ N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='40) First, using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='40), we can apply Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 to the first term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='39) to obtain � a̸=y YaxGαa �Gaβ = δαβmα �mα([(1 − Sm �m)−1Y ]αx − δαyYαx) + O≺ � Ψ2 �Ψ + Ψ�Ψ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='41) Using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='36), we proceed by computing mα �mα[(1 − Sm �m)−1Y ]αx = � m �m � 1 − Sm �m �−1 (W − 1s∗) � αx = � m �m � 1 − Sm �m �−1W � αx − � m �m � 1 − Sm �m �−11 � α¯sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='42) Finally, it follows from subtracting the vector Dyson equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) for z and ζ that m �m � 1 − Sm �m �−11 = m − �m z − ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='43) Next, we estimate the second term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Applying the local law in the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4), we obtain gy αβ = δαβ mα − �mα z − ζ − δαβδαymα �mα + O≺ � (|η| + |�η|)−1(Ψ + �Ψ) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='44) where we used that |z − ζ| ≥ |η| + |�η|, since η�η < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Combining (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='38), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='39), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='41)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='44) yields � a̸=y WaxGαa �Gaβ = δαβ � m �m � 1 − Sm �m �−1W � αx − δαβδαy[m �mW]αx + O≺ � (Ψ + �Ψ)(Ψ�Ψ + min{Θ, �Θ}) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='45) 16 which proves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) by setting Wax := wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' To prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7), we observe that by setting x = y = α = β = b in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='39) and summing over b yields � b � a̸=b WabGba �Gab = � b � a̸=b YabGaa �Gab+⟨s, g⟩, gb := Gbb − �Gbb z − ζ −Gbb �Gbb, b ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='46) To estimate ⟨s, g⟩, we use (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='38) and the averaged local law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) to obtain � s, g � = � s, m − �m z − ζ − m �m � + O≺ � (|η| + |�η|)−1(Θ + �Θ) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='47) where we used that |z − ζ| ≥ |η| + |�η|, since η�η < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Setting x = y = α = β = b in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='41), summing over b, using the identities (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='42) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='43), and combining the result with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='47), we deduce that � b � a̸=b WabGba �Gab = Tr � m �mSm �m � 1 − Sm �m �−1W � + NO≺ � Ψ�Ψ(Ψ + �Ψ) + Θ�Θ � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='48) where we used that (|η| + |�η|)−1(Θ + �Θ) = NΘ�Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This establishes (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) and concludes the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We outline the steps needed to achieve the optimal error estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' First, one needs to adapt the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' More specifically, replace the decomposition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='36) with W = Y + 1s∗ + q1∗, such that Π(z, ζ)Y = Y Πt(ζ, z) = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='49) where Π(z, ζ) is the destabilizing eigenprojector defined in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The terms involving s and q are handled using the averaged local law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3), similarly to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For the remaining term, R := � y Fyy yy , we adapt the mechanism of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 by using the following iterative scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In the first step, we apply an expansion similar to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) to the partial derivative ∂jkR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This improves the error in the estimate on R by a factor of (Ψ + �Ψ)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' If we expand ∂lp∂jkR in a similar manner, we gain another (Ψ + �Ψ)1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Iterating this approach we can estimate R with an error stochastically dominated by NΨ�Ψ(Ψ + �Ψ)2−2−d for any given integer d (where d is the maximal order of expanded partial derivatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' By Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1, this is sufficient to establish (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Similar arguments in the context of random band matrices can be found in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) on Txy(ζ, z) follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) by setting α = β = y and wa := Sxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) on Tr[AT (z, ζ)] follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) by setting W := SAt, which satisfies |Wab| ≲ N −1 ∥A∥ℓ∞→ℓ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This concludes the proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Note that estimates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) (also with the improved error term (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12)) hold without omission of indices in the a summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Indeed, it follows from Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 that � a waGαa �Gaβ = δαβ � m �m � 1 − Sm �m �−1w � α + O≺ � (Ψ + �Ψ)(Ψ�Ψ + 1{η�η<0} min{Θ, �Θ}) � , � a,b WabGba �Gab = Tr � m �m � 1 − Sm �m �−1W � + O≺ � N(Ψ + �Ψ)Ψ�Ψ + 1{η�η<0}NΘ�Θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='50) 7 Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 In this section, we compute the variance V (f) defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) for mesoscopic C2 c test functions f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In [17], the limiting variance was computed for several types of C∞ test functions, including compactly supported ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' however, V (f) is computed with an O(1) error (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7 in [17]), which is not negligible in the setting of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' To obtain effective error bounds, we augment the proof laid out in [17] by performing further integration by parts in the integral representation of V (f), thus eliminating the f ′ terms, improving the error by a factor of O(η0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Throughout this section, we adhere to the notation m ≡ m(z), �m ≡ m(ζ), η := Im z, �η := Im ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 17 The stability operator (1 − Sm �m) can be expressed in terms of the self-saturated energy operator F, defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6), via the following identity 1 − Sm �m = |m �m|−1/2 (U∗ − F(z, ζ)) |m �m|1/2U, U := m �m |m �m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) Furthermore, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9), the operator F can be decomposed such that F(z, ζ) = ψ1(z, ζ) v(z, ζ) � v(z, ζ) �∗ + A(z, ζ), A(z, ζ)v(z, ζ) = 0, ∥A(z, ζ)∥ℓ2→ℓ2 ≤ 1 − �δ, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) where ψ1, v is the principal eigenvalue-eigenvector pair of F, and �δ is the constant in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let R ≡ R(z, ζ) denote (U∗(z, ζ) − A(z, ζ))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In the sequel, we drop the arguments and write A ≡ A(z, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lower bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8) and the inequality in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) imply that ∥R∥ℓ2→ℓ2 + ∥R∥ℓ∞→ℓ∞ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) In the following lemma, we collect the perturbative estimates on the saturated self-energy operator F and related quantities established in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='52), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='60), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='71), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='67) in [17]) Let w, ζ1, ζ2 be spectral parameters in Iκ + i[−1, 1], and let F be the operator defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6), then the principal eigenvalue- eigenvector pair ψ1, v of F satisfies ∥v(w, ζ1) − v(w, ζ2)∥ℓ2→ℓ2 + |ψ1(w, ζ1) − ψ1(w, ζ2)| ≲ |ζ1 − ζ2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) Furthermore, for operator A defined in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2), we have the estimate ∥F(w, ζ1) − F(w, ζ2)∥ℓ2→ℓ2 + ∥A(w, ζ1) − A(w, ζ2)∥ℓ2→ℓ2 ≲ |ζ1 − ζ2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5) Let z := x + iη, ζ := y − iη, with x, y ∈ Iκ, 0 ≤ η ≤ 1, then ψ1 � v, Rm′ m U∗Rv � = ψ1(z, z) � v(z, z)m′ m v(z, z) � + O(|x − y|) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) Let ω ≡ ω(z, ζ) := 1 − ψ1⟨v, Rv⟩, then ω(z, ζ) = 1 − ψ1(z, z) + ψ1(z, z)(x − y) � v(z, z)m′ m v(z, z) � + O(|x − y|2), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) Moreover, there exists ε > 0 independent of N, such that for all x, y ∈ Iκ satisfying |x − y| ≤ ε, |ω(z, ζ)| ≳ η + |x − y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8) Finally, for z := x + iη with x ∈ Iκ, the following identity holds lim η→+0 � v(z, z)m′ m v(z, z) � = iπ 2 ρ(x) ���� Im m(x + i0) |m(x)| ���� −2 2 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) By our choice of κ, E0 is in the interior of the bulk interval Iκ, defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5) , hence if we define ˆε := min{ε/4, dist(E0, R\\Iκ)}, then ˆε ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Furthermore, since the function g is compactly supported, we assume that supp(f) ⊂ [E0 − ˆε, E0 + ˆε] for large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let η∗ ≡ η∗(N) satisfy 0 < η∗ ≤ N −100, then V (f), defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3), admits the estimate V (f) = 1 4π2 �� [E0−ˆε,E0+ˆε]2 (f(y) − f(x))2 �K(x + iη∗, y − iη∗)dxdy + O � η0 + N −ε0� , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) where �K(z, ζ) := −2 Re Tr �m′ m (1 − Sm �m)−1Sm �m′(1 − Sm �m)−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) 18 In preparation for the proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 we define an auxiliary function L(z, ζ) L(z, ζ) := Llog(z, ζ) + L1(z, ζ), Llog(z, ζ) := −2 log det {1 − Sm �m} , L1(z, ζ) := − Tr [Sm �m] + 1 2 � m �m, C(4)m �m � , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) where log is the principal branch of the complex logarithm, and C(4) is the matrix of the fourth cumu- lants of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' By Jacobi’s formula for the derivative of the determinant, it follows from the definitions of L and K, that for all z, ζ ∈ C\\R ∂2 ∂ζ∂z L(z, ζ) = K(z, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='13) Furthermore, by condition (A) and the upper bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2), it follows that |Llog(z, ζ)| ≤π + log |det {1 − Sm �m}| ≲ 1 + Tr � (1 − Sm �m)∗ (1 − Sm �m) − I � ≲ 1, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) where in the last line we used � (1 − Sm �m)∗ (1 − Sm �m) − I � jj ≲ N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The partial derivatives of L1 contribute only sub-leading terms to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Indeed, we have the estimates L1(z, ζ) ≲ 1, ∂ ∂zL1(z, ζ) ≲ 1, ∂2 ∂ζ∂z L1(z, ζ) ≲ 1, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='15) where we used the moment condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) to bound Sjk and C(4) jk , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) to get the upper bound m, �m ≲ 1, and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) to obtain m′, �m′ ≲ 1, since [E0 + ˆε, E0 − ˆε] ⊂ Iκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The following claim collects the bounds on K and ∂zL that together with (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) enable integration by parts in the definition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) of the variance V (f), which is the essence of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6 in [17]) Let K(z, ζ) and L(z, ζ) be as defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) (with β = 1) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) respectively, then for all z, ζ ∈ C\\R with Re z, Re ζ ∈ [E0 − ˆε, E0 + ˆε] and | Im z|, | Im ζ| ≤ 1 we have K(z, ζ) ≲ 1 + 1{η�η<0}(|η| + |�η|)−2, ∂ ∂z L(z, ζ) ≲ 1 + (| Re z − Re ζ| + |η| + |�η|)−1, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16) where η := Im z, �η := Im ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Define Ω∗ := {z ∈ C : 1 > | Im z| > η∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Recall the definition of V (f) from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' First, we prove that V (f) = 1 π2 � Ω∗ � Ω∗ ∂ �f(ζ) ∂¯ζ ∂ �f(z) ∂¯z K(z, ζ)d¯ζdζd¯zdz + O � N −ε0� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17) It follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) that ∂ �f ∂¯z = 1 2 � −ηχ′(η)f ′(x) + i � ηχ(η)f ′′(x) + χ′(η)f(x) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='18) Moreover, for all z with | Im z| < 1/2, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='18) and the properties of χ in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) imply ∂ �f ∂¯z = i Im z 2 f ′′(Re z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='19) Let V∗(f) denote the integral on right hand side of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17), and define η1 := N −ε0/2η0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows from the first inequality in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16), and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='19) that |V (f) − V∗(f)| ≲ �� R2 |f ′′(x)f ′′(y)| dxdy η1 � η∗ 2η1 � η∗ η�η (η + �η)2 d�ηdη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='20) 19 Note that η�η ≤ (η + �η)2/4, hence the integral over d�ηdη is bounded by η2 1/2, and since ∥f ′′∥1 ∼ η−1 0 , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17) is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We write z := x + iη, ζ := y + i�η and plug (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='13) into the expression (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17) for V (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Using the fact that ∂zu = −i∂ηu for any holomorphic function u(z), and integrating by parts in η, we obtain V (f) = i π2 �� R2 dxdy � |�η|>η∗ ∂ �f(ζ) ∂¯ζ � |η|>η∗ ∂2 �f(z) ∂η∂¯z ∂ ∂ζ L(z, ζ)d�ηdη − i π2 �� R2 dxdy � |�η|>η∗ ∂ �f(ζ) ∂¯ζ � η=±η∗ ∂ �f ∂¯z (x + iη) ∂ ∂ζ L(z, ζ)d�η + O � N −ε0� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21) The second estimate in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16), expression (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='18) and the estimates ∥f ′′∥1 ∼ η−1 0 , ∥f ′∥1 ∼ 1, ∥f∥1 ∼ η0 imply that the boundary term in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21) is dominated by O≺(η∗η−2 0 ), which is smaller than O (N −ε0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Similarly, integrating the first term on the right hand side of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21) by parts in �η we get V (f) = − 1 π2 � Ω∗ � Ω∗ ∂2 �f(z) ∂¯z∂η ∂2 �f(ζ) ∂¯ζ∂�η L(z, ζ)d¯ζdζd¯zdz + 1 π2 �� R2 dxdy � |η|>η∗ ∂2 �f(z) ∂η∂¯z � �η=±η∗ ∂ �f ∂¯ζ (y + i�η)L(z, y + i�η)dη + O � N −ε0� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='22) It follows from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) and the expression (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='18) that the boundary term (the second line of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='22)) is again dominated by O≺(N −ε0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We apply Stokes’ theorem to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='22) twice: once in z and once in ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Considering that ∂η �f(z) vanishes on the boundary of Ω∗ except for the lines {Im z = ±η∗}, this results in V (f) = 1 4π2 �� R2 � η,�η=±η∗ sign (η�η) ∂ �f(x + iη) ∂η ∂ �f(y + i�η) ∂�η L(x + iη, y + i�η)dxdy + O � N −ε0� = − 1 2π2 �� R2 f ′(x)f ′(y) �L(x, y)dxdy + O � N −ε0� , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='23) where �L(x, y) := Re [L(x + iη∗, y + iη∗) − L(x + iη∗, y − iη∗)] (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='24) We restrict the integrations in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='23) to [E0 − ˆε, E0 + ˆε], since this interval contains the support of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Furthermore, for all y ∈ supp(f), y − E0 ≲ η0, hence |y − E0 ± ˆε| ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' By symmetry of L(z, ζ), and the second estimate in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16) it follows that ∂ ∂y �L(E0 ± ˆε, y) ≲ 1, y ∈ supp(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='25) We write f ′(y) = ∂y (f(y) − f(x)), perform integration by parts in y and integrate the boundary term by parts in x to obtain V (f) = 1 2π2 E0+ˆε � E0−ˆε E0+ˆε � E0−ˆε f ′(x) (f(y) − f(x)) ∂ ∂y �L(x, y)dxdy + 1 4π2 E0+ˆε � E0−ˆε (f(x))2 ∂ ∂x � �L(x, E0 + ˆε) − �L(x, E0 − ˆε) � dx + O � N −ε0� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='26) Since ∥f∥2 2 ≲ η0, it follows from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='25) that the second integral in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='26) is O (η0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Similarly, integrating (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='26) by parts in x and using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='26) to substitute one of the emerging itegrals for −V (f) + O (N −ε0 + η0), we get 2V (f) = 1 2π2 E0+ˆε � E0−ˆε E0+ˆε � E0−ˆε (f(y) − f(x))2 ∂2 ∂x∂y �L(x, y)dxdy + O � η0 + N −ε0� , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='27) 20 where we again used (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='25) to estimate the boundary term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For any holomorphic function u(z) of z = x + iη, we have ∂xu = Re[∂zu], hence ∂x∂y �L(x, y) = Re [K(x + iη∗, y + iη∗) − K(x + iη∗, y − iη∗)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Finally, in view of in view of the first estimate in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16), ∂z∂ζLlog(x + iη∗, y + iη∗) ≲ 1, so its contribution is also bounded by O≺(η0 ∥g∥2 2 + η2 0 ∥g∥2 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Moreover, it follows from the last estimate in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='15) that we can replace K(x+iη∗, y −iη∗) by ∂z∂ζLlog(x+iη∗, y −iη∗), since the contribution of the remaining terms is bounded by O≺(η0 ∥g∥2 2 + η2 0 ∥g∥2 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This concludes the proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Once Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 is established, we can follow the method of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7 in [17] to finish the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Fix x, y ∈ [E0 − ˆε, E0 + ˆε] and write z := x + iη∗, ζ := y − iη∗, as in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) that the kernel �K(z, ζ) can be written as �K(z, ζ) = −2 Re Tr �m′ m U∗� R + ψ1 ω Rvv∗R � F �m′ �m � R + ψ1 ω Rvv∗R �� , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='28) where ω is defined in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Expanding the brackets in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='28), collecting like terms according to the powers of ω−1, and using the cyclic property of trace yields �K(z, ζ) = −2 Re �ψ2 1 ω2 � v, Rm′ m U∗Rv �� v, RF �m′ �m Rv �� + O � 1 + ω−1� , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='29) since Tr � m′ m U∗RF � m′ � m R � , Tr � m′ m U∗RF � m′ � m Rvv∗R � , and Tr � m′ m U∗Rvv∗RF � m′ � m R � are all O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The first scalar product in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='29) can be estimated using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We compute the second scalar product in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows from uniform bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) that ∥m(z)− m(¯ζ)∥∞ ≲ |x− y|, and hence ∥U(z, ζ) − 1∥ℓ2→ℓ2 ≲ |x− y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Together with estimates (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4), this yields ψ1 � v, RF �m′ �m Rv � = ⟨v(ζ, ζ), F(ζ, ζ) � m′ �m v(ζ, ζ) � + O(|x − y|), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='30) where we used the identity R(¯ζ, ζ)v(ζ, ζ) = (1 − A(ζ, ζ))−1v(ζ, ζ) = v(ζ, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows from the estimate on v in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) that ∥v(ζ, ζ) − v(y, y)∥2 ≲ η∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Vector v(y, y) is the ℓ2- normalization of |m(y)|−1 Im m(y + i0), hence it satisfies F(y, y)v(y, y) = v(y, y) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Therefore using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) and the lower bound in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5), we obtain ∥F(ζ, ζ)v(ζ, ζ) − v(ζ, ζ)∥2 ≲ η∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='31) Substituting (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='31) into (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='30) yields ψ1 � v, RF �m′ �m Rv � = ⟨v(ζ, ζ), �m′ �m v(ζ, ζ) � + O(|x − y| + η∗), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='32) Combining (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='28) with estimates (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='32) yield �K(z, ζ) = −2 Re �ψ1(z, z)ψ1(ζ, ζ) ω2 � v(z, z)m′ m v(z, z) � ⟨v(ζ, ζ), �m′ �m v(ζ, ζ) �� + O(1 + ω−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='33) It follows by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) that lim η∗→+0 �K(x + iη∗, y − iη∗) = 2|x − y|−2 + O(|x − y|−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='34) Since f ∈ C2 c (R), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='33) implies that the integrand in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) is uniformly bounded in η∗ ∈ [0, N −100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Therefore, we can take the limit η∗ → 0 in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10), and apply the boundary estimate (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='34) to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' V (f) = 1 2π2 �� [E0−ˆε,E0+ˆε]2 (f(x) − f(y))2 (x − y)2 dxdy + O � η0 log N + N −ε0� , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='35) because the contribution of O(|x − y|−1) to the integral (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) is bounded by O(η0 log N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 21 Finally, the contribution of the regime (x, y) /∈ [E0 − ˆε, E0 + ˆε]2 to the integral �� R2 (f(x) − f(y))2 (x − y)2 dxdy = ∥f∥2 ˙H1/2 = ∥g∥2 ˙H1/2 , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='36) is bounded by O≺(η0), therefore V (f) = 1 2π2 ∥g∥2 ˙H1/2 + O � η0 log N + N −ε0� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='37) This concludes the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Appendix A Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4 We use the Helffer–Sj¨ostrand representation to express the linear eigenvalue statistics in terms of the resolvent of H (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 in [19] for references), {1 − E} [Tr f(H)] = 1 2π � C ∂ �f ∂¯z {1 − E} [Tr G(z)] d¯zdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) The characteristic function φ then admits the form φ(λ) = E [e(λ)] , e(λ) := exp � iλ 1 2π � C ∂ �f ∂¯z {1 − E} [Tr G(z)] d¯zdz � , λ ∈ R, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) and its derivative φ′ is given by φ′(λ) = E � e(λ) i 2π � C ∂ �f ∂¯z {1 − E} [Tr G(z)] d¯zdz � , λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) As observed in [19], the regime | Im z| ≤ N −ε0/2η0, referred to as the ultra-local scales, does not contribute to the integrals in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This yields the estimates (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) (see equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='22) in [19] for further detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It remains to show that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Applying the cumulant expansion formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='15) to the quantity E [�e(λ) {1 − E} [Gjj(z)]] yields the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7 in [17]) For all z ∈ D defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) and j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N} we have −1 mj(z) E [�e(λ) {1 − E} [Gjj(z)]] = − mj(z) N � k=1 Sjk E [�e(λ) {1 − E} [Gkk(z)]] − E [�e(λ) {1 − E} [Tjj(z, z)]] + E � N � k=1 SjkGkj(z)∂�e(λ) ∂Hjk � − 1 2 N � k=1 C(4) jk mj(z)mk(z) E �∂2�e(λ) ∂H2 jk � + O≺ � (1 + |λ|4) � Ψ(z)Θ(z) + N −1Ψ(z)η−1/2 0 �� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) where η0 is from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3), and for a, b ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}, z, ζ ∈ C\\R, Txy(z, ζ) is defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let gj := E [�e(λ) {1 − E} [Gjj(z)]] and let rj denote the right-hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) without the first term, then (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) reads �� 1 − Sm2(z) �g � j = −mj(z)rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The operator � 1 − Sm2(z) � can be inverted to 22 deduce that gj = − �� 1 − Sm2(z) �−1 m(z)r � j, where m(z) is interpreted as a multiplication operator acting on the vector r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Summing over j, we obtain E [�e(λ) {1 − E} [Tr G(z)]] = N � j=1 gj = − N � j,k=1 �� 1 − Sm2(z) �−1� jk mk(z)r k = − N � j=1 m′ j(z) mj(z)rj, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5) where in the last step we applied the identity m′(z)/m2(z) = (1 − Sm2(z))−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The second term on the right-hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) contributes the first term to the right hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11), which, as we show in Section 6, is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Therefore, it suffices to estimate the contribution of the third and fourth terms on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The necessary estimates on the partial derivatives of �e(λ) are collected in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6 in [17]) For all j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' , N} we have ∂�e(λ) ∂Hjk = −iλ π 2 1 + δjk �e(λ) � Ω′ 0 ∂ �f ∂¯ζ ∂Gkj(ζ) ∂ζ d¯ζdζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) Moreover, for all p ∈ N, the following bound holds ���� ∂p�e(λ) ∂Hp jk ���� = O≺ � (1 + |λ|)p� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) and for k ̸= j ���� ∂�e(λ) ∂Hjk ���� = O≺ � N −1/2(1 + |λ|)η−1/2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8) Second derivatives with k ̸= j are given by ∂2�e(λ) ∂H2 jk = 2iλ π �e(λ) � Ω′ 0 ∂ �f ∂¯ζ ∂ {mj(ζ)mk(ζ)} ∂ζ d¯ζdζ + O≺ � N −1/2(1 + |λ|)2η−1/2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) The form in which we write the error terms in Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 slightly differs from their original form in [17] because we have already applied the estimate ∥f ′′∥1 ∼ η−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The leading term in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) results in the third line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Using Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6 we proceed to estimate the third term on the right hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='65) of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8 in [17]) For all z ∈ D defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) and all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' , N} we have E � N � k=1 SjkGkj(z)∂�e(λ) ∂Hjk � = − 2iλ π E � �e(λ) � Ω′ 0 ∂ �f ∂¯ζ ∂Tjj(z, ζ) ∂ζ d¯ζdζ � − iλ π Sjj E [�e(λ)] � Ω′ 0 ∂ �f ∂¯ζ m′ j(ζ)mj(z)d¯ζdζ + O≺ �Ψ(z)(1 + |λ|) Nη1/2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In view of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1), multiplying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) by SjkGkj(z), summing over k ̸= j and taking expectations gives the first term on the right hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For the remaining k = j term, observe that the function K(ζ) := Gjj(ζ) − mj(ζ) is analytic in C\\R and is stochastically dominated by Ψ(ζ) in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5 with p = 1 to K(ζ), we obtain ∂Gjj(ζ) ∂ζ = m′ j(ζ) + O≺ � | Im ζ|−1Ψ(ζ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) Plugging (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) into (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) with k = j and applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6 with K(ζ) := ∂ζGjj(ζ) − m′ j(ζ) with s = 3/2, we get ∂�e(λ) ∂Hjj = −iλ π �e(λ) � Ω′ 0 ∂ �f ∂¯ζ m′ j(ζ)d¯ζdζ + O≺ � 1 + |λ|)N −1/2η−1/2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) 23 where we used the the fact that |e(λ)| = 1 and the first line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) to bound |�e(λ)| by O≺(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Multiplying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) by SjjGjj(z) and using the local law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) to estimate Gjj(z) gives the second term on the right hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Application of the local law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) is justified by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) with p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This concludes the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Summing up the leading terms in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) results in the second and third terms on the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Collecting all the error terms, the estimate in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) now follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='13), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This concludes the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' References [1] Oskari Ajanki, L´aszl´o Erd˝os, and Torben Kr¨uger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Quadratic Vector Equations On Complex Upper Half-Plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Memoirs of the American Mathematical Society 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1261 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [2] Oskari Ajanki, L´aszl´o Erd˝os, and Torben Kr¨uger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Universality for general Wigner-type matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Probability Theory and Related Fields 169 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 667–727.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [3] Oskari Ajanki, Torben Kr¨uger, and L´aszl´o Erd˝os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half-Plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Communications on Pure and Applied Mathematics 70 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 1672–1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [4] Zhidong Bai and Jian-Feng Yao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' On the convergence of the spectral empirical process of Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Bernoulli 11 (2005), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 1059–1092.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [5] Zhigang Bao, Kevin Schnelli, and Yuanyuan Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Central Limit Theorem for Mesoscopic Eigen- value Statistics of the Free Sum of Matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' International Mathematics Research Notices 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 5320–5382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [6] Zhigang Bao and Junshan Xie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' CLT for Linear Spectral Statistics of Hermitian Wigner Matrices with General Moment Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Theory of Probability & Its Applications 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 187– 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [7] Anne Marie Boutet de Monvel and Alexei Khorunzhy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Asymptotic distribution of smoothed eigenvalue density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Gaussian random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Random Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Stochastic Equations 7 (1999), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 1–22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [8] Anne Marie Boutet de Monvel and Alexei Khorunzhy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Asymptotic distribution of smoothed eigenvalue density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Wigner random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Random Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Stochastic Equations 7 (1999), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 149–168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [9] L´aszl´o Erd˝os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The matrix Dyson equation and its applications for random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' IAS/Park City Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 75–158.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [14] Yukun He and Matteo Marcozzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Diffusion profile for random band matrices: a short proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 177 (2019), pp.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='01178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 24 [18] Benjamin Landon and Philippe Sosoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Almost-optimal bulk regularity conditions in the CLT for Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 2022.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [20] Yiting Li, Kevin Schnelli, and Yuanyuan Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Central limit theorem for mesoscopic eigenvalue statistics of deformed Wigner matrices and sample covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Annales de l’Institut Henri Poincar´e, Probabilit´es et Statistiques 57 (2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 506–546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [21] Yiting Li and Yuanyuan Xu.' metadata={'source': 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Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Advances in Mathematics 249 (2013), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 37–87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [25] Eugene Wigner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Characteristics Vectors of Bordered Matrices with Infinite Dimensions II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Annals of Mathematics 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 (1957), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 203–207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} diff --git a/BtAyT4oBgHgl3EQfR_cQ/content/tmp_files/2301.00075v1.pdf.txt b/BtAyT4oBgHgl3EQfR_cQ/content/tmp_files/2301.00075v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7600da35d3247dfae0a1b5b8b0378a12bde6756b --- /dev/null +++ b/BtAyT4oBgHgl3EQfR_cQ/content/tmp_files/2301.00075v1.pdf.txt @@ -0,0 +1,597 @@ + s +er +ne +ngi +E +al +ci +han +c +e +M +of Iranian Society of + +ce +en +er +f +Con + +al +on +i +nat +nter +I +ual +n +An + +th +30 +The + +10 to 12 May, 2022, Tehran, Iran. + + +ISME2022-IC1332 + + +10 to 12 May, 2022 + + +Optimal Motion Generation of the Bipedal Under-Actuated Planar Robot for Stair Climbing + +Aref Amiri 1, Hassan Salarieh 2 + +1Graduate Student, Sharif University of Technology, Tehran; aref.amiri@mech.sharif.edu +2Professor, Sharif University of Technology, Tehran; salarieh@sharif.edu + +Abstract +The importance of humanoid robots in today's world is +undeniable, one of the most important features of +humanoid robots is the ability to maneuver in +environments such as stairs that other robots can not +easily cross. A suitable algorithm to generate the path +for the bipedal robot to climb is very important. In this +paper, an optimization-based method to generate an +optimal stairway for under-actuated bipedal robots +without an ankle actuator is presented. The generated +paths are based on zero and non-zero dynamics of the +problem, and according to the satisfaction of the zero +dynamics constraint in the problem, tracking the path is +possible, in other words, the problem can be +dynamically feasible. The optimization method used in +the problem is a gradient-based method that has a +suitable +number +of +function +evaluations +for +computational processing. This method can also be +utilized to go down the stairs. + +Keywords: Bipedal robot, under-actuated, optimization, +motion planning + +Introduction +Inspired by human body physics, bipedal robots have +many degrees of freedom and can perform various +actions with their joint movements. Bipedal robots can +adapt to different environments that other wheeled +robots are unable to move. The study of path (trajectory) +generation methods as a reference for the output of the +control problem of bipedal robots in this regard is +essential. For the bipedal robot to climb the stairs, it is +necessary to analyze the movement of them ascending +the stairs and to examine the method of planning the +bipedal robot to move and to determine the position of +feet for walking on the stairs [1]. + So far, researches have been done on how to go up +and downstairs and find a suitable or optimal path for +bipedal robots. Various papers using optimization +algorithms and considering the robot angles as +polynomial functions tried to design an optimal path for +a 6-degree bipedal robot [2]. Some articles have even +paths planned for multi-legged robots to cross the stairs +[3]. Some articles also used stability criteria such as +ZMP in designing their paths [4-7]. But this method is +only appliable for robots that have feet (soles) with ankle +joint actuators, which often have much lower speed in +maneuvering than under-actuated robots without feet, +and of course, due to the relatively large feet have more +wasted energy. Some articles also derive their initial path +using data based on motion capturing and then try to +optimize their results by combining optimization +methods [8]. However, according to the existing +literature, few articles have attempted to design a +holonomic path for under-actuated bipedal robots +without feet. Due to the importance of optimal motion +planning, a lot of work has been done in recent years in +this area. + In this paper, the problem of motion planning is +investigated to find the optimal paths for under-actuated +bipedal robots to step on the stairs, the results obtained +as a control output will cause the robot to move properly +and optimally. This article consists of three sections. In +the first part, the dynamic model of the bipedal robot is +derived. In the second part, the constraints of the +optimization problem are examined, in the third part, the +cost function and method of optimal problem solving +and finding a suitable movement gate are examined. In +the fourth section, the results are presented and +discussed, and at the end, the research of this article is +summarized as the conclusion. + +Dynamics equation +The dynamic model of the robot is shown in Figure +1. The robot has 7 degrees of freedom and 5 links, +each leg has two joints (one in the knee and the +other in the hip) and 3 degrees of freedom. We +assume that the contact of the tip of the leg is the +point. + +Figure 1. Planar bipedal robot + + +0.2 +10 to 12 May, 2022 + The robot's motion is planar and the robot has 4 +actuators, two actuators at the knees and two actuators +at the junction of the hip and the trunk so that there is +one actuator between each leg and trunk. It is assumed +that by hitting the tip of the swing leg on the ground, the +other leg rises from the ground, in other words, the +robot has no double support phase. So, when moving on +the stairs, no time is wasted for placing both feet on the +ground. Therefore, the hybrid dynamic equations of a +robot are a combination of a single support phase and +collision phase. The equations of the hybrid model are +as follows: +( ) +( ) +: +( +) +x +f x +g x u +x +x +x +x + + + + +  + + + +  +  + + + (1) + The vector +: ( +, +) +T +T T +x +q q + + consists of the vector of +generalized coordinates and their derivatives.  is a map +to find the states of the system exactly after the collision, +and the positive and negative symbols indicate the states +of the system before and after the collision. The switch +condition is as follows: + + +2 +2 +( , ) +| +( ) +0, +( ) +0 +v +h +q q +x P q +P +q +  + + + + (2) + In equation (2), +2 +h +P represents the horizontal position +of the swing leg and +2 +v +P represents its vertical position. + The dynamic equations of the robot before and after +the collision and in the single support phase can be +written as follows: +  +( ) +( , ) +( ) +q +q q +q +q +M +q +C +q +G +B u + + + + (3) + Matrix B is also a pre-multiplication matrix in the +torque vector and is not a square matrix due to the +under-actuation of the system. + In Equation 1, there is an expression called zero +dynamics, and it is easy to separate this term if the +generalized coordinates of the system are written in +relative terms (as has been done in this paper). The +satisfaction of this constraint is important in two ways. +First, if this constraint is not satisfied, the problem of +optimizing the input torques is practically ambiguous, +because these torques are not really applicable to the +problem. Although it may lead to a feasible kinematic +equation (kinematically possible), it is not feasible in +terms of control (open-loop), i.e. it is not dynamically +possible. + +Optimization problem +The most important constraint of the problem, called +zero dynamics, was introduced in the previous section. +Other constraints in this issue are important to plan the +robot movement in the best way; the constraints of the +optimization problem are generally classified into two +general modes of constraints based on dynamics and +constraints based on kinematics. + +1. Dynamic constraints: + Torque limit: because the torque generators have a +certain limit (inequality constraint). + Zero dynamic: the importance of which was +mentioned earlier (equality constraint). + Coefficient of friction limit: for the robot to move on +real environments, the ratio of horizontal force to +vertical force should not be more or less than a certain +limit. In other words, the coefficient of friction required +for stepping should not exceed a certain limit that can +not be implemented in real environments. (inequality +constraint). +2. Kinematic constraints: + Configuration: As an initial and final condition, the +robot needs to move from an initial configuration to a +final configuration. The best option is for the initial and +final state to be the same so that the robot has +periodicity in its movement and the best footprint is in +the middle of each stair Figure 2 (equality constraint). + +height +width +clearance +best +footprint +r1 +r2 + +Figure 2. Stair properties + + Angular velocity limit: Because motors have limited +angular velocity production. (inequality constraint) + Contact in single support phase: The robot is in +contact with the ground during the single support phase +and the acceleration of the contact point in the +horizontal and vertical direction during this period is +zero. (equality constraint) + Swing leg collision: The robot swing leg during the +single-phase phase, except at the beginning and end of +the phase, should not collide with the ground, on the +other hand, should have a suitable distance to the +obstacles. + Knees movement limitation: To create maximum +similarity to human movement, the robot knee should +not be opened and closed too much. +Failure to satisfy any of the above constraints will cause +problems +in +creating +optimal +and +appropriate +movement. + +Optimization method +This optimization is a nonlinear, constrained, and single- +objective problem. + Cost function: To find the optimal path, various cost +functions are considered, for example, the norm of +torque input, system input energy, and cost of transport +are common options. In this paper, we consider the +norm of torque inputs as the cost function. By this +choice, the torques are rational in size and will have +proper distribution (If the optimization problem is +solved properly). +4 +2 +0 +0 +( +( )) +T +i +i +J +u +d + + + + + + + (4) + In the above equation, T is the length of the time +period. + Selection of optimization variables: Optimization +variables can have different types, one of the best +choices +is +the +paths +followed +by +generalized +coordinates. Here our choice is a time-varying path as a +function of polynomials. The polynomial functions are + + +10 to 12 May, 2022 +uniform and smooth, and they are also simple for +deriving. +4 +, +0 +( ) +n +i +k +k i +i +q t +t + + + + + (5) + The degree of this polynomial must be chosen in +such a way that the number of optimization parameters, +which are the same as the number of polynomial +coefficients, are appropriate (minimum value to have a +smooth motion satisfied the mentioned constraints). In +this article, we choose the function of order 4 to have +freedom of action in terms of the optimization problem +and also not to make the number of optimization +parameters of the problem irrational and complicated. + Method of solving the optimization problem: This +optimization problem is solved by Variable Metric +methods for constrained optimization. This method is a +gradient-based method, which provides a desirable and +fast solution. Another advantage of this method is to not +get out easily from the feasible area [9]. + +Results and Discussion +Following the model and algorithm presented above, a +bipedal robot has been simulated to climb the stairs. The +height of the stairs is considered 20cm and the width of +the stairs is 40cm. The robot model specifications are in +accordance with Table 1. The initial and final angles of +the bipedal robot as a configuration are given in Table +2. Here the initial and final configurations are intuitively +obtained from the human configuration. The speed of +crossing each step is .5 seconds. The torque limit +applied to the system is 150 N.m and the maximum +angular velocity of the motors 10 rad/sec can be. + + +Table 1. Rabbit robot properties [10] +Symbol +Value +m1, m5 +3.2 kg +m2, m4 +6.8 kg +m3 +20 kg +I1, I5 +0.93 kg-m2 +I2, I4 +1.08 kg-m2 +I3 +2.22 kg-m2 +l1, l5 +0.4 m +l2, l4 +0.4 m +l3 +0.625 m +d1, d5 +0.128 m +d2, d4 +0.163 m +d3 +0.2 m + + + +Table 2. The initial and final configuration +Parameters +Initial value(rad) +Final value(rad) +q1 +0.2618 +0.1964 +q2 +1.3140 +0 +q3 +-1.2267 +0.0219 + q4 +-0.0219 +1.2267 +q5 +0 +1.3140 + + +Figure 3. Input torques + + According to Figure 3, the torques have a good +margin from the saturation and compared to other +articles and research reviewed in the introduction, more +optimal results have been obtained, also zero dynamics +( +v ) in a very good way is satisfied. + +Figure 4. Friction coefficient + + According to Figure 4, it is clear that the generated +path needs the maximum coefficient of friction .69 to +slip, so on all surfaces that have a coefficient of friction +higher than .69 there is the ability to move. + +Figure 5. Angles vs. angular velocities + + According to Figure 5, the generated paths, due to +the nature of the polynomial functions, have a smooth + + +10 to 12 May, 2022 +and non-breaking behavior, and the angular velocities +are far from their saturation limit. + + +Figure 6. Stick diagram of the climbing a stair up + + As can be seen in Figure 6, the robot's movement is +quite normal and very similar to human movement. The +trunk is kept in a good position and also the tip of the +feet and other links do not touch the surfaces except at +the beginning and at the end of the movement. +According to the sum of the presented results, the +generated path is an optimal path for the proper gait of +the under-actuated bipedal robot. + +Conclusions +In this article, we present a method to generate optimal +motion for a bipedal robot, we used this method to find +the paths that the 'rabbit' robot by tracking them can +optimally climb stairs. This process consists of 3 parts: +robot dynamic extraction (because optimization is based +on the model), design of constraints based on dynamics +and kinematics, and optimization. As a result of the +problem, a series of virtual holonomic paths were +extracted in which the zero hybrid dynamics of the +problem is also satisfied, so tracking the paths are +possible for under-actuated robots. +In the future, we plan to use a new method called impact +invariance to design the above path, which guarantees +the periodicity of the proposed paths. + + +References + +[1] Goldfarb, Nathaniel, Charles Bales, and Gregory S. +Fischer. "Toward Generalization of Bipedal Gait +Cycle During Stair Climbing Using Learning From +Demonstration." IEEE Transactions on Medical +Robotics and Bionics 3.2 (2021): 446-454. +[2] Kweon Soo Jeon, Ohung Kwon, Jong Hyeon Park. +Optimal trajectory generation for a biped robot +walking +a +staircase +based +on +genetic +algorithms[C]//Proceedings +of +2004 +IEEE/RSJ +International Conference on Intelligent Robots and +Systems, Sendai, Japan: IEEE, 2004: 2837- 2842. +[3] Cebe, Oguzhan, et al. "Online dynamic trajectory +optimization and control for a quadruped robot." +2021 IEEE International Conference on Robotics +and Automation (ICRA). IEEE, 2021. +[4] Kim, Eun-Su, Jo-Hwan Kim, and Jong-Wook Kim. +"Generation of optimal trajectories for ascending +and descending a stair of a humanoid based on +uDEAS." 2009 IEEE International Conference on +Fuzzy Systems. IEEE, 2009. +[5] Sugahara, Yusuke, et al. "Walking up and down +stairs carrying a human by a biped locomotor with +parallel mechanism." 2005 IEEE/RSJ International +Conference on Intelligent Robots and Systems. +IEEE, 2005. +[6] Zhang, Qin, et al. "Action generation of a biped +robot climbing stairs." 2013 IEEE International +Conference on Mechatronics and Automation. +IEEE, 2013. +[7] Kim, E., T. Kim, and J-W. Kim. "Three-dimensional +modelling of a humanoid in three planes and a +motion scheme of biped turning in standing." IET +control theory & applications 3.9 (2009): 1155- +1166. +[8] Powell, Matthew J., Huihua Zhao, and Aaron D. +Ames. "Motion primitives for human-inspired +bipedal robotic locomotion: walking and stair +climbing." 2012 IEEE International Conference on +Robotics and Automation. IEEE, 2012.Misc, A., +2003. Miscellaneous Title. On the WWW, May. +URL http://www.abc.edu. +[9] Powell, Michael JD. "A fast algorithm for +nonlinearly constrained optimization calculations." +Numerical analysis. Springer, Berlin, Heidelberg, +1978. 144-157.. +[10] Chevallereau, Christine, et al. "Rabbit: A testbed for +advanced control theory." IEEE Control Systems +Magazine 23.5 (2003): 57-79. + + + + + + + + + +2 +1.5 +0.5 +-0.5 +-0.5 +0 +0.5 +1 +1.5 +2 \ No newline at end of file diff --git a/BtAyT4oBgHgl3EQfR_cQ/content/tmp_files/load_file.txt b/BtAyT4oBgHgl3EQfR_cQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8bd8e470942bb3ba827f64ba36708b9ffe3e97bb --- /dev/null +++ b/BtAyT4oBgHgl3EQfR_cQ/content/tmp_files/load_file.txt @@ -0,0 +1,187 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf,len=186 +page_content='s er ne ngi E al ci han c e M of Iranian Society of ce en er f Con al on i nat nter I ual n An th 30 The 10 to 12 May, 2022, Tehran, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' ISME2022-IC1332 10 to 12 May, 2022 Optimal Motion Generation of the Bipedal Under-Actuated Planar Robot for Stair Climbing Aref Amiri 1, Hassan Salarieh 2 1Graduate Student, Sharif University of Technology, Tehran;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' aref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='amiri@mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='edu 2Professor, Sharif University of Technology, Tehran;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' salarieh@sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content="edu Abstract The importance of humanoid robots in today's world is undeniable, one of the most important features of humanoid robots is the ability to maneuver in environments such as stairs that other robots can not easily cross." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' A suitable algorithm to generate the path for the bipedal robot to climb is very important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' In this paper, an optimization-based method to generate an optimal stairway for under-actuated bipedal robots without an ankle actuator is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' The generated paths are based on zero and non-zero dynamics of the problem, and according to the satisfaction of the zero dynamics constraint in the problem, tracking the path is possible, in other words, the problem can be dynamically feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' The optimization method used in the problem is a gradient-based method that has a suitable number of function evaluations for computational processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' This method can also be utilized to go down the stairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Keywords: Bipedal robot, under-actuated, optimization, motion planning Introduction Inspired by human body physics, bipedal robots have many degrees of freedom and can perform various actions with their joint movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Bipedal robots can adapt to different environments that other wheeled robots are unable to move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' The study of path (trajectory) generation methods as a reference for the output of the control problem of bipedal robots in this regard is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' For the bipedal robot to climb the stairs, it is necessary to analyze the movement of them ascending the stairs and to examine the method of planning the bipedal robot to move and to determine the position of feet for walking on the stairs [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' So far, researches have been done on how to go up and downstairs and find a suitable or optimal path for bipedal robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Various papers using optimization algorithms and considering the robot angles as polynomial functions tried to design an optimal path for a 6-degree bipedal robot [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Some articles have even paths planned for multi-legged robots to cross the stairs [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Some articles also used stability criteria such as ZMP in designing their paths [4-7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' But this method is only appliable for robots that have feet (soles) with ankle joint actuators, which often have much lower speed in maneuvering than under-actuated robots without feet, and of course, due to the relatively large feet have more wasted energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Some articles also derive their initial path using data based on motion capturing and then try to optimize their results by combining optimization methods [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' However, according to the existing literature, few articles have attempted to design a holonomic path for under-actuated bipedal robots without feet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Due to the importance of optimal motion planning, a lot of work has been done in recent years in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' In this paper, the problem of motion planning is investigated to find the optimal paths for under-actuated bipedal robots to step on the stairs, the results obtained as a control output will cause the robot to move properly and optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' This article consists of three sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' In the first part, the dynamic model of the bipedal robot is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' In the second part, the constraints of the optimization problem are examined, in the third part, the cost function and method of optimal problem solving and finding a suitable movement gate are examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' In the fourth section, the results are presented and discussed, and at the end, the research of this article is summarized as the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Dynamics equation The dynamic model of the robot is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' The robot has 7 degrees of freedom and 5 links, each leg has two joints (one in the knee and the other in the hip) and 3 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' We assume that the contact of the tip of the leg is the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Planar bipedal robot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content="2 10 to 12 May, 2022 The robot's motion is planar and the robot has 4 actuators, two actuators at the knees and two actuators at the junction of the hip and the trunk so that there is one actuator between each leg and trunk." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' It is assumed that by hitting the tip of the swing leg on the ground, the other leg rises from the ground, in other words, the robot has no double support phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' So, when moving on the stairs, no time is wasted for placing both feet on the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Therefore, the hybrid dynamic equations of a robot are a combination of a single support phase and collision phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' The equations of the hybrid model are as follows: ( ) ( ) : ( ) x f x g x u x x x x \uf02d \uf02b \uf02d \uf02d \uf0ec \uf03d \uf02b \uf0cf\uf047 \uf0ef \uf053 \uf0ed \uf03d \uf044 \uf0ce\uf047 \uf0ef\uf0ee (1) The vector : ( , ) T T T x q q \uf03d consists of the vector of generalized coordinates and their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' \uf044 is a map to find the states of the system exactly after the collision, and the positive and negative symbols indicate the states of the system before and after the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' The switch condition is as follows: \uf07b \uf07d 2 2 ( , ) | ( ) 0, ( ) 0 v h q q x P q P q \uf047 \uf03d \uf0ce \uf03d \uf03e (2) In equation (2), 2 h P represents the horizontal position of the swing leg and 2 v P represents its vertical position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' The dynamic equations of the robot before and after the collision and in the single support phase can be written as follows: \uf028 \uf029 ( ) ( , ) ( ) q q q q q M q C q G B u \uf02b \uf02b \uf03d (3) Matrix B is also a pre-multiplication matrix in the torque vector and is not a square matrix due to the under-actuation of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' In Equation 1, there is an expression called zero dynamics, and it is easy to separate this term if the generalized coordinates of the system are written in relative terms (as has been done in this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' The satisfaction of this constraint is important in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' First, if this constraint is not satisfied, the problem of optimizing the input torques is practically ambiguous, because these torques are not really applicable to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Although it may lead to a feasible kinematic equation (kinematically possible), it is not feasible in terms of control (open-loop), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' it is not dynamically possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Optimization problem The most important constraint of the problem, called zero dynamics, was introduced in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Other constraints in this issue are important to plan the robot movement in the best way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' the constraints of the optimization problem are generally classified into two general modes of constraints based on dynamics and constraints based on kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Dynamic constraints: Torque limit: because the torque generators have a certain limit (inequality constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Zero dynamic: the importance of which was mentioned earlier (equality constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Coefficient of friction limit: for the robot to move on real environments, the ratio of horizontal force to vertical force should not be more or less than a certain limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' In other words, the coefficient of friction required for stepping should not exceed a certain limit that can not be implemented in real environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' (inequality constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Kinematic constraints: Configuration: As an initial and final condition, the robot needs to move from an initial configuration to a final configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' The best option is for the initial and final state to be the same so that the robot has periodicity in its movement and the best footprint is in the middle of each stair Figure 2 (equality constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' height width clearance best footprint r1 r2 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Stair properties Angular velocity limit: Because motors have limited angular velocity production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' (inequality constraint) Contact in single support phase: The robot is in contact with the ground during the single support phase and the acceleration of the contact point in the horizontal and vertical direction during this period is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' (equality constraint) Swing leg collision: The robot swing leg during the single-phase phase, except at the beginning and end of the phase, should not collide with the ground, on the other hand, should have a suitable distance to the obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Knees movement limitation: To create maximum similarity to human movement, the robot knee should not be opened and closed too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Failure to satisfy any of the above constraints will cause problems in creating optimal and appropriate movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Optimization method This optimization is a nonlinear, constrained, and single- objective problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Cost function: To find the optimal path, various cost functions are considered, for example, the norm of torque input, system input energy, and cost of transport are common options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' In this paper, we consider the norm of torque inputs as the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' By this choice, the torques are rational in size and will have proper distribution (If the optimization problem is solved properly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' 4 2 0 0 ( ( )) T i i J u d \uf074 \uf074 \uf03d \uf03d \uf0e5 \uf0f2 (4) In the above equation, T is the length of the time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Selection of optimization variables: Optimization variables can have different types, one of the best choices is the paths followed by generalized coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Here our choice is a time-varying path as a function of polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' The polynomial functions are 10 to 12 May, 2022 uniform and smooth, and they are also simple for deriving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' 4 , 0 ( ) n i k k i i q t t \uf061 \uf03d \uf03d \uf03d\uf0e5 (5) The degree of this polynomial must be chosen in such a way that the number of optimization parameters, which are the same as the number of polynomial coefficients, are appropriate (minimum value to have a smooth motion satisfied the mentioned constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' In this article, we choose the function of order 4 to have freedom of action in terms of the optimization problem and also not to make the number of optimization parameters of the problem irrational and complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Method of solving the optimization problem: This optimization problem is solved by Variable Metric methods for constrained optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' This method is a gradient-based method, which provides a desirable and fast solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Another advantage of this method is to not get out easily from the feasible area [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Results and Discussion Following the model and algorithm presented above, a bipedal robot has been simulated to climb the stairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' The height of the stairs is considered 20cm and the width of the stairs is 40cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' The robot model specifications are in accordance with Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' The initial and final angles of the bipedal robot as a configuration are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Here the initial and final configurations are intuitively obtained from the human configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' The speed of crossing each step is .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='5 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' The torque limit applied to the system is 150 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='m and the maximum angular velocity of the motors 10 rad/sec can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Rabbit robot properties [10] Symbol Value m1, m5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='2 kg m2, m4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='8 kg m3 20 kg I1, I5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='93 kg-m2 I2, I4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='08 kg-m2 I3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='22 kg-m2 l1, l5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='4 m l2, l4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='4 m l3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='625 m d1, d5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='128 m d2, d4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='163 m d3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='2 m Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' The initial and final configuration Parameters Initial value(rad) Final value(rad) q1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='2618 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='1964 q2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='3140 0 q3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='2267 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='0219 q4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='0219 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='2267 q5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='3140 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Input torques According to Figure 3, the torques have a good margin from the saturation and compared to other articles and research reviewed in the introduction, more optimal results have been obtained, also zero dynamics ( v\uf074 ) in a very good way is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Friction coefficient According to Figure 4, it is clear that the generated path needs the maximum coefficient of friction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='69 to slip, so on all surfaces that have a coefficient of friction higher than .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='69 there is the ability to move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Angles vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' angular velocities According to Figure 5, the generated paths, due to the nature of the polynomial functions, have a smooth 10 to 12 May, 2022 and non-breaking behavior, and the angular velocities are far from their saturation limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=" Stick diagram of the climbing a stair up As can be seen in Figure 6, the robot's movement is quite normal and very similar to human movement." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' The trunk is kept in a good position and also the tip of the feet and other links do not touch the surfaces except at the beginning and at the end of the movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' According to the sum of the presented results, the generated path is an optimal path for the proper gait of the under-actuated bipedal robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=" Conclusions In this article, we present a method to generate optimal motion for a bipedal robot, we used this method to find the paths that the 'rabbit' robot by tracking them can optimally climb stairs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' This process consists of 3 parts: robot dynamic extraction (because optimization is based on the model), design of constraints based on dynamics and kinematics, and optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' As a result of the problem, a series of virtual holonomic paths were extracted in which the zero hybrid dynamics of the problem is also satisfied, so tracking the paths are possible for under-actuated robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' In the future, we plan to use a new method called impact invariance to design the above path, which guarantees the periodicity of the proposed paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' References [1] Goldfarb, Nathaniel, Charles Bales, and Gregory S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Fischer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' "Toward Generalization of Bipedal Gait Cycle During Stair Climbing Using Learning From Demonstration.' metadata={'source': 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IEEE, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='Misc, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=', 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Miscellaneous Title.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' On the WWW, May.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' URL http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='abc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' [9] Powell, Michael JD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' "A fast algorithm for nonlinearly constrained optimization calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='" Numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' Springer, Berlin, Heidelberg, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' 144-157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='. [10] Chevallereau, Christine, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' "Rabbit: A testbed for advanced control theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='" IEEE Control Systems Magazine 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='5 (2003): 57-79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} +page_content='5 2' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'} diff --git a/DtE4T4oBgHgl3EQf6Q5X/vector_store/index.pkl b/DtE4T4oBgHgl3EQf6Q5X/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..5cb05f9cb55598e6a84412338794211f57baa391 --- /dev/null +++ b/DtE4T4oBgHgl3EQf6Q5X/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:cf0d72c9bccb4522409c7786e281982bd48758a4e9b8d380d8450d0a67dcd820 +size 98708 diff --git a/GNE0T4oBgHgl3EQfRQAC/content/tmp_files/2301.02203v1.pdf.txt b/GNE0T4oBgHgl3EQfRQAC/content/tmp_files/2301.02203v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c5f5d5a78081a35af224a8d3bf15553ab361c631 --- /dev/null +++ b/GNE0T4oBgHgl3EQfRQAC/content/tmp_files/2301.02203v1.pdf.txt @@ -0,0 +1,1210 @@ +arXiv:2301.02203v1 [math.CO] 5 Jan 2023 +DIVISIBILITY OF CHARACTER VALUES OF THE SYMMETRIC +GROUP BY PRIME POWERS +SARAH PELUSE AND KANNAN SOUNDARARAJAN +In memory of Chandra Sekhar Raju +Abstract. Let k be a positive integer. We show that, as n goes to infinity, almost every +entry of the character table of Sn is divisible by k. This proves a conjecture of Miller. +1. Introduction +It is a standard fact that the irreducible characters of Sn take only integer values for every +natural number n. In 2017, Miller [11] computed the character tables of Sn for all n ≤ 38 +and looked at various statistical properties of these integers as n grew. His computations +suggested that +(1) the density of even entries seemed to tend to 1, +(2) the density of entries divisible by 3, the density of entries divisible by 5, and the +density of entries divisible by 7 seemed to increase as n grew, +(3) about half of the nonzero entries were positive, +(4) and the density of zeros in the character table seemed to decrease as n grew, but not +very quickly. +Based on this first observation, Miller [11, 13] conjectured that as n goes to infinity, almost +every entry of the character table of the symmetric group Sn is even. +Following partial +progress due to McKay [10], Gluck [5], and Morotti [14], the first author proved this conjec- +ture in [15]. Based on the second observation, Miller [11, 13] also conjectured, more generally, +that for any fixed prime p, almost every entry of the character table of Sn is a multiple of +p as n goes to infinity. We proved this conjecture in [16], with a uniform upper bound for +the number of entries not divisible by a fixed prime. Recently, Miller [12] conjectured, even +more generally, that for any fixed prime power q, almost every entry of the character table +of Sn is a multiple of q as n goes to infinity. In this paper, we prove this most general of +Miller’s conjectures. +Theorem 1.1. Let n be large and q ≤ 10−3 log n/(log log n)2 be a prime power. The number +of entries in the character table of Sn that are not divisible by q is at most +O +� +p(n)2 exp(−(log log n)2) +� +. +It follows immediately from Theorem 1.1 and the union bound that almost every entry of +the character table of Sn is divisible by any fixed integer as n goes to infinity. +Corollary 1.2. Let k be any positive integer. Then, as n goes to infinity, the proportion of +entries in the character table of Sn that are not divisible by k tends to 0. +Our methods do not seem to shed any light on Miller’s third and fourth observations. +Most interesting to us is the question of what proportion of character table entries are zero, +1 + +2 +SARAH PELUSE AND KANNAN SOUNDARARAJAN +and it is not clear from Miller’s data whether the proportion is decreasing to zero or some +positive constant. Combining the Murnaghan–Nakayama rule and an old result of Erd˝os +and Lehner [2] on the distribution of the largest part of a uniformly random partition of n +produces a proportion of +1 +log n zeros in the character table of Sn, and it appears that no lower +bound of a larger order of magnitude is known. In the related setting of finite simple groups +of Lie type, Larsen and Miller [7] have shown that almost every character table entry is zero +as the rank goes to infinity. +Acknowledgments. +The first author is partially supported by the NSF Mathematical +Sciences Postdoctoral Research Fellowship Program under Grant No. DMS-1903038 and by +the Oswald Veblen Fund. The second author is partially supported by a grant from the +National Science Foundation, and a Simons Investigator Grant from the Simons Foundation. +We thank David Speyer for drawing our attention to Lemma 2.1. +2. Proof outline +For any partitions λ and µ of n, let χλ +µ denote the value of the irreducible character of +Sn corresponding to λ on the conjugacy class of elements with cycle type corresponding to +µ. In [16], our argument proceeded by combining two key facts: (i) if µ contains a part +substantially larger than the typical largest part of a random partition, then χλ +µ = 0 for +almost every λ, and (ii) if ν is another partition of n that is obtained from µ by combining +p parts of the same size m into one part of size pm, then χλ +µ ≡ χλ +ν (mod p) for every λ. +We showed that, for almost every µ, repeatedly combining p parts of the same size in this +manner produces a partition �µ containing a very large part, large enough so that χλ +�µ must +be zero for almost every λ. Our main result on the divisibility of character values by primes +then followed from the fact that χλ +µ ≡ χλ +�µ (mod p) for every λ. +The second key fact generalizes to a congruence of character value modulo prime powers +in a straightforward manner. +Lemma 2.1. Let pr be a power of the prime p. Suppose that µ is a partition of n, and that +ν is another partition of n obtained from µ by replacing pr parts of the same size m by pr−1 +parts of size pm. Then for all partitions λ of n, we have +χλ +µ ≡ χλ +ν +(mod pr). +However, when r > 1, it is no longer the case that starting from a typical partition µ of n +and repeatedly combining pr parts of the same size m into pr−1 parts of size pm produces a +partition �µ containing a part substantially larger than the largest part of a typical partition +of n. The argument from [16] that worked for primes thus breaks down for all other prime +powers. +The key idea used to overcome this barrier is a new condition for character values of the +symmetric group to be divisible by a fixed prime power, which we prove by exploiting certain +symmetries that appear after applying the Murnaghan–Nakayama rule multiple times. +Theorem 2.2. Let n, m1, . . . , mr be distinct positive integers. Let µ be a partition of n +containing parts of size m1, . . . , mr, each appearing at least pr−1 times. If λ is a (�r +i=1 kimi)- +core partition of n for all r-tuples (k1, . . . , kr) of integers 0 ≤ k1, . . . , kr ≤ pr−1 for which +some ki = pr−1, then +pr | χλ +µ. + +DIVISIBILITY OF CHARACTER VALUES OF THE SYMMETRIC GROUP +3 +Starting with a partition µ of n, repeatedly combine pr parts of the same size m into pr−1 +parts of size pm, until the process terminates in a partition �µ where no part appears more +than pr − 1 times. As a preliminary to applying Theorem 2.2 we show that for a typical +partition µ, the resulting partition �µ will have r parts that are suitably large, and with each +appearing at least pr−1 times. +Proposition 2.3. Starting with a partition µ of n, repeatedly replace every occurrence of pr +parts of the same size m by pr−1 parts of size pm until we arrive at a partition ˜µ where no +part appears more than pr − 1 times. Then, except for +O +� +p(n) exp +� +−n1/20pr�� +initial partitions µ, the partition �µ contains at least r distinct parts m1, . . . , mr, each appear- +ing at least pr−1 times and satisfying +pr−1mj > +� +1 + 1 +6pr +�√ +6 +2π +√n log n. +This holds uniformly for pr ≤ 10−3 log n/(log log n)2. +The significance of the lower bound on pr−1mj in Proposition 2.3 is that it lies beyond the +threshold of values t such that almost every partition of n is a t-core. +Lemma 2.4. Let 1 ≤ L ≤ log n/ log log n be a real number. Then, for any given integer t +with +t ≥ +� +1 + 1 +L +�√ +6 +2π +√n log n, +all but +O +� +p(n) log n +n1/2L +� +partitions of n are t-cores. +We can swiftly deduce our main result, Theorem 1.1, from the results stated above. +Deducing Theorem 1.1. Let µ be a partition of n, and suppose that �µ is as in Proposition 2.3. +Then, for all but at most +O +� +p(n) exp +� +−n1/20pr�� +choices of µ, the partition �µ contains at least r distinct parts m1, . . . , mr, each appearing at +least pr−1 times and satisfying +(2.1) +pr−1mj > +� +1 + 1 +6pr +�√ +6 +2π +√n log n. +Consider any r-tuple (k1, . . . , kr) with 0 ≤ k1, . . . , kr ≤ pr−1 and ki = pr−1 for some i. +Then k1m1 + . . . + krmr also exceeds the bound in (2.1), so that by Lemma 2.4 all but +O(p(n)(log n)/n +1 +2L) partitions λ of n are (k1m1 + . . . + krmr)-cores. Since there are at most +r(pr−1 + 1)r−1 such r-tuples (k1, . . . , kr), by the union bound we see that all but at most +O +� +p(n) log n +n1/12pr r +� +pr−1 + 1 +�r−1 +� +partitions λ of n are (k1m1 + . . . + krmr)-cores for all choices of the r-tuple (k1, . . . , kr). + +4 +SARAH PELUSE AND KANNAN SOUNDARARAJAN +Theorem 2.2 now shows that pr divides χλ +�µ, and since χλ +µ ≡ χλ +�µ (mod pr) by Lemma 2.1, it +also follows that pr divides χλ +µ. Putting everything together, we conclude that the number +of partitions λ and µ with pr ∤ χλ +µ is at most +O +� +p(n)2� +exp(−n1/(20pr)) + +1 +n1/13pr r +� +pr−1 + 1 +�r−1 �� += O +� +p(n)2 exp(−(log log n)2) +� +, +in the range pr ≤ 10−3 log n/(log log n)2. +□ +The rest of the paper is organized as follows. +We will prove Lemmas 2.1 and 2.4 in +Section 3, Theorem 2.2 in Sections 4, 5, 6, and 7, and Proposition 2.3 in Sections 8 and 9. +3. Proofs of Lemmas 2.1 and 2.4 +We begin by proving the two lemmas stated in the previous section. +Proof of Lemma 2.1. We claim that if Q ∈ Z[x1, . . . , xk] is a polynomial with integer coeffi- +cients, then +Q(x1, . . . , xk)pr ≡ Q(xp +1, . . . , xp +k)pr−1 +(mod pr). +As is well known, we may write +(3.1) +Q(x1, . . . , xk)p = Q(xp +1, . . . , xp +k) + p · R(x1, . . . , xk) +for some R ∈ Z[x1, . . . , xk], which establishes the claim when r = 1. For r > 1, raise both +sides of (3.1) to the power pr−1, and expand using the binomial theorem: +Q(x1, . . . , xk)pr = (Q(xp +1, . . . , xp +k) + p · R(x1, . . . , xk))pr−1 += Q(xp +1, . . . , xp +k)pr−1 + +pr−1 +� +ℓ=1 +�pr−1 +ℓ +� +Q(xp +1, . . . , xp +k)pr−1−ℓ(pR(x1, . . . , xk))ℓ. +Note that for 1 ≤ ℓ ≤ pr−1 +pℓ +�pr−1 +ℓ +� += pℓpr−1 +ℓ +�pr−1 − 1 +ℓ − 1 +� +≡ 0 +(mod pr), +since the power of p dividing ℓ is certainly at most ℓ − 1. This establishes our claim. +The lemma now follows by applying this observation to the polynomials appearing in +Frobenius’s formula for the character values χλ +µ and χλ +ν (see Chapter 4 of [4]). +□ +Proof of Lemma 2.4. The proof is essentially identical to that of Proposition 1 of [16], but +we include the short argument for completeness. Since every partition of n is a t-core for +t > n, we may naturally assume that t ≤ n. From Lemma 5 of [14], we know that at most +(t + 1)p(n − t) partitions of n are not t-cores. By the asymptotic formula +p(m) ∼ +1 +4 +√ +3m exp +� 2π +√ +6 +√m +� +for the partition function, we have +(t + 1)p(n − t) ≪ +t + 1 +n − t + 1 exp +� 2π +√ +6 +√ +n − t +� +≤ +t + 1 +n − t + 1 exp +� 2π +√ +6 +√n − +πt +√ +6n +� +. + +DIVISIBILITY OF CHARACTER VALUES OF THE SYMMETRIC GROUP +5 +In the range n ≥ t ≥ (1 + 1/L) +√ +6 +2π +√n log n, the right-hand side above is maximized at the +lower endpoint t = (1 + 1/L) +√ +6 +2π +√n log n. It follows that the number of partitions of n that +are not t-cores is +≪ log n +√n n−(1+1/L)/2 exp +� 2π +√ +6 +√n +� +≪ p(n) log n +n1/2L , +where the last step uses again the asymptotic for the partition function. +□ +4. Partitions and Abaci +The proof of Theorem 2.2 requires the machinery of the abacus associated to a partition. +A good reference for this theory is Section 2.7 of of [6], and we recall some salient facts +below. +4.1. The notion of an abacus. An abacus is a bi-infinite sequence of 0’s and 1’s beginning +with an infinite sequence of 1’s and ending with an infinite sequence of 0’s. +More formally, let +S := {s : Z → {0, 1} : there exists a k ≥ 0 such that s(−i) = 1 and s(i) = 0 for all i ≥ k} +denote the set of all sequences of 0’s and 1’s indexed using the integers, that begin with an +infinite sequence of 1’s and end with an infinite sequence of 0’s. For example, +. . . , 1, . . . , 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, . . ., 0, . . . +is in S. We consider two sequences s and s′ in S to be equivalent if there is some integer j +such that s(i) = s′(i − j) for all i, that is, if s′ can be produced by shifting the terms in s by +j. This is an equivalence relation, and an abacus refers to an equivalence class in S under +this relation. We denote by A the set of such abaci, so that by an element a of A we mean +the equivalence class consisting of some sequence s ∈ S together with all its shifts. +4.2. The abacus associated to a partition. We now show how abaci are in one-to-one +correspondence with partitions of integers. Starting with an integer partition λ, we construct +an abacus aλ ∈ A as follows. Draw the Young diagram of λ, and trace out the boundary of +the diagram, moving from the lower left-hand corner to the upper right-hand corner, writing +a 0 for each horizontal move and a 1 for each vertical move. Then prepend an infinite string of +1’s and append an infinite string of 0’s to find a representative of the corresponding element +aλ of A. +This procedure is easily reversed, and starting with an abacus a in A we obtain a Young +diagram, which corresponds to a partition λ. If s ∈ S is a representative of a, then the +partition λ is a partition of the integer n(a) which counts the number of pairs of indices (i, j) +with i < j such that s(i) = 0 and s(j) = 1. +To illustrate, consider the partition (6, 5, 3, 1, 1, 1), whose Young diagram is pictured in +Figure 1. +If we start in the lower left-hand corner of this diagram and move along the +boundary to the upper right-hand corner, we move right, up three times, right twice, up, +right twice, up, right, and up. The correspondence described above produces the string +(4.1) +0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, +which we can turn into a bi-infinite sequence by adding an infinite sequence of 1’s to the +beginning and an infinite sequence of 0’s to the end: +(4.2) +. . . , 1, . . . , 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, . . ., 0, . . . . + +6 +SARAH PELUSE AND KANNAN SOUNDARARAJAN +Figure 1. The Young diagram of (6, 5, 3, 1, 1, 1) +The equivalence class of this sequence is the abacus associated to (6, 5, 3, 1, 1, 1). +4.3. Hooks and border strips. Let λ be a partition. The hook h associated to a box b +in the Young diagram of λ consists of the box b together with all the boxes directly to its +right and directly below it. The hook-length of h, denoted by ℓ(h), is the number of boxes +contained in the hook. The height of the hook h, denoted by ht(h), is one less than the +number of rows in the Young diagram of λ that contain a box of h. Associated to each hook +is a border strip (also known as a skew hook), denoted bs(h), which is the connected region +of boundary boxes of the Young diagram running from the rightmost to the bottommost box +of h. Removing such a border strip leaves behind a smaller Young diagram. These notions +play a prominent role in the representation theory of the symmetric group, and in particular +feature in the Murnaghan–Nakayama rule for computing character values, which we next +recall (see Theorem 2.4.7 of [6], and also Chapter 4 of [4]). +Theorem 4.1 (The Murnaghan–Nakayama rule). Let n and t be positive integers, with +t ≤ n. Let σ ∈ Sn be of the form σ = τ · ρ, where ρ is a t-cycle, and τ is a permutation of +Sn with support disjoint from ρ. Let λ be a partition of n. Then +(4.3) +χλ(σ) = +� +h∈λ +ℓ(h)=t +(−1)ht(h)χλ\bs(h)(τ). +Above, χλ(σ) denotes the value of the character of the irreducible representation of Sn +corresponding to the partition λ, evaluated on the conjugacy class of σ, λ \ bs(h) denotes +the partition of n − t obtained by removing the border strip bs(h) from the Young diagram +of λ, and χλ\bs(h)(τ) denotes the character value of the irreducible representation of Sn−t +corresponding to the partition λ \ bs(h) evaluated on the conjugacy class of τ. +The abacus notation helps with thinking about hook lengths and border strips. Let λ be a +partition, let aλ denote the corresponding abacus, and let s be a representative in S for the +abacus aλ. Each hook h in the Young diagram of λ is in natural one-to-one correspondence +with a pair of indices (i, j), i < j, with s(i) = 0 and s(j) = 1. The length of the hook h is +j − i. In particular, the partition λ contains no hooks of length t (that is, λ is a t-core) if +and only if there is no pair of indices (i, i + t) with s(i) = 0 and s(i + t) = 1. The height of +the hook h equals the number of 1’s in the sequence s lying strictly between the 0 at index +i and the 1 at index j: +ht(h) = # {i < k < j : s(k) = 1} . + +DIVISIBILITY OF CHARACTER VALUES OF THE SYMMETRIC GROUP +7 +11 7 +6 +4 +3 +1 +9 +5 +4 +2 +1 +6 +2 +1 +3 +2 +1 +Figure 2. Hook-lengths for (6, 5, 3, 1, 1, 1) +, +Figure 3. The Young diagram of (6, 2, 1, 1, 1, 1) +Further, the abacus notation gives an easy description of the result of removing a border +strip from a partition. Define, for any pair of distinct integers (i, j) the operator Tij : S → S +that swaps the terms indexed by i and j in a bi-infinite sequence s ∈ S and leaves all other +entries fixed. Thus for s ∈ S +(Tijs)(k) = + + + + + +s(k) +k ̸= i, j +s(j) +k = i +s(i) +k = j. +With this notation in place, suppose λ is a partition, and s ∈ aλ is a representative of the +abacus of λ. Let h be a hook of λ, corresponding to the pair of indices (i, j) (with i < j) in +s. Then Tijs is a representative of the abacus associated to λ \ bs(h). +Returning to our example of the partition (6, 5, 3, 1, 1, 1), Figure 2 contains its Young +diagram again, but now with each box filled in with the corresponding hook-length. The +unique hook h of length 5 in the diagram corresponds to the pair of indices (5, 10) of the +sequence (4.1). If we remove the corresponding border strip, we obtain the diagram pictured +in Figure 3, which corresponds to the partition (6, 5, 3, 1, 1, 1) \ bs(h) = (6, 2, 1, 1, 1, 1) and +the bi-infinite sequence +. . . , 1, . . . , 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0 . . ., 0, . . . +of 0’s and 1’s. +Note that if we swap the 0 and 1 corresponding to the hook h in the +representative (4.2) of a(6,5,3,1,1,1), then we get an equivalent bi-infinite sequence. +4.4. Removing several hooks in succession. In our work below, we will need to remove +several hooks (more precisely, the border strips corresponding to those hooks) in succession + +8 +SARAH PELUSE AND KANNAN SOUNDARARAJAN +from a partition. By removing a sequence of hooks h1, . . ., hR from a partition λ, we mean +the following: h1 is a hook of λ, h2 is a hook of λ \ bs(h1), h3 is a hook of λ \ bs(h1) \ bs(h2), +and so on, until we arrive at hR which is a hook of λ \ bs(h1) . . . \ bs(hR−1), and when this +is removed we obtain the final partition λ′ = λ \ bs(h1) . . . \ bs(hR). +Let s be a representative of the abacus aλ associated to λ. Let (i1, j1) denote the pair of +indices in s corresponding to the hook h1, (i2, j2) the corresponding pair to h2 (which, +recall, is a hook of λ \ bs(h1) corresponding to the bi-infinite sequence Ti1,j1s), and so +on. +Thus, the sequence of hooks h1, . . ., hR may be encoded by the R-tuple of pairs +((i1, j1), (i2, j2), . . . , (iR, jR)), and the process of removing these hooks results in the sequence +s′ = TiR,jRTiR−1,jR−1 · · · Ti1,j1s. +The sequence s′ is a representative of the abacus aλ′ associated to the partition λ′. +Of particular interest for us will be the situation where all the hooks have the same length, +m say. Here jk = ik + m for all 1 ≤ k ≤ R, and we may encode the sequence of hooks by +simply the R-tuple (i1, . . . , iR). Note that the indices i1, . . ., iR may contain repeats, but +there are also constraints, such as i2 ̸= i1 (since (i1, i1 + m) is a hook in s and so it cannot +be a hook in Ti1,i1+ms). +5. Plan of the proof of Theorem 2.2 +We begin by restating Theorem 2.2 in terms of values of irreducible characters at elements +of Sn, which will make the notation involved in its proof cleaner. +Theorem 5.1 (An equivalent formulation of Theorem 2.2). Let n, m1, . . . , mr be distinct +positive integers. Let σ ∈ Sn be a permutation of the form +σ = τ · +r� +i=1 +pr−1 +� +j=1 +ρ(j) +i , +where each ρ(j) +i +is a cycle of length mi, the supports of all the cycles ρ(j) +i +are disjoint, and +τ ∈ Sn is a permutation with support disjoint from those of the ρ(j) +i ’s. Suppose that λ is a +(�r +i=1 kimi)-core partition of n for all r-tuples (k1, . . . , kr) of integers 0 ≤ k1, . . . , kr ≤ pr−1 +for which some ki = pr−1. Then +pr | χλ(σ). +The proof of Theorem 5.1 rests on the following crucial proposition, which is based on +applying the Murnaghan–Nakayama rule pr−1 times. +Proposition 5.2. Let r, m and n be positive integers. Let σ ∈ Sn be of the form +σ = τ · +pr−1 +� +j=1 +ρ(j), +where each ρ(j) is an m-cycle, with all the cycles ρ(j) being disjoint, and with τ ∈ Sn being +a permutation whose support is disjoint from all the cycles ρ(j). Denote by L the set of +partitions of n − pr−1m that can be obtained from λ by removing, in succession, pr−1 border +strips of length m. If λ is a pr−1m-core partition of n, then +χλ(σ) = p +� +λ′∈L +ǫλ′χλ′(τ), + +DIVISIBILITY OF CHARACTER VALUES OF THE SYMMETRIC GROUP +9 +where each ǫλ′ is an integer. +We will quickly deduce Theorem 5.1 (and hence Theorem 2.2) from Proposition 5.2 and +the following simple observation. +Lemma 5.3. Let n, t and m be positive integers. Let λ be a partition of n which is both a +t-core and a (t + m)-core. Let λ′ be a partition of n − m that can be obtained by removing a +border strip of length m from λ. Then λ′ is a t-core. +Proof. If λ has no hook (and thus no border strip) of length m then the lemma holds +vacuously. Suppose that λ′ arises from removing the border strip corresponding to the hook +h of length m in λ. Let aλ be the abacus of λ, and s be a representative bi-infinite sequence +in aλ. Suppose the hook h corresponds to the pair of indices (i, i + m) with s(i) = 0 and +s(i + m) = 1, so that the partition λ′ corresponds to the abacus containing s′ = Ti,i+ms. +If λ′ is not a t-core, then there must exist a pair of indices (j, j + t) with s′(j) = 0 and +s′(j + t) = 1. Since the entries of s and s′ differ only at the indices i and i + m, and since +λ is a t-core, we must have either j = i + m, or j + t = i. If j = i + m, then s(i) = 0 and +s(i + t + m) = s′(j + t) = 1 which contradicts the assumption that λ is a (t + m)-core. If +j = i−t, then s(i−t) = s′(j) = 0 and s(i+m) = 1, which again contradicts the assumption +that λ is a (t + m)-core. +□ +Deducing Theorem 5.1 from Proposition 5.2. Apply Proposition 5.2 first with m = mr to +obtain +χλ(σ) = p +� +λ′∈L +ǫλ′χλ′� +τ +r−1 +� +i=1 +pr−1 +� +j=1 +ρ(j) +i +� +. +If t is any number of the form t = �r−1 +i=1 kimi where the ki lie in [0, pr−1] with at least one +of them being pr−1, then λ is a (t + krmr)-core for all 0 ≤ kr ≤ pr−1. Since any λ′ ∈ L +arises from λ by removing pr−1 border strips of length mr, it follows by pr−1 applications of +Lemma 5.3 that λ′ is a t-core. +We may now repeat this argument, applying Proposition 5.2 to each λ′ ∈ L and now +removing pr−1 border strips of length mr−1. Applications of Lemma 5.3 show that the new +partitions λ′′ that arise are (�r−2 +i=1 kimi)-cores for all choices of 0 ≤ ki ≤ pr−1 with some +ki = pr−1. +Carrying this argument out r times, we obtain the desired result. +□ +The proof of Proposition 5.2 depends on the following two lemmas, which we shall prove +in the next two sections. +Lemma 5.4. Let λ be a partition, and let λ′ be obtained from λ by removing a sequence of +R border strips of the same length m. Let h1, . . ., hR be a sequence of R hooks of length m +which may be removed from the initial partition λ to result in the final partition λ′. Then +(−1)ht(h1)+...+ht(hR) = ǫ(λ, λ′) +where the sign ǫ(λ, λ′) = ±1 depends only on the initial and final partitions λ and λ′ and is +the same for all such possible sequences of hooks. +Lemma 5.5. Let λ be a pr−1m-core partition, and let λ′ be a partition that can be obtained +from λ by removing R = pr−1 border strips of length m. The number of tuples (i1, . . . , iR) +such that +s′ = TiR,iR+mTiR−1,iR−1+m · · ·Ti1,i1+ms + +10 +SARAH PELUSE AND KANNAN SOUNDARARAJAN +is a multiple of p. Here s is a representative of the abacus of λ, and the partition λ′ corre- +sponds to the abacus containing s′ +Once Lemmas 5.4 and 5.5 are in place, it is a simple matter to deduce Proposition 5.2. +Deducing Proposition 5.2. We apply the Murnaghan–Nakayama rule repeatedly while re- +moving in succession R = pr−1 hooks of length m from λ. This will result in an expression +for χλ(σ) of the form � +λ′∈L cλ′χλ′(τ), for suitable integers cλ′ which we must show are +multiples of p. Now +cλ′ = +� +(i1,...,iR) +(−1)ht(h1)+...+ht(hR) +where the sum is over all R-tuples (i1, . . . , iR) corresponding to hooks h1, . . ., hR, which +when removed from λ in order result in the partition λ′. Lemma 5.4 tells us that the sign +(−1)ht(h1)+...+ht(hR) is the same for all suitable tuples (i1, . . . , iR), and Lemma 5.5 tells us that +the number of such R-tuples is a multiple of p. +□ +6. Parity of heights of hooks: Proof of Lemma 5.4 +Let λ be a partition, and s a representative of the abacus aλ associated to λ. Augment s +by coloring a finite number N of 1’s in s with distinct colors, taking care to color all the 1’s +appearing to the right of the first zero in s. The 1’s appearing to the left of the first 0 are +unimportant, but we allow the flexibility of coloring some of them since this situation may +arise at an intermediate step when we remove hooks from λ. Note that the number of 1’s +appearing to the right of the first zero equals the number of rows in the partition λ. Thus +N is at least the number of rows in λ. Color these 1’s in the order of their appearance in s +using the colors c1, . . ., cN. Call the augmented sequence �s. +We begin with a general observation on removing hooks. Suppose (i, j) is a pair of indices +corresponding to a hook h in s (at the moment the hook can have any length j−i). Removing +this hook produces the sequence Ti,js. +Considering the augmented sequence �s, we have +the corresponding augmented sequence Ti,j�s after removing this hook. If we consider the +sequence of colors among the 1’s in this sequence, we obtain a permutation πij, say, of the +original sequence of colors (c1, . . . , cN) — the 1 appearing in (Ti,j�s)(i) has the color of the 1 +in �s(j), and all other 1’s in Tij(�s) retain their color in �s. If the height of the hook removed +is k, then note that �s had k colored 1’s between s(i) = 0 and s(j) = 1 and the permutation +πij can be obtained by making k-transpositions, each time swapping the color of the 1 at +position j by the color immediately preceding it. Thus (−1)k = (−1)ht(h) equals the sign of +the permutation πij. +If we remove hooks h1, . . ., hℓ in succession (again, their lengths could be arbitrary), then +the associated permutations of colors multiply, and therefore so do the signs of these permu- +tations. Thus, after removing these hooks in succession we would arrive at a permutation π +of the sequence of colors (c1, . . . , cN) and +(−1)ht(h1)+ht(h2)+...+ht(hℓ) = sgn(π). +We now turn to the situation of the lemma, where a sequence h1, . . ., hR of R hooks +is removed all of length m. +Our observation above shows that removing these hooks in +order leads to the sequence �s ′ where the color of the 1’s is given by a permutation π of the +original sequence of colors c1, . . ., cN. Further the sign of this permutation sgn(π) equals +(−1)ht(h1)+...+ht(hR). + +DIVISIBILITY OF CHARACTER VALUES OF THE SYMMETRIC GROUP +11 +To complete the proof, we will show that every way of removing R hooks of length m +that leads to the partition λ′ results in the same permutation of colors π. Consider the +subsequence of �s obtained by restricting to a progression (mod m): namely, (�s(a + ℓm))ℓ∈Z. +There are m such subsequences corresponding to a = 1, . . ., m. Since the hooks removed all +have length m, each removal of a hook affects only the terms within one of these subsequences, +leaving all the other subsequences unaltered. Further within any particular subsequence +(�s(a + ℓm))ℓ∈Z, it is impossible to alter the original sequence of colors by removing any +sequence of hooks of length m. +Therefore we can determine uniquely the color of any +element in �s ′: the 1’s appearing in this sequence in the progression a (mod m) have colors +determined by their order of appearance in the original sequence s. +7. Proof of Lemma 5.5 +Let λ be a pr−1m-core partition, and let s be a representative of its abacus. Let s′ be the +sequence obtained by removing a sequence of R = pr−1 border strips of length m from λ, +and let λ′ be the partition associated to s′. Our goal is to show that the number of ways of +reaching λ′ starting from λ is a multiple of p. +Let us first note that when r = 1, it is impossible to remove a border strip of length m +from λ, since λ is m-core by assumption. Thus the number of ways here is 0, and the lemma +holds (vacuously). Henceforth, assume that r ≥ 2. +For each a = 1, . . ., m, consider the subsequences of s and s′ obtained by restricting to +the progression a (mod m): thus, set +s(a; m) = (s(a + ℓm))ℓ∈Z, +s′(a; m) = (s′(a + ℓm))ℓ∈Z. +We may think of s(a; m) and s′(a; m) as corresponding to partitions λ(a; m) and λ′(a; m), and +note that a hook of length m in the partition λ corresponds to a hook of length 1 (or simply +a border square) in the partition λ(a; m) (for some choice of a). Since λ′(a; m) arises from +λ(a; m) by removing some number of hooks of length 1, the Young diagram of the partition +λ′(a; m) is contained in the Young diagram of the partition λ(a; m) (that is, λi(a; m) ≥ +λ′ +i(a; m) for all i). The difference between the Young diagram of λ(a; m) and λ′(a; m) (in +other words, the boxes in λ(a; m) that are not in λ′(a; m)) is a skew diagram, which we +denote by λ(a; m)/λ′(a; m). Let ℓa denote the size of this skew diagram |λ(a; m)/λ′(a; m)|, +so that ℓa hooks of length 1 must be removed from λ(a; m) to reach λ′(a; m). Since a total +of R = pr−1 hooks of length m are removed to go from λ to λ′, note that +R = pr−1 = +m +� +a=1 +ℓa. +The number of ways to go from λ(a; m) to λ′(a; m) by removing successively ℓa hooks +of length 1 equals the number of standard Young tableaux of skew shape λ(a; m)/λ′(a; m), +which we denote (in the usual notation) by fλ(a;m)/λ′(a;m). Recall that a standard Young +tableau of this skew shape is a numbering of the boxes in the skew diagram using the +numbers 1 to ℓa such that the entries are increasing from left to right in each row, and +increasing down each column. Each such tableau corresponds to a way of removing hooks, +by removing boxes in descending order of their entries. +We can now count the number of ways of going from λ to λ′ by removing R hooks of +length m. Note that removing a hook from one subsequence s(a; m) has no impact on the + +12 +SARAH PELUSE AND KANNAN SOUNDARARAJAN +hooks in any of the other subsequences. Therefore the desired number of ways to proceed +from λ to λ′ equals +� +pr−1 +ℓ1, ℓ2, . . . , ℓm +� m +� +a=1 +fλ(a;m)/λ′(a;m). +The multinomial coefficient +� +pr−1 +ℓ1,ℓ2,...,ℓm +� +is a multiple of p, except in the situation where +ℓa = pr−1 for some a (and all other ℓj are 0). Thus we are left with the case when all the +hooks of length m in going from λ to λ′ are confined to one subsequence s(a; m). So far, we +have not made use of the condition that λ is a pr−1m-core, and it is only in this case that +we need this assumption. The assumption implies that λ(a; m) is pr−1-core, and so the skew +diagram λ(a; m)/λ′(a; m) (which has size ℓa = pr−1) cannot be a border strip of λ(a; m). In +this situation, it turns out that fλ(a;m)/λ′(a;m) is a multiple of p. This is implied by our next +lemma, which is perhaps of independent interest. +Lemma 7.1. Let π and τ be two partitions, with the Young diagram of π containing the +Young diagram of τ (thus πi ≥ τi for all i). Suppose the skew diagram π/τ is not a border +strip of the partition π (equivalently, either π/τ is disconnected, or it contains a 2×2 square), +and that |π/τ| = pt is a prime power. Then the number of standard Young tableaux of skew +shape π/τ, denoted fπ/τ, is a multiple of p. +Proof. First suppose that π/τ is disconnected, and is composed of k ≥ 2 maximally connected +skew shapes S1, . . ., Sk, with |Sj| = sj ≥ 1. Then +fπ/τ = +� +pt +s1, . . . , sk +� +fS1 · · · fSk, +is clearly a multiple of p. +Now suppose that π/τ is a connected skew shape, but contains a 2 × 2 square so that it +is not a border strip of π. Since fπ/τ depends only on the shape π/τ, we may assume that π +is minimal, having only as many rows and columns as needed for the skew shape π/τ. Then +the maximal hook length of π equals the number of border squares of π, which is strictly +smaller than |π/τ| = pt (since π/τ is not a border strip by assumption). +It is a basic fact (see Section I.9 of [8], for example — the identity below follows from +equation (9.1) of [8] by taking the Hall inner product of both sides with the symmetric +function ept +1 ) that +fπ/τ = +� +ν⊢pt +fνcπ +τν, +where the sum is over partitions ν of |π/τ| = pt, fν = χν +(1,...,1) is the degree of the irreducible +character corresponding to ν and the cπ +τν are the Littlewood–Richardson coefficients (which +are integers). By Lemma 2.1, fν ≡ χν +(pt) (mod p), so that p | fν unless ν is a hook of length +pt. Suppose now that ν is a hook of length pt. Here we use that the Littlewood–Richardson +coefficient cπ +τν is zero unless the Young diagram of the partition ν is contained in that of π +(see Section I.9 of [8] once again). But all the hooks of π have length < pt, and therefore π +cannot contain a hook ν of length pt. Thus either cπ +τν = 0 or p|fν, and therefore the lemma +follows. +□ + +DIVISIBILITY OF CHARACTER VALUES OF THE SYMMETRIC GROUP +13 +8. Preliminaries for the proof of Proposition 2.3 +As in [16], let �p(k) denote the number of partitions of a nonnegative integer k into powers +of p, with the convention that �p(0) = 1. Denote by Fp(t) the associated generating function +Fp(t) := +∞ +� +k=0 +�p(k)e−k/t = +∞ +� +j=0 +� +1 − e−pj/t�−1 +, +where t > 0 is a real number. We begin by recalling some estimates from our prior work [16]. +Lemma 8.1 (Lemma 2 of [16]). When 0 < t ≤ 1, we have Fp(t) = O(1), and when t ≥ 1, +we have +(log t)2 +2 log p + 1 +2 log t + O(1) ≤ log Fp(t) ≤ (log t)2 +2 log p + 1 +2 log t + 1 +8 log p + O(1). +More precise results are known for fixed primes p, as partitions into prime powers have +been studied extensively since the work of Mahler [9] and de Bruijn [1]. We will only require +the estimates of Lemma 8.1, which are cruder but uniform in p. +Given a partition µ of k into powers of p, let �µ denotes the partition obtained by repeatedly +replacing every occurrence of pr parts of the same size pj by pr−1 parts of size pj+1 until no +part appears more than pr − 1 times. For every nonnegative integer s, define �p(k; s) to be +the number of partitions µ of k into powers of p such that �µ does not contain (at least) pr−1 +parts of the same size pj for any j ≥ s. The second lemma of this section gives a useful lower +bound for the difference between �p(k) and �p(k; s). +Lemma 8.2. For all s ≥ 2 and k ≥ pr+s−1(1 + 4/s), we have +�p(k) − �p(k; s) ≥ ps(s−1)/2 +(s − 1)s−1. +Proof. We will construct at least ps(s−1)/2/(s − 1)s−1 partitions of k counted in �p(k) but not +in �p(k; s). For each 1 ≤ i ≤ s − 1, pick an integer ai in the range +0 ≤ ai ≤ ps−i +s − 1. +Each choice of a1, . . . , as−1 gives a partition µ counted in �p(k) by using ai copies of pi for +1 ≤ i ≤ s − 1 and k − �s−1 +i=1 aipi copies of 1. The number of such partitions is +s−1 +� +i=1 +� ps−i +s − 1 +� +≥ +s−1 +� +i=1 +ps−i +s − 1 = +ps(s−1)/2 +(s − 1)s−1. +Note that if i > s − log(s − 1)/ log p, then ai must be zero, so that all of these partitions +have largest part at most +ps +s−1. +We must check that each such µ is not counted in �p(k; s); that is, that the corresponding +�µ contains at least pr−1 copies of some part pj with j ≥ s. Suppose that this is not the +case. Notice that, by construction, the number of times any part appears in µ is congruent +modulo pr−1 to the number of times it appears in �µ. Since no part can appear more than +pr − 1 times in �µ, it follows that any part that appears fewer than pr−1 times or more than +pr − pr−1 times in �µ must have appeared in the original partition µ. Since all the parts of µ +are below ps/(s − 1), we conclude that �µ can contain (i) at most pr − 1 copies of any part pj + +14 +SARAH PELUSE AND KANNAN SOUNDARARAJAN +with pj ≤ ps/(s −1), (ii) at most pr −pr−1 copies of any part pj with ps/(s −1) < pj ≤ ps−1, +and (iii) no parts of size pj with j ≥ s. But these constraints imply that +k = |�µ| ≤ (pr − 1) +� +pj≤ps/(s−1) +pj + (pr − pr−1) +� +ps/(s−1) 0 if ∥x − x′∥ ≤ D for any x, x′ ∈ X . Given a closed and convex set X ⊆ Rn, +the Euclidean projection operator is ΠX : Rn → Rn such that ΠX [x] = argminx′∈X ∥x − x′∥. For +closed and convex set X , Euclidean projection is non-expansive, i.e., ∥ΠX [x] − ΠX [x′]∥ ≤ ∥x − x′∥. +For a closed convex set X , the normal cone of x ∈ X is defined as NX (x) := {v : ⟨v, x′ − x⟩ ≤ 0}. +We make use of the following properties of the normal cone: (i) for any v ∈ NX (x), x = ΠX [x + v]; +(ii) if x = ΠX [x′], then x′ − x ∈ NX (x). +2.1 +Monotone Games and Nash Equilibria +A (continuous) multi-player game is denotes as G = ([N], (X i)i∈[N], (ℓi)i∈[N]) where [N] = {1, 2, . . . , N} +denotes the set of players. Each player i chooses action from a compact and convex set X i ∈ Rni +and we write X = ∏N +i=1 X i ∈ Rn where n = n1 + · · · + nN. We always use x−i to denote the +4 + +actions of all players except player i and write x = (xi, x−i) = (x1, x2 . . . , xN) as players’ action +profile or strategy profile. Note that we reserve the bold x to denote the players’ action profile and +use the normal x to denote a single player’s action. Each player i wishes to minimize a loss func- +tion ℓi(xi, x−i) : X → R which is continuous in x and convex in xi. In this paper, we study +learning in multi-player games with gradient feedback where after playing action profile x, each +player i receives Vi(x) := ∇xiℓi(xi, x−i). We define the gradient operator V : X → Rn to be +V(·) = (V1(·) · · · , VN(·)). The widely used solution concept for a game is Nash equilibrium, an +action profile where no player gains from unilateral deviation. Formally, a Nash equilibrium of a +game G is an action profile x⋆ ∈ X such that for each player i, it holds that ℓi(x⋆) ≤ ℓi(xi, x−i +⋆ ) for +any xi ∈ X i. +In this paper, we study smooth monotone games where the gradient operator V is L-Lipschitz for +L > 0: +��V(x) − V(x′) +�� ≤ L · +��x − x′��, ∀x, x′ ∈ X , +and monotone [Rosen, 1965] : +� +V(x) − V(x′), x − x′� ≥ 0, ∀x, x′ ∈ X . +It is not hard to see that for smooth monotone games, a Nash equilibrium always exists. If x⋆ +is a Nash equilibrium, then a simple characterization of x⋆ is that, for any x ∈ X , it holds that +⟨V(x⋆), x⋆ − x⟩ ≤ 0. +Monotone games include many well-studied games, e.g., two-player zero-sum games, convex- +concave games, λ-cocoercive games [Lin et al., 2020], strongly monotone games (such as Kelly +auctions), zero-sum polymatrix games [Bregman and Fokin, 1987, Daskalakis and Papadimitriou, +2009, Cai and Daskalakis, 2011, Cai et al., 2016], and zero-sum socially-concave games [Even-Dar +et al., 2009]. +Example 1 (Convex-Concave Min-Max Optimization). Given a function f (x, y) : X × Y → R that +is convex in x and concave in y, find a saddle point z = (x, y) such that +f (x, y′) ≤ f (x, y) ≤ f (x′, y), ∀x′ ∈ X , y′ ∈ Y. +It is not hard to see that the set of Nash equilibria of a two-player zero-sum game G = {[2], (X , Y), ( f, − f )} +corresponds to the set of saddle points of f. Thus convex-concave min-max optimization is a special case of +monotone games. +For a monotone game G and an action profile x, two standard measures of proximity to Nash +equilibrium are the gap function and the total gap function. +Definition 1. Let G = ([N], (X i)i∈[N], (ℓi)i∈[N]) be a monotone game. The gap function for x ∈ X is +GAP(x) = max +x′∈X +� +V(x), x − x′� +. +The total gap function for x ∈ X is +TGAP(x) = +N +∑ +i=1 +� +ℓi(x) − min +x′∈X i ℓi(x′, x−i) +� +. +Since ℓi is convex in xi for all i ∈ N, we have TGAP(x) ≤ GAP(x) for all x ∈ X . +5 + +A stronger measure of proximity to Nash equilibrium is the tangent residual defined as rtan(x) = +minc∈NX (x) ∥V(x) + c∥. The tangent residual is an upper bound for both the gap and the total gap. +Lemma 1 (Cai et al. [2022b]). Let G = ([N], (X i)i∈[N], (ℓi)i∈[N]) be a monotone game where X = +∏i∈[N] X i is bounded by D. For any x ∈ X , we have TGAP(x) ≤ GAP(x) ≤ D · rtan(x). +2.2 +Online Learning and Regret +A central theme of online learning is to design learning algorithms that minimize the regret. For +each time t = 1, 2, . . . , T, suppose the environment generates convex loss function ft : Ω → R and +the algorithm chooses action xt ∈ Ω where Ω ⊆ Rd is a compact convex set. The external regret +is defined as the gap between the algorithm’s realized cumulative loss and the cumulative loss of +the best fixed action in hindsight: +Reg(T) := +T +∑ +t=1 +ft(xt) − min +x∈Ω +T +∑ +t=1 +ft(x). +By the convexity of ℓt, we can bound the external regret by +Reg(T) ≤ max +x∈Ω +T +∑ +t=1 +⟨∇ ft(xt), xt − x⟩. +We will simply call the external regret as regret and any algorithm achieving sub-linear regret +Reg(T) = o(T) as a no-regret algorithm. +A much stronger performance measure of an online algorithm is the (worst-case) dynamic regret +[Zinkevich, 2003]: +DynamicReg(T) := +T +∑ +t=1 +ft(xt) − +T +∑ +t=1 +min +x∈Ω ft(x), +where the algorithm is competing with the best action in each round. It is not hard to see that in +adversarial setting, DynamicReg(T) must be linear in T. +3 +No-Regret Learning Algorithms and Games +In this section, we first review some background of gradient-based algorithms from both the on- +line learning and optimization. +We start with online gradient descent (GD) [Zinkevich, 2003]: the algorithm produces iterates +xt ∈ Ω defined by xt+1 = ΠΩ[xt − ηtgt] where we write gt := ∇ ft(xt) as the gradient of the loss +function ft. Online gradient descent is a no-regret algorithm in the adversarial setting. When +employed by all players, however, it diverges in last-iterate even for simple two-player zero-sum +games. +6 + +Optimism in Online Learning +A modification of online gradient descent is the Optimistic Gradi- +ent (OG) [Popov, 1980, Rakhlin and Sridharan, 2013, Daskalakis et al., 2018]: the algorithm chooses +action xt+ 1 +2 in each round t and updates iterates: +xt+ 1 +2 = ΠΩ +� +xt − ηtgt− 1 +2 +� +, +xt+1 = ΠΩ +� +xt − ηtgt+ 1 +2 +� +. +(OG) +Compared to online gradient descent, OG also achieves optimal regret in the single-agent ad- +versarial setting. Moreover, OG converges in the last-iterate sense as optimism stabilizes the +trajectory. When employed by all players in monotone games, their trajectory of play (xt+ 1 +2 )t≥1 +converges to a Nash equilibrium with an O( 1 +√ +T) last-iterate convergence rate [Cai et al., 2022b]. +Unfortunately, the O( 1 +√ +T) rate is tight for OG and more generally all p-SCLI algorithms [Golowich +et al., 2020a]. New ideas are needed to further sharpen the convergence rate. +Acceleration in Optimization +We are inspired by a technique from optimization for accelerating +first-order methods known as the Halpern iteration [Halpern, 1967] or Anchoring. The technique is +closely related to Nesterov’s accelerated method [Tran-Dinh, 2022] and has received extensive at- +tention from the optimization community recently [Diakonikolas, 2020, Yoon and Ryu, 2021, Lee +and Kim, 2021, Cai et al., 2022a]. When the Halpern iteration is applied to the classical extragradi- +ent (EG) algorithm [Korpelevich, 1976], which belongs to the p-SCLI family and also has an O( 1 +√ +T) +last-iterate convergence rate [Cai et al., 2022b], the resulting extra anchored gradient (EAG) algo- +rithm achieves an O( 1 +T) last-iterate convergence rate [Yoon and Ryu, 2021, Cai et al., 2022a]. Cai +and Zheng [2023] obtain a single-call algorithm – Accelerated Reflected Gradient (ARG) that also +achieves the same optimal last-iterate convergence rate. However, EAG is not suitable for multi- +player games, as it could exhibit linear regret as we demonstrated in Appendix D. ARG requires +evaluating the gradient at points outside of the feasible domain, thus it is also incompatible with +multi-player games. Our analysis is based on a construction from [Golowich et al., 2020a], where +they show that EG has linear regret in multi-player games. +3.1 +Accelerated Optimistic Gradient +We propose the following algorithm – the accelerated optimistic gradient (AOG) algorithm. The +central idea is to combine optimism with Halpern iteration: in round t, the algorithm chooses action +xt+ 1 +2 and updates as follows. +xt+ 1 +2 = ΠΩ +� +xt − ηtgt− 1 +2 + +1 +t + 1(x1 − xt) +� +, +xt+1 = ΠΩ +� +xt − ηtgt+ 1 +2 + +1 +t + 1(x1 − xt) +� +. +(AOG) +7 + +Double Optimality. +Our main result is that (AOG) is a doubly optimal online algorithm: with +ηt = Θ( 1 +√ +t), (AOG) achieves optimal O( +√ +T) regret in adversarial setting (Theorem 1); when all +players employ (AOG) with constant step size in a monotone game, their trajectory of play enjoys +optimal O( 1 +T) last-iterate convergence rate (Theorem 2). +Step-Size Adaptation +We also present an implementation of (AOG) in Algorithm 1 with a step- +size adaptation procedure (Line 7-11). This procedure uses the player’s own second-order gradient +variation St+1 = ∑t +s=2 ∥gs+ 1 +2 − gs− 1 +2 ∥2 as a proxy for the environment and adapts the step-size ac- +cordingly. The high level idea is that if all players use Algorithm 1 in a smooth monotone game, +then each player’s second-order gradient variation remains to be bounded by a constant that only +depends on L and D (Theorem 4), so the algorithm will keep a constant learning rate and achieve +an O( 1 +T) last-iterate convergence (Theorem 2); if the player’s second-order gradient variation ex- +ceeds a certain constant threshold, then Algorithm 1 decreases the learning rate according to the +second-order gradient variation, and by the standard argument of ”regret is bounded by stability”, +we can essentially bound the player’s regret by the the second-order gradient variation, which is +at most O( +√ +T) even in the adversarial setting (Theorem 1). +Algorithm 1 AOG with step-size adaptation +1: Input: L, D > 0. +2: Initialize g 1 +2 =⃗0, η1 = η = +1 +3L, and choose an arbitrary x1 ∈ Ω. +3: for t = 1, 2, · · · do +4: +xt+ 1 +2 = ΠΩ[xt − ηtgt− 1 +2 + +1 +t+1(x1 − xt)] +5: +Play xt+ 1 +2 and receive feedback gt+ 1 +2 . +6: +xt+1 = ΠΩ[xt − ηtgt+ 1 +2 + +1 +t+1(x1 − xt)] +7: +if St+1 := ∑t +s=2 ∥gs+ 1 +2 − gs− 1 +2 ∥2 > 4500πD2L2 then +8: +ηt+1 = +1 +√1+St+1 . +9: +else +10: +ηt+1 = ηt. +11: +end if +12: end for +Remark 1. In the adversarial setting, L and D can be any positive real numbers. If all players use Al- +gorithm 1, L should be an upper bound of the Lipschitz constant of the game, and D should be an upper +bound of the diameter ∥x − x′∥ ≤ D for x, x′ ∈ X . In other words, the players do not need to know exactly +the environment that they are interacting with to carefully pick the learning rate. As long as they know an +upper bound for the Lipschitz constant and the diameter of all games that they could potentially participate +in, Algorithm 1 will successfully choose the appropriate learning rate for them. +8 + +4 +Worst-Case Regret in the Adversarial Environment +In this section, we view Algorithm 1 as a single-agent online learning algorithm in the adversarial +setting. One could also interpret the result in the game setting, where we make no assumption +on how the other players choose their actions. We show in Theorem 1 that Algorithm 1 achieves +min-max optimal O( +√ +T) regret when the gradient feedback is bounded. It shows that AOG is an +optimal no-regret algorithm in the adversarial setting. +Theorem 1 (Regret Bound of Algorithm 1). Let G = maxt ∥gt+ 1 +2 ∥2 and suppose the action set Ω is +bounded by D. The regret of Algorithm 1 is bounded by O(D2G +√ +T + G2). +We first establish a single-step regret inequality in Lemma 2. +Lemma 2 (Single-Step Regret Inequality). Suppose the action set Ω is bounded by D. For all t ≥ 1 and +any x′ ∈ X , the iterates of AOG satisfies +� +xt+ 1 +2 − x′, gt+ 1 +2 +� +≤ +1 +2ηt +���x′ − xt +��2 − +��x′ − xt+1 +��2� ++ ηt +���gt+ 1 +2 − gt− 1 +2 +��� +2 ++ +D2 +ηt(t + 1). +The main idea behind Lemma 2 is to view the update rule of AOG as a standard update rule +of OG with modified gradients gt− 1 +2 − +1 +ηt(t+1)(x1 − xt) and gt+ 1 +2 − +1 +ηt(t+1)(x1 − xt), which allows us +to apply the classical analysis of OG [Rakhlin and Sridharan, 2013]. Equipped with Lemma 2, we +can bound the regret of Algorithm 1 even with adaptive size. We defer the proofs of Lemma 2 and +Theorem 1 to Appendix B. +5 +Last-Iterate Convergence Rate to a Nash Equilibrium in Monotone +Games +In this section, we consider a multi-player learning setting where each player follows AOG with +constant step size in smooth monotone games: each player i plays xi +t+ 1 +2 , receives gradient Vi(xt+ 1 +2 ), +and updates +xi +t+ 1 +2 = ΠX i +� +xi +t − ηVi(xt− 1 +2 ) + +1 +t + 1(xi +1 − xi +t) +� +, +xi +t+1 = ΠX i +� +xi +t − ηVi(xt+ 1 +2 ) + +1 +t + 1(xi +1 − xi +t) +� +. +We show in Theorem 2 that the trajectory of the action profile (xt+ 1 +2 )t∈[T] converges to Nash +equilibrium in last-iterate with an O( 1 +T) rate. Our convergence rate result matches the Ω( 1 +T) lower +bound by Yoon and Ryu [2021] and thus establishes that AOG is doubly optimal. +9 + +Theorem 2 (Optimal Last-Iterate Convergence Rate). Let G = {N, (X i)i∈[N], (ℓi)i∈[N]} be a L-smooth +monotone game, where the diameter of X = ∏i∈[N] X i is bounded by D. When all players employ AOG +with a constant step size η ≤ +1 +√ +6L in G, then for any T ≥ 2, we have +• rtan(xT+ 1 +2 ) ≤ 55D +ηT ; +• TGAP(xT+ 1 +2 ) ≤ GAP(xT+ 1 +2 ) ≤ 55D2 +ηT . +A Sketch of the Proof. +First, recall that the tangent residual provides upper bounds for both +the gap function and the total gap function due to Lemma 1, so it suffices to prove a last-iterate +convergence rate with respect to the tangent residual. For x ∈ X , its tangent residual is defined +as rtan(x) = minc∈NX (x) ∥V(x) + c∥. The definition itself contains an optimization problem, thus +is not explicit and difficult to directly work with. We relax the tangent residual by choosing an +explicit c ∈ NX (x) as follows: for each player i ∈ [N] and iteration t ≥ 2, we define +ci +t = +xi +t−1 − ηVi(xt− 1 +2 ) + 1 +t (xi +1 − xi +t−1) − xi +t +η +. +According to the update rule of AOG, ci +t ∈ NX i(xi +t). Define ct = (c1 +t , c2 +t , · · · , cN +t ) and we have +ct ∈ NX (xt). Thus rtan(xt) = minc∈NX (xt) ∥V(xt) + c∥ ≤ ∥V(xt) + ct∥. +Using ∥V(xt) + ct∥ as a proxy of the tangent residual rtan(xt), we construct a potential function +of Pt in the order of Θ(t2 · ∥V(xt) + ct∥2). Although the potential function might increase between +consecutive iterates, we manage prove that in Lemma 3 that the increment is sufficiently small: +Pt+1 ≤ Pt + O(∥V(xt+1) + ct+1∥2) for any t ≥ 2. Using the approximate monotonicity of Pt, we +derive the following inequality for the sequence (∥V(xt) + ct∥2)t≥2 +Θ(t2 · ∥V(xt) + ct∥2) ≤ O(1) + O( +t−1 +∑ +s=2 +∥V(xs) + cs∥2). +Based on the above inequality, we show in Lemma 4 that ∥V(xt) + ct∥2 = O( 1 +t2 ) for any t ≥ 2, +which implies O( 1 +T) last-iterate convergence rate for xt. The final step is to relate the convergence +on xt to the convergence of the action profile xt+ 1 +2 . +5.1 +Proof of Theorem 2 +Some of the proofs are postponed to Appendix C. We also defer some auxiliary propositions to +Appendix G. +Potential Function +We first formally define our potential function Pt: for t ≥ 2, let Pt be +t(t + 1) +2 +� +∥ηV(xt) + ηct∥2 + +���ηV(xt) − ηV(xt− 1 +2 ) +��� +2� ++ t⟨ηV(xt) + ηct, xt − x1⟩. +10 + +We first provide an upper bound on P2. +Proposition 1. In the same setup of Theorem 2, P2 ≤ 9D2. +Now we present the main technical lemma of this section, where we show the potential func- +tion Pt is approximately non-increasing. +Lemma 3. In the same setup of Theorem 2, if we choose η = +√q +L for any q ∈ (0, 1 +4), then for all t ≥ 2, +Pt+1 ≤ Pt + +3q +2(1 − 4q)∥ηV(xt+1) + ηct+1∥2. +Proof. We show Pt − Pt+1 minus a few non-negative terms is at least − +3q +2(1−4q)∥ηV(xt+1) + ηct+1∥2. +Here we present the list of non-negative terms that we use in the proof. +Non-Negative Terms +Since the game is monotone, we have +⟨ηV(xt+1) − ηV(xt), xt+1 − xt⟩ ≥ 0. +(1) +Using the L-Lipschitzness of V and the fact that (ηL)2 ≤ q, we have +q +���xt+1 − xt+ 1 +2 +��� +2 +− +���ηV(xt+1) − ηV(xt+ 1 +2 ) +��� +2 +≥ 0. +(2) +Since ct lies in the normal cone NX (xt) and ct+1 lies in the normal cone NX (xt+1), by the definition +of normal cone we have +⟨ηct+1, xt+1 − xt⟩ ≥ 0 +(3) +� +ηct, xt − xt+ 1 +2 +� +≥ 0 +(4) +⟨ηct, xt − xt+1⟩ ≥ 0 +(5) +As xt − ηV(xt− 1 +2 ) + +1 +t+1(x1 − xt) − xt+ 1 +2 lies in the normal cone NX (xt+ 1 +2 ), we also have +� +xt − ηV(xt− 1 +2 ) + x1 − xt +t + 1 − xt+ 1 +2 , xt+ 1 +2 − xt+1 +� +≥ 0. +(6) +11 + +Descent Identity +For convenience, we denote LHSI as “left-hand side of inequality”. We have +the following identity by Proposition 3: +Pt − Pt+1 − t(t + 1) · LHSI (1) − t(t + 1) +4q +· LHSI (2) +− t(t + 1) · LHSI (3) − t(t + 1) +2 +· (LHSI (4) + LHSI (5) + LHSI (6)) += t(t + 1) +2 +���� +xt+ 1 +2 − xt+1 +2 ++ ηV(xt) − ηV(xt+ 1 +2 ) +���� +2 ++ t(t + 1) +2 +���� +xt+ 1 +2 + xt+1 +2 +− xt + ηV(xt) + ct − x1 − xt +t + 1 +���� +2 ++ (1 − 4q)t − 4q +4q +(t + 1) +���ηV(xt+ 1 +2 ) − ηV(xt+1) +��� +2 +� +�� +� +I ++ (t + 1) · +� +ηV(xt+ 1 +2 ) − ηV(xt+1), ηV(xt+1) + ηct+1 +� +. +� +�� +� +II +Further using identity ∥a∥2 + ⟨a, b⟩ = ∥a + b +2∥ +2 − 1 +4∥b∥2, we can simplify the last two terms: +I + II += +���A(ηV(xt+ 1 +2 ) − ηV(xt+1)) + B(ηV(xt+1) + ηct+1) +��� +2 +− +q(t + 1) +(1 − 4q)t − 4q∥ηV(xt+1) + ct+1∥2 +≥ − +3q +2(1 − 4q)∥ηV(xt+1) + ct+1∥2, +where A = +� +(1−4q)t−4q +4q +(t + 1) , B = +� +q +(1−4q)t−4q(t + 1), and we use the fact that t+1 +t +≤ +3 +2 for +t ≥ 2 in the last inequality. Combining the above two inequalities and the fact that we only add +non-positive terms to Pt − Pt+1, we conclude that Pt+1 ≤ Pt + +3q +2(1−4q)∥ηV(xt+1) + ct+1∥2. +Using the fact that the potential function Pt is approximately non-increasing, we are able to +use induction to show last-iterate convergence rate of the sequence (xt)t≥2. +Lemma 4. If X is bounded by D and η ∈ (0, +1 +√ +6L), then we have for all T ≥ 2, +∥V(xT) + cT∥ ≤ 13D +ηT +and +���V(xT) − V(xT− 1 +2 ) +��� ≤ 13D +ηT . +12 + +Proof. Let x⋆ be a Nash equilibrium of G. For any t ≥ 2, we have +Pt = t(t + 1) +2 +� +∥ηV(xt) + ηct∥2 + +���ηV(xt) − ηV(xt− 1 +2 ) +��� +2� ++ t⟨ηV(xt) + ηct, x⋆ − x1⟩ + t⟨ηV(xt) + ηct, xt − x⋆⟩ +≥ t(t + 1) +2 +� +∥ηV(xt) + ηct∥2 + +���ηV(xt) − ηV(xt− 1 +2 ) +��� +2� ++ t⟨ηV(xt) + ηct, x⋆ − x1⟩ +≥ t(t + 1) +4 +� +∥ηV(xt) + ηct∥2 + 2 +���ηV(xt) − ηV(xt− 1 +2 ) +��� +2� +− +t +t + 1∥x⋆ − x1∥2 +≥ t(t + 1) +4 +� +∥ηV(xt) + ηct∥2 + 2 +���ηV(xt) − ηV(xt− 1 +2 ) +��� +2� +− ∥x⋆ − x1∥2. +In the first inequality, we drop a positive term where ⟨V(xt), xt − x⋆⟩ ≥ ⟨V(x⋆), xt − x⋆⟩ ≥ 0 since +x⋆ is Nash equilibrium, and ⟨ct, xt − x⋆⟩ ≥ 0 as ct ∈ NX (xt). In the second inequality, we apply +inequality ⟨a, b⟩ ≥ − α +4∥a∥2 − 1 +α∥b∥2 with a = +√ +tη(V(xt) + ct), b = x⋆ − x1, and α = √ +t + 1; we +use +t +t+1 ≤ 1 in the last inequality. Combing the above inequality with Lemma 3 and Proposition 1, +we get for any t ≥ 2, +t(t + 1) +4 +� +∥ηV(xt) + ηct∥2 + 2 +���ηV(xt) − ηV(xt− 1 +2 ) +��� +2� +≤ ∥x⋆ − x1∥2 + Pt +≤ ∥x⋆ − x1∥2 + P2 + 1 +3 +t−1 +∑ +s=2 +∥ηV(xs) + ηcs∥2 +≤ 10D2 + 1 +3 +t−1 +∑ +s=2 +∥ηV(xs) + ηcs∥2. +By Proposition 4, we can conclude that for any t ≥ 2, +∥ηV(xt) + ηct∥2 + 2 +���ηV(xt) − ηV(xt− 1 +2 ) +��� +2 +≤ 160D2 +t2 +. +This completes the proof as 132 = 169 ≥ 160. +Using the last-iterate convergence rate on (xt)t≥2, we only need to bound the distance between +xt and xt+ 1 +2 . +Lemma 5. In the same setup of Theorem 2, we have for any t ≥ 2, +���xt+ 1 +2 − xt +��� ≤ 27D +t +. +13 + +Proof of Theorem 2 +Given Lemma 4 that proves the last-iterate convergence rate on the sequence +(xt)t≥2, and Lemma 5 that upper bounds the distance between xt and xt+ 1 +2 , we are now ready to +prove the last-iterate convergence rate for (xt+ 1 +2 )t≥2. +Note that xt − ηV(xt− 1 +2 ) + x1−xt +t+1 − xt+ 1 +2 ∈ NX (xt+ 1 +2 ), thus we can upper bound the tangent +residual at xt+ 1 +2 by +rtan(xt+ 1 +2 ) = 1 +η +min +c∈NX (xt+ 1 +2 +) +���ηV(xt+ 1 +2 ) + c +��� +≤ 1 +η +����ηV(xt+ 1 +2 ) + xt − ηV(xt− 1 +2 ) + x1 − xt +t + 1 − xt+ 1 +2 +���� +≤ +���V(xt) − V(xt− 1 +2 ) +��� + 1 + ηL +η +���xt+ 1 +2 − xt +��� + +D +η(t + 1) +≤ 13D +ηt + +3 +2 · 27D +ηt ++ +D +η(t + 1) +(Lemma 4, 5 and ηL ≤ 1 +2) +≤ 55D +ηt , +where we use the triangle inequality and the L-Lipschitzness of V in the second inequality. This +completes the first part of Theorem 2. The second part of Theorem 2 follows directly from the first +part of Theorem 2 and Lemma 1. +6 +Dynamic Regret and Second-Order Gradient Variation +Recent works on no-regret learning in games have provided near-optimal bounds for players’ in- +dividual external or swap regret. In particular, Daskalakis et al. [2021], Anagnostides et al. [2022a,b] +achieve logarithmic regret bounds for general-sum games, and the bound can be sharpen to O(1) +if the games are monotone [Hsieh et al., 2021]. However, dynamic regret is a much stronger concept, +which is impossible to achieve in the single-agent adversarial setting and tightly relates to the con- +cept of last-iterate convergence in game settings. For example, the O( 1 +√ +T) last-iterate convergence +rate of OG implies a O( +√ +T) individual dynamic regret bound in monotone games. To the best of +our knowledge, O( +√ +T) is the best bound for dynamic regret even in two-player zero-sum games. +We significantly improve the bound and show that the individual dynamic regret is at most +O(log T) if each player employs AOG in monotone games. This is made possible by the fast O( 1 +T) +last-iterate convergence rate of AOG. +Theorem 3 (Individual Dynamic Regret Bound). In the same setup of Theorem 2, for any i ∈ [N] and +T ≥ 2, +DynamicRegi(T) ≤ O(log T). +14 + +Proof. By the definition of dynamic regret and total gap function, for any T ≥ 2, we have +DynamicRegi(T) = +T +∑ +t=1 +� +ℓi(xt+ 1 +2 ) − min +x′∈X i ℓi(x′, x−i +t+ 1 +2 ) +� +≤ O(1) + +T +∑ +t=2 +TGAP(xt+ 1 +2 ) ≤ +T +∑ +t=2 +O(1 +t ) = O(log T). +Last-iterate convergence rate of AOG also implies each player’s bounded second-order gradi- +ent variation. We defer the proof of Theorem 4 to Appendix E. +Theorem 4 (Bounded Second-Order Gradient Variation). In the same setup of Theorem 2 but with +η = +1 +3L, for any player i and time t ≥ 2, we have Si +T ≤ 4500πD2L2. +Bounded second-order gradient variation guarantees when each player employs Algorithm 1 +with the step-size adaptation procedure, they will always use constant step size. Combining The- +orem 1, Theorem 2, and Theorem 4, we conclude that Algorithm 1 is doubly optimal. +Theorem 5. Algorithm 1 automatically adapts to the environment and achieves O( +√ +T) regret in the +adversarial setting and O( 1 +T) last-iterate convergence rate in smooth monotone games. +7 +Illustrative Experiments +100 +101 +102 +103 +104 +105 +Iteration +10 +2 +10 +1 +100 +101 +102 +Tangent residual +AOG +OG +5000/(t+1) +0 +20000 +40000 +60000 +80000 +100000 +Iteration +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Dynamic regret +1e8 +AOG +OG +Figure 1: Numerical Results of AOG and OG. +In this section, we numerically verify our theoretical results through Example 1. Let A ∈ Rn×n, +b, h ∈ Rn, and X , Y ⊆ Rn, and f : X × Y → R be of the form f (x, y) = +1 +2x⊤Hx − h⊤x − +⟨Ax − b, y⟩ [Ouyang and Xu, 2021]. We consider a convex-concave min-max optimization prob- +lem minx∈X maxy∈Y f (x, y), which is also a two-player zero-sum game G = ([2], (X , Y), ( f, − f )). +Details of the choices of H, A, b, h, X , Y and step size η are deferred to Appendix F. +15 + +The numerical result is shown in Figure 1. We use z to denote (x, y). When players use AOG, +the tangent residual of players’ action profile rtan(zt+ 1 +2 ) decreases at a rate of O( 1 +T), and corrobo- +rates our theoretical results (Theorem 2). Moreover, AOG significantly outperforms OG in terms +of both the last-iterate convergence rate and the individual dynamic regret. +8 +Conclusion and Discussion +In this paper, we propose the first doubly optimal online learning algorithm, the accelerated opti- +mistic gradient (AOG) algorithm, which achieves optimal O( +√ +T) regret bound in the adversarial +setting and optimal O( 1 +T) last-iterate convergence rate in smooth monotone games. Extending our +results in settings where players only receive noisy gradient or even bandit feedback is an interest- +ing and challenging future direction. Finally, We significantly improve the state-of-the-art upper +bound of the individual dynamic regret from O( +√ +T) to O(log T). 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In +Proceedings of the 20th international conference on machine learning (ICML), 2003. +20 + +Contents +1 +Introduction +1 +1.1 +Our Contributions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +2 +Preliminaries +4 +2.1 +Monotone Games and Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.2 +Online Learning and Regret . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +3 +No-Regret Learning Algorithms and Games +6 +3.1 +Accelerated Optimistic Gradient +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +4 +Worst-Case Regret in the Adversarial Environment +9 +5 +Last-Iterate Convergence Rate to a Nash Equilibrium in Monotone Games +9 +5.1 +Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +6 +Dynamic Regret and Second-Order Gradient Variation +14 +7 +Illustrative Experiments +15 +8 +Conclusion and Discussion +16 +A Related Works +21 +B +Missing proofs in Section 3 +22 +C Missing proofs in Section 5 +24 +D Linear Regret of EAG +25 +E +Proof of Theorem 4 +26 +F +Details on Numerical Experiments +27 +G Auxiliary Results +27 +A +Related Works +Last-Iterate Convergence of No-regret learning in Games +There is a vast literature on no-regret +learning in games. For strongly monotone games, linear last-iterate convergence rate is known [Tseng, +1995, Liang and Stokes, 2019, Mokhtari et al., 2020b, Zhou et al., 2020]. Even under bandit feed- +back or noisy gradient feedback, optimal sub-linear last-iterate convergence rate is achieved by +no-regret learning algorithms for strongly monotone games [Lin et al., 2022, Jordan et al., 2022]. + +Obtaining last-iterate convergence rate to Nash equilibria beyond strongly monotone games +received extensive attention recently. Daskalakis and Panageas [2018] proved asymptotic conver- +gence of the optimistic gradient (OG) algorithm in zero-sum games. Asymptotic convergence was +also achieved in variationally stable games [Zhou et al., 2017b,a, Mertikopoulos and Zhou, 2019, +Hsieh et al., 2021] even with noisy feedback [Hsieh et al., 2022]. Finite time O( 1 +√ +T) convergence +was shown for unconstrained cocoercive games [Lin et al., 2020] and unconstrained monotone +games [Golowich et al., 2020a]. For bilinear games over polytopes, Wei et al. [2021] show linear +convergence rate of OG but this rate depends on a problem constant c which can be arbitrarily +large. Recently, Cai et al. [2022b] proved a tight O( 1 +√ +T) last-iterate convergence rate of OG and the +extragradient (EG) algortihm for constrained monotone games, matching the lower bound of p- +SCIL algorithms by Golowich et al. [2020a]. We remark that for general gradient-based algorithms, +the lower bound is Ω( 1 +T) [Ouyang and Xu, 2021, Yoon and Ryu, 2021]. +Regret Minimization in Games +There is a large body of works on minimizing individual re- +gret in games, from early results in two-player zero-sum games [Daskalakis et al., 2011, Kangar- +shahi et al., 2018] to more recent works on general-sum games [Syrgkanis et al., 2015, Chen and +Peng, 2020, Daskalakis et al., 2021, Anagnostides et al., 2022a,b]. Among them, Daskalakis et al. +[2021], Anagnostides et al. [2022a,b] achieves O(log T) regret for general-sum games and Hsieh +et al. [2021] achieves O(1) regret for variationally stable games. Little is known, however, for the +stronger notion of dynamic regret except for O( +√ +T) bound of OG in monotone games [Cai et al., +2022b]. +B +Missing proofs in Section 3 +Proof of Lemma 2: Let us view the update rule of AOG as standard update rule of OG with mod- +ified gradients gt− 1 +2 − +1 +ηt(t+1)(x1 − xt) and gt+ 1 +2 − +1 +ηt(t+1)(x1 − xt). Thus by the standard analysis +of OG (see Rakhlin and Sridharan [2013][Lemma 1]), we have for any t ≥ 1 and any x′ ∈ X , +� +gt+ 1 +2 − +1 +ηt(t + 1)(x1 − xt), xt+ 1 +2 − x′ +� +≤ +1 +2ηt +���xt − x′��2 − +��xt+1 − x′��2� ++ +���gt+ 1 +2 − gt− 1 +2 +��� · +���xt+ 1 +2 − xt+1 +��� +≤ +1 +2ηt +���xt − x′��2 − +��xt+1 − x′��2� ++ ηt +���gt+ 1 +2 − gt− 1 +2 +��� +2 +, +where in the second inequality we use the following inequality: +���xt+ 1 +2 − xt+1 +��� ≤ +����ΠX +� +xt − ηtgt− 1 +2 − +1 +t + 1(x1 − xt) +� +− ΠX +� +xt − ηtgt+ 1 +2 − +1 +t + 1(x1 − xt) +����� +≤ +���gt− 1 +2 − gt+ 1 +2 +��� +(ΠX is non-expansive) +22 + +Therefore, we can bound the single-step regret by +� +gt+ 1 +2 , xt+ 1 +2 − x′� +≤ +1 +2ηt +���xt − x′��2 − +��xt+1 − x′��2� ++ ηt +���gt+ 1 +2 − gt− 1 +2 +��� +2 ++ +� +1 +ηt(t + 1)(x1 − xt), xt+ 1 +2 − x′ +� +≤ +1 +2ηt +���xt − x′��2 − +��xt+1 − x′��2� ++ ηt +���gt+ 1 +2 − gt− 1 +2 +��� +2 ++ +D2 +ηt(t + 1). +(Cauchy-Schwarz/ inequality and X is bounded by D) +Thi completes the proof. +□ +Proof of Theorem 1: Let T1 ≥ 2 be the last time the player uses constant step size η. By line +7 of Algorithm 1, we know the the second-order gradient variation ST1+1 ≤ ST1 + 2G2 is upper +bounded by a constant. By telescoping the inequality from Lemma 2, we know that the player’s +regret up to time T1 is at most +T1 +∑ +t=1 +� +gt+ 1 +2 , xt+ 1 +2 − x′� +≤ ∥x1 − x′∥2 +2η ++ ηST1+1 + G2 + +T1 +∑ +t=1 +D2 +η(t + 1) +≤ O(G2 + log T1). +Now we consider t ≥ T1 + 1 when the player switches to an adaptive step size. Using Lemma 2, +for any T ≥ T1 + 1, we have +T +∑ +t=T1+1 +� +gt+ 1 +2 , xt+ 1 +2 − x′� +≤ +T +∑ +t=T1+1 +1 +ηt +(∥xt − x∗∥2 − ∥xt+1 − x∗∥2) +� +�� +� +I ++ +T +∑ +t=T1+1 +ηt +���gt+ 1 +2 − gt− 1 +2 +��� +2 +� +�� +� +II ++ +T +∑ +t=T1+1 +D2 +ηt(t + 1) +� +�� +� +III +. +Since for any t ≥ 1, ∥gt+ 1 +2 − gt− 1 +2 ∥2 ≤ 2∥gt+ 1 +2 ∥2 + 2∥gt− 1 +2 ∥2 ≤ 4G2. We have St ≤ 4G2t and +ηt = +1 +√1+St ≥ +1 +2G +√ +t for any t ≥ T1 + 1. We now proceed to bound each terms as follows. +I ≤ +D2 +ηT1+1 ++ +T +∑ +t=T1+2 +D2 +� 1 +ηt +− +1 +ηt−1 +� +≤ D2 +ηT +≤ O(D2G +√ +T). +23 + +II = +T +∑ +t=T1+1 +(ηt+1 + ηt − ηt+1) +���gt+ 1 +2 − gt− 1 +2 +��� +2 +≤ +T +∑ +t=T1+1 +� +�∥gt+ 1 +2 − gt− 1 +2 ∥2 +√1 + St+1 ++ 4G2(ηt − ηt+1) +� +� +≤ +T +∑ +t=T1+1 +(√1 + St+1 − √1 + St)(√1 + St+1 + √1 + St) +√1 + St+1 ++ 4G2 +≤ +T +∑ +t=T1+1 +2( +� +1 + St+1 − +� +1 + St) + 4G2 +≤ 2 +� +� +� +�1 + +T +∑ +t=1 +���gt+ 1 +2 − gt− 1 +2 +��� +2 ++ 4G2 = O(G +√ +T + G2). +III ≤ D2 +t +∑ +i=1 +√1 + St +t + 1 +≤ D2 +T +∑ +t=1 +O( G +√ +t) = O(D2G +√ +T). +Combing the above inequalities, we get the regret between T1 and T is at most O(D2G +√ +T + G2). +□ +C +Missing proofs in Section 5 +Proof of Proposition 1: Note that x3/2 = x1 and ηc2 = x1 − ηV(x1) − x2. Thus +∥ηV(x2) + ηc2∥ = ∥ηV(x2) + x1 − ηV(x1) − x2∥ +≤ η∥V(x2) − V(x1)∥ + ∥x1 − x2∥ +≤ (1 + ηL)∥x1 − x2∥ +(V is L-Lipschitz) +≤ 3D +2 . +(ηL ≤ 1 +2) +Using the above inequality, we can bound P2 as follows: +P2 = 3 +� +∥ηV(x2) + ηc2∥2 + ∥ηV(x2) − ηV(x1)∥2� ++ 2⟨ηV(x2) + ηc2, x2 − x1⟩ +≤ 3 +� +∥ηV(x2) + ηc2∥2 + ηL∥x2 − x1∥2� ++ 2∥ηV(x2) + ηc2∥∥x2 − x1∥ +≤ 3 +�9D2 +4 ++ D2 +4 +� ++ 3D2 +(ηL ≤ 1 +2) += 33D2 +4 +≤ 9D2. +24 + +This completes the proof of Proposition 1. +□ +Proof of Lemma 5: Fix any t ≥ 2. Using triangle inequality, we have +���xt+ 1 +2 − xt +��� ≤ +���xt+ 1 +2 − ΠX [xt − ηV(xt)] +��� + ∥ΠX [xt − ηV(xt)] − xt∥. +We can bound the first term as follows: +���xt+ 1 +2 − ΠX [xt − ηV(xt)] +��� = +����ΠX +� +xt − ηV(xt− 1 +2 ) + +1 +t + 1(x1 − xt) +� +− ΠX [xt − ηV(xt)] +���� +≤ +����ηV(xt) − ηV(xt− 1 +2 ) + +1 +t + 1(x1 − xt) +���� +(ΠX is non-expansive) +≤ +���ηV(xt) − ηV(xt− 1 +2 ) +��� + ∥x1 − xt∥ +t + 1 +≤ 14D +t +. +(Lemma 4) +Since ct ∈ NX (xt), we have xt = ΠX [xt + ηct]. Using this fact we can bound the second term: +∥ΠX [xt − ηV(xt)] − xt∥ = ∥ΠX [xt − ηV(xt)] − ΠX [xt + ηct]∥ +≤ ∥ηV(xt) + ηct∥ +(ΠX is non-expansive) +≤ 13D +t +. +(Lemma 4) +Combing the above inequalities, we have ∥xt+ 1 +2 − xt∥ ≤ 27D +t . This completes the proof of Lemma 5. +□ +D +Linear Regret of EAG +In this section, we review the definition of the Extra Anchored Gradient (EAG) algorithm and +show that it is not a no-regret algorithm when implemented it in the online learning setting. The +proof is similar to the linear regret proof of EG Golowich et al. [2020a] and we include it for +completeness. Given a game G with gradient operator V, initial point x1 ∈ X , the Extra Anchored +Gradient algorithm updates as follows: +xt+ 1 +2 = ΠX +� +xt − ηV(xt) + +1 +t + 1(x1 − xt) +� +, +xt+1 = ΠX +� +xt − ηV(xt+ 1 +2 ) + +1 +t + 1(x1 − xt) +� +. +(EAG) +The key difference of EAG compared to AOG is that in one iteration, the update of EAG requires +two gradients V(xt) and V(xt+ 1 +2 ). Since in online learning setting, players only see the gradients +corresponding to the action they play, players must play both xt and xt+ 1 +2 using EAG. Thus to +implement EAG in standard online learning setting, we need two iterations for each iteration +25 + +of EAG. Specifically, each player i plays yi +t for t ≥ 1, while yi +2t−1 = xi +t and yi +2t = xi +t+ 1 +2 . The +corresponding update is for t ≥ 1, +yi +2t = ΠX i +� +yi +2t−1 − ηVi(y2t−1) + +1 +t + 1(yi +1 − yi +2t−1) +� +, +(7) +yi +2t+1 = ΠX i +� +yi +2t−1 − ηVi(y2t) + +1 +t + 1(yi +1 − yi +2t−1) +� +. +(8) +We will show when the other players’ action y−i +t +is adversarial, EAG has linear regret and is not +no-regret. +Proposition 2. There exits a two-player zero-sum 1-smooth game G = ([2], {X1, X2}, ( f, − f )), such that +for an adversarial choice of (y2 +t )t∈[T], the EAG updates (7) and (8) for the first player has Ω(T) regret for +any T ≥ 1. +Proof. We use exactly the same construction as Golowich et al. [2020a][Proposition 10]. We take +X 1 = X 2 = [−1, 1] and f : X → R to be f (y1, y2) = y1 · y2. Player 2 play the following sequence +of actions: +y2 +t = +� +1 +t is odd +0 +t is even +Then for any t ≥ 1, we have +V1(y2t−1) = y2 +2t−1 = 1, +V1(y2t) = y2 +2t = 0. +Suppose y1 +1 = 0. Then we have y1 +2t−1 = 0 and y1 +2t = max{−η, −1} for any t ≥ 1. Thus the +accumulative loss for player 1 until T ≥ 1 round is ∑T +t=1 f (y1 +t , y2 +t ) = 0. However, the accumulative +loss of action y1 = −1 is only ∑T +t=1 f (−1, y2 +t ) ≤ − T +2 . Thus the regret is at least T +2 = Ω(T) +E +Proof of Theorem 4 +Proof of Theorem 4: In the game setting, player i’s second-order gradient variation is Si +T = +∑T +t=2 ∥Vi(xt+ 1 +2 ) − Vi(xt− 1 +2 )∥ +2. Using Lemma 4 and Lemma 5, we have +���Vi(xt+ 1 +2 ) − Vi(xt− 1 +2 ) +��� +2 +≤ +���V(xt+ 1 +2 ) − V(xt− 1 +2 ) +��� +2 +≤ 2L2���xt+ 1 +2 − xt +��� +2 ++ 2 +���V(xt) − V(xt− 1 +2 ) +��� +2 +(L-Lipschitzness of V) +≤ 2L2 · 272D2 +t2 ++ 2 · 132D2 +η2t2 += +(1458L2 + 338 +η2 )D2 +t2 +. +26 + +For a choice of η = +1 +3L, we have +���Vi(xt+ 1 +2 ) − Vi(xt− 1 +2 ) +��� +2 +≤ 4500D2L2 +t2 +and Si +T ≤ 4500πD2L2. +□ +F +Details on Numerical Experiments +We choose +A = 1 +4 +� +����� +−1 +1 +· · · +· · · +−1 +1 +−1 +1 +1 +� +����� +∈ Rn×n, +b = 1 +4 +� +����� +1 +1 +· · · +1 +1 +� +����� +∈ Rn, +h = 1 +4 +� +����� +0 +0 +· · · +0 +1 +� +����� +∈ Rn, +and H = 2A⊤A. As shown in [Ouyang and Xu, 2021], ∥A∥ ≤ 1 +2 and ∥H∥ ≤ 1 +2 which implies f = +1 +2x⊤Hx − h⊤x − ⟨Ax − b, y⟩ is 1-smooth. We choose n = 100, X = Y = [−200, 200]n. We run both +AOG and OG with step size η = 0.3 and initial points x1 = y1 = 1 +n1 for 105 iterations. The code can +be found at https://github.com/weiqiangzheng1999/Doubly-Optimal-No-Regret-Learning. +G +Auxiliary Results +Proposition 3. In the setup of Lemma 3, the following identity holds. +Pt − Pt+1 − t(t + 1) · LHSI (1) − t(t + 1) +4q +· LHSI (2) +− t(t + 1) · LHSI (3) − t(t + 1) +2 +· (LHSI (4) + LHSI (5) + LHSI (6)) += t(t + 1) +2 +���� +xt+ 1 +2 − xt +2 ++ ηV(xt) − ηV(xt+ 1 +2 ) +���� +2 ++ (1 − 4q)t − 4q +4q +(t + 1) +���ηV(xt+ 1 +2 ) − ηV(xt+1) +��� +2 ++ (t + 1) · +� +ηV(xt+ 1 +2 ) − ηV(xt+1), ηV(xt+1) + ηct+1 +� +. +Proof. We use MATLAB to verify the following inequality, which implies the claim by suitable +change of variables. For any vectors a0, a1, a2, a3, a4, b1, b2, b3, b4, u2, u4 ∈ Rn, any real numbers +t ≥ 1 and q > 0, if +a4 = a2 − b3 + +1 +t + 1(a0 − a2) − u4, +27 + +then the following identity holds +t(t + 1) +2 +� +∥a2 + u2∥2 + ∥b2 − b1∥2� ++ t⟨b2 + u2, a2 − a0⟩ +− (t + 1)(t + 2) +2 +� +∥a4 + u4∥2 + ∥b4 − b3∥2� ++ t⟨b4 + u4, a4 − a0⟩ +− t(t + 1)⟨b4 − b2, a4 − a2⟩ − t(t + 1) +4q +� +q∥a4 − a3∥2 − ∥b4 − b3∥2� +− t(t + 1)⟨u4, a4 − a2⟩ − t(t + 1) +2 +⟨u2, a2 − a3⟩ − t(t + 1) +2 +⟨u2, a2 − a4⟩ +− t(t + 1) +2 +� +a2 − b1 + +1 +t + 1(a0 − a2) − a3, a3 − a4 +� +=t(t + 1) +2 +���� +a3 − a4 +2 ++ b1 − b2 +���� +2 ++ t(t + 1) +2 +���� +a3 + a4 +2 +− a2 + b2 + u2 − a0 − a2 +t + 1 +���� +2 ++ (1 − 4q)t − 4q +4q +(t + 1)∥b3 − b4∥2 ++ (t + 1)⟨b3 − b4, b4 + u4⟩. +The MATLAB code for verification of the above identity is available at https://github.com/ +weiqiangzheng1999/Doubly-Optimal-No-Regret-Learning. To see how the above identity im- +plies the claimed identity, we replace a0 with x1; replace ak with xt−1+ k +2 for k ∈ [4]; replace bk with +ηV(xt−1+ k +2 ) for k ∈ [4]; replace u2 with ηct; replace u4 with ηct+1; and note that by the definition +of ct+1, we have +xt+1 = xt − ηV(xt+ 1 +2 ) + +1 +t + 1(x1 − xt) − ηct+1. +This completes the proof. +Proposition 4 (Cai et al. [2022a]). Let {ak ∈ R+}k≥2 be a sequence of real numbers. Let C1 ≥ 0 and +p ∈ (0, 1 +3) be two real numbers. If the following condition holds for every k ≥ 2, +k2 +4 · ak ≤ C1 + +p +1 − p · +k−1 +∑ +t=2 +at, +then for each k ≥ 2 we have +ak ≤ 4 · C1 +1 − 3p · 1 +k2 . +28 + diff --git a/MtFPT4oBgHgl3EQflDVE/content/tmp_files/load_file.txt b/MtFPT4oBgHgl3EQflDVE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bf54a369fb501898d463c84f6417f2ac60df1979 --- /dev/null +++ b/MtFPT4oBgHgl3EQflDVE/content/tmp_files/load_file.txt @@ -0,0 +1,891 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf,len=890 +page_content='Doubly Optimal No-Regret Learning in Monotone Games Yang Cai* Yale University yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='cai@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='edu Weiqiang Zheng Yale University weiqiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='zheng@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='edu January 31, 2023 Abstract We consider online learning in multi-player smooth monotone games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Existing algorithms have limitations such as (1) being only applicable to strongly monotone games;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' (2) lacking the no-regret guarantee;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' (3) having only asymptotic or slow O( 1 √ T ) last-iterate convergence rate to a Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' While the O( 1 √ T ) rate is tight for a large class of algorithms including the well-studied extragradient algorithm and optimistic gradient algorithm, it is not optimal for all gradient-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We propose the accelerated optimistic gradient (AOG) algorithm, the first doubly optimal no- regret learning algorithm for smooth monotone games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Namely, our algorithm achieves both (i) the optimal O( √ T) regret in the adversarial setting under smooth and convex loss functions and (ii) the optimal O( 1 T ) last-iterate convergence rate to a Nash equilibrium in multi-player smooth monotone games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' As a byproduct of the accelerated last-iterate convergence rate, we further show that each player suffers only an O(log T) individual worst-case dynamic regret, providing an exponential improvement over the previous state-of-the-art O( √ T) bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 1 Introduction We consider multi-agent online learning in games [Cesa-Bianchi and Lugosi, 2006].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We focus on a rich family of multi-player games – monotone games that has been the central object of a series of recent studies in online learning and optimization [Hsieh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2019, Golowich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2020a,a, Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2020, Hsieh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2021, Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2022, Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2022b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Monotone games, first introduced by Rosen [1965], encompass many commonly studied games as special cases such as two-player zero-sum games, convex-concave games, λ-cocoercive games [Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2020], zero- sum polymatrix games [Bregman and Fokin, 1987, Daskalakis and Papadimitriou, 2009, Cai and Daskalakis, 2011, Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2016], and zero-sum socially-concave games [Even-Dar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In Supported by a Sloan Foundation Research Fellowship and the NSF Award CCF-1942583 (CAREER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='13120v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='LG] 30 Jan 2023 this paper, we investigate the following fundamental question: How fast can the players’ day-to-day behavior convergeto a Nash equilibrium in monotone games if players act according to a no-regret learning algorithm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' (*) Regret is the central metric used in online learning to measure the performance of a learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In the classical single-agent setting, online learning considers the following repeated interaction between a player and the environment: (i) at day t, the player chooses an action xt ∈ Ω ⊆ Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' (ii) the environment selects a loss function ft(·), and the player receives the loss ft(xt) along with some feedback (such as the loss function ft(·), the gradient ∇ ft(xt), or just the loss ft(xt)) and the process repeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The regret is defined as the difference between the cumu- lative loss of the player ∑T t=1 ft(xt) and the cumulative loss of the best fixed action in hindsight minx∈Ω ∑T t=1 ft(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' A single-agent online learning algorithm is considered no-regret if, even under an adversarially chosen sequence of loss functions, its regret at the end of round T is sub-linear in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Arguably, a most common scenario, where the above online learning model instantiates, is multi-agent online learning in games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Namely, every player makes an online decision on their action and receives a loss that is determined based on their own action, as well as the actions chosen by the other players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' A well-known result states that if all players use no-regret learning algorithms to determine their joint actions, the empirical frequency of their joint actions will con- verge to a coarse correlated equilibrium (CCE) [Cesa-Bianchi and Lugosi, 2006].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' However, this general convergence result has two caveats: (i) the guaranteed convergence is only the empirical frequency of the players’ actions rather than the actual, day-to-day play;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' and (ii) the concept of CCE has limitations and may violate even the most basic rationalizability axioms [Viossat and Zapechelnyuk, 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='1 Motivated by these two weaknesses, a substantial body of works [Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2017a,b, 2018, Daskalakis and Panageas, 2019, Mokhtari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2020a, Hsieh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2019, Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2021, Golowich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2020a,a, Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2020, 2022, Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2022b] aim to identify types of games as well as no-regret learning algorithms such that the convergence can be strengthened in two ways: (a) by achieving convergence to the more robust solution concept of Nash equilib- rium, and (b) by ensuring convergence in the players’ day-to-day behavior rather than just their empirical frequency of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In other words, the goal is to identify games and develop no- regret learning algorithms so that players’ action profile converges to a Nash equilibrium in the last-iterate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Monotone games emerge as the most general class of games where such strengthened conver- gence result is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='2 Unlike in the general convergence to CCE that holds for any no-regret learning algorithms, the last-iterate convergence to Nash equilibria is more subtle and demands a careful design of the learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' For example, as demonstrated by Mertikopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' [2018], the well-known family of no-regret learning algorithms – follow-the-regularized-leader fails to converge even in two-player zero-sum games (a special case of monotone games), as the action profile of the players may cycle in space perpetually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The key to correct such cycling behavior is 1For instance, a CCE may put positive weight only on strictly dominated actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 2For the more general family of variationally stable games, only asymptotic convergence to Nash equilibria is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 2 to introduce optimism in the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Indeed, the optimistic gradient (OG) algorithm by Popov [1980], a optimistic variant of the gradient descent algorithm, has recently been shown to exhibit an O( 1 √ T) last-iterate convergence rate to a Nash equilibrium in monotone games [Golowich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2020a, Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2022b, Golowich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2020b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' As shown by Golowich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' [2020a], this rate is tight for OG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' However, it is not clear if O( 1 √ T) is the optimal rate achievable by a no-regret algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='1 Our Contributions We consider multi-agent online learning in monotone games with gradient feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' More con- cretely, each player i at day t not only observes their loss ℓi(xi t, x−i t ) but also receives the gradient ∇xi tℓi(xi t, x−i t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Main Contribution: We answer question (*) by presenting a new single-agent online learning algorithm – the Accelerated Optimistic Gradient (AOG) that is doubly optimal (The- orem 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' More specifically, Optimal regret: AOG achieves the optimal O( √ T)-regret in the adversarial environ- ment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Optimal last-iterate convergence rate: If all players use AOG to determine their actions in a monotone game, the action profile has the optimal O( 1 T) last-iterate conver- gence rate to a Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Note that O( 1 T) is the fastest rate possible for solving monotone games using any gradient- based methods [Ouyang and Xu, 2021, Yoon and Ryu, 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='3 Since the players only receive gra- dient feedback in our setting, this lower bound also applies to our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Step-size adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We provide an implementation of AOG (Algorithm 1) that can automat- ically adapt to the environment and achieves a best-of-both-world guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' When deploy in an adversarial setting, Algorithm 1 obtains at most O( √ T)-regret;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' when deploy in a monotone game where other players also play according to Algorithm 1, the action profile converges to a Nash equilibrium at a O( 1 T) rate in the last-iterate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Importantly, the adaptation does not require any communication between the players and only uses the the player’s local information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We be- lieve such guarantee is crucial as even in a game setting, other players may not follow the same algorithm and might act arbitrarily, in which case, our algorithm still provides a guarantee on the worst-case regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Dynamic regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' As an interesting byproduct of our last-iterate convergence rate, we further show that each player suffers only an O(log T) individual dynamic regret, when all players play according to Algorithm 1 (Theorem 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The dynamic regret of an algorithm is defined as the difference between the algorithm’s cumulative loss and the cumulative loss of the best action 3These lower bounds apply to general first-order methods that produce their iterates in an arbitrary manner based on past gradient information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 3 every day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The dynamic regret is notoriously difficulty to minimize, and it is well-known that a linear dynamic regret is unavoidable in the adversarial setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In the game setting, results on dynamic regret are also sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' To the best of our knowledge, the only sub-linear dynamic regret bound we are aware of is the O( √ T) dynamic regret of OG for monotone games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Our accelerated algorithm obtains an exponential improvement on the dynamic regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' See Table 1 for comparison with other well-studied learning algorithms in monotone games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Algorithm Adversarial Setting Monotone Games No-Regret?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Convergence Rate∗ Dynamic Regret∗∗ GD \x13 \x17 Ω(T) EG \x17 O( 1 √ T) O( √ T) OG \x13 O( 1 √ T) O( √ T) EAG \x17 O( 1 T) O(log T) This paper \x13 O( 1 T) O(log T) Table 1: Existing results on learning in monotone games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' (*) last-iterate convergence rate with respect to the gap function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' (**) individual worst-case dynamic regret in monotone games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The key of our new algorithm is combining optimism with Halpern iteration [Halpern, 1967], a mechanism used in optimization to design accelerated methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In our setting, Halpern iteration can be viewed as adding a diminishing strongly convex loss to the player’s loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The schedule used to decrease the added loss must be crafted carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' If the added loss di- minishes too slowly, the adversarial regret would be sub-optimal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' if the added loss decreases too quickly, the algorithm may converge at a slower rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The Halpern iteration provides a schedule that strikes the right balance and allows us to obtain the doubly optimal algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 2 Preliminaries Basic Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We consider Euclidean space (Rn, ∥ · ∥) where ∥ · ∥ is ℓ2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We say a set X ⊆ Rn is bounded by D > 0 if ∥x − x′∥ ≤ D for any x, x′ ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Given a closed and convex set X ⊆ Rn, the Euclidean projection operator is ΠX : Rn → Rn such that ΠX [x] = argminx′∈X ∥x − x′∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' For closed and convex set X , Euclidean projection is non-expansive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', ∥ΠX [x] − ΠX [x′]∥ ≤ ∥x − x′∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' For a closed convex set X , the normal cone of x ∈ X is defined as NX (x) := {v : ⟨v, x′ − x⟩ ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We make use of the following properties of the normal cone: (i) for any v ∈ NX (x), x = ΠX [x + v];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' (ii) if x = ΠX [x′], then x′ − x ∈ NX (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='1 Monotone Games and Nash Equilibria A (continuous) multi-player game is denotes as G = ([N], (X i)i∈[N], (ℓi)i∈[N]) where [N] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' , N} denotes the set of players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Each player i chooses action from a compact and convex set X i ∈ Rni and we write X = ∏N i=1 X i ∈ Rn where n = n1 + · · · + nN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We always use x−i to denote the 4 actions of all players except player i and write x = (xi, x−i) = (x1, x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' , xN) as players’ action profile or strategy profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Note that we reserve the bold x to denote the players’ action profile and use the normal x to denote a single player’s action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Each player i wishes to minimize a loss func- tion ℓi(xi, x−i) : X → R which is continuous in x and convex in xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In this paper, we study learning in multi-player games with gradient feedback where after playing action profile x, each player i receives Vi(x) := ∇xiℓi(xi, x−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We define the gradient operator V : X → Rn to be V(·) = (V1(·) · · · , VN(·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The widely used solution concept for a game is Nash equilibrium, an action profile where no player gains from unilateral deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Formally, a Nash equilibrium of a game G is an action profile x⋆ ∈ X such that for each player i, it holds that ℓi(x⋆) ≤ ℓi(xi, x−i ⋆ ) for any xi ∈ X i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In this paper, we study smooth monotone games where the gradient operator V is L-Lipschitz for L > 0: ��V(x) − V(x′) �� ≤ L · ��x − x′��, ∀x, x′ ∈ X , and monotone [Rosen, 1965] : � V(x) − V(x′), x − x′� ≥ 0, ∀x, x′ ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' It is not hard to see that for smooth monotone games, a Nash equilibrium always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' If x⋆ is a Nash equilibrium, then a simple characterization of x⋆ is that, for any x ∈ X , it holds that ⟨V(x⋆), x⋆ − x⟩ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Monotone games include many well-studied games, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', two-player zero-sum games, convex- concave games, λ-cocoercive games [Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2020], strongly monotone games (such as Kelly auctions), zero-sum polymatrix games [Bregman and Fokin, 1987, Daskalakis and Papadimitriou, 2009, Cai and Daskalakis, 2011, Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2016], and zero-sum socially-concave games [Even-Dar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Example 1 (Convex-Concave Min-Max Optimization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Given a function f (x, y) : X × Y → R that is convex in x and concave in y, find a saddle point z = (x, y) such that f (x, y′) ≤ f (x, y) ≤ f (x′, y), ∀x′ ∈ X , y′ ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' It is not hard to see that the set of Nash equilibria of a two-player zero-sum game G = {[2], (X , Y), ( f, − f )} corresponds to the set of saddle points of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Thus convex-concave min-max optimization is a special case of monotone games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' For a monotone game G and an action profile x, two standard measures of proximity to Nash equilibrium are the gap function and the total gap function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Let G = ([N], (X i)i∈[N], (ℓi)i∈[N]) be a monotone game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The gap function for x ∈ X is GAP(x) = max x′∈X � V(x), x − x′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The total gap function for x ∈ X is TGAP(x) = N ∑ i=1 � ℓi(x) − min x′∈X i ℓi(x′, x−i) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Since ℓi is convex in xi for all i ∈ N, we have TGAP(x) ≤ GAP(x) for all x ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 5 A stronger measure of proximity to Nash equilibrium is the tangent residual defined as rtan(x) = minc∈NX (x) ∥V(x) + c∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The tangent residual is an upper bound for both the gap and the total gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Lemma 1 (Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' [2022b]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Let G = ([N], (X i)i∈[N], (ℓi)i∈[N]) be a monotone game where X = ∏i∈[N] X i is bounded by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' For any x ∈ X , we have TGAP(x) ≤ GAP(x) ≤ D · rtan(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='2 Online Learning and Regret A central theme of online learning is to design learning algorithms that minimize the regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' For each time t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' , T, suppose the environment generates convex loss function ft : Ω → R and the algorithm chooses action xt ∈ Ω where Ω ⊆ Rd is a compact convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The external regret is defined as the gap between the algorithm’s realized cumulative loss and the cumulative loss of the best fixed action in hindsight: Reg(T) := T ∑ t=1 ft(xt) − min x∈Ω T ∑ t=1 ft(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' By the convexity of ℓt, we can bound the external regret by Reg(T) ≤ max x∈Ω T ∑ t=1 ⟨∇ ft(xt), xt − x⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We will simply call the external regret as regret and any algorithm achieving sub-linear regret Reg(T) = o(T) as a no-regret algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' A much stronger performance measure of an online algorithm is the (worst-case) dynamic regret [Zinkevich, 2003]: DynamicReg(T) := T ∑ t=1 ft(xt) − T ∑ t=1 min x∈Ω ft(x), where the algorithm is competing with the best action in each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' It is not hard to see that in adversarial setting, DynamicReg(T) must be linear in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 3 No-Regret Learning Algorithms and Games In this section, we first review some background of gradient-based algorithms from both the on- line learning and optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We start with online gradient descent (GD) [Zinkevich, 2003]: the algorithm produces iterates xt ∈ Ω defined by xt+1 = ΠΩ[xt − ηtgt] where we write gt := ∇ ft(xt) as the gradient of the loss function ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Online gradient descent is a no-regret algorithm in the adversarial setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' When employed by all players, however, it diverges in last-iterate even for simple two-player zero-sum games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 6 Optimism in Online Learning A modification of online gradient descent is the Optimistic Gradi- ent (OG) [Popov, 1980, Rakhlin and Sridharan, 2013, Daskalakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2018]: the algorithm chooses action xt+ 1 2 in each round t and updates iterates: xt+ 1 2 = ΠΩ � xt − ηtgt− 1 2 � , xt+1 = ΠΩ � xt − ηtgt+ 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' (OG) Compared to online gradient descent, OG also achieves optimal regret in the single-agent ad- versarial setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Moreover, OG converges in the last-iterate sense as optimism stabilizes the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' When employed by all players in monotone games, their trajectory of play (xt+ 1 2 )t≥1 converges to a Nash equilibrium with an O( 1 √ T) last-iterate convergence rate [Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2022b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Unfortunately, the O( 1 √ T) rate is tight for OG and more generally all p-SCLI algorithms [Golowich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2020a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' New ideas are needed to further sharpen the convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Acceleration in Optimization We are inspired by a technique from optimization for accelerating first-order methods known as the Halpern iteration [Halpern, 1967] or Anchoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The technique is closely related to Nesterov’s accelerated method [Tran-Dinh, 2022] and has received extensive at- tention from the optimization community recently [Diakonikolas, 2020, Yoon and Ryu, 2021, Lee and Kim, 2021, Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2022a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' When the Halpern iteration is applied to the classical extragradi- ent (EG) algorithm [Korpelevich, 1976], which belongs to the p-SCLI family and also has an O( 1 √ T) last-iterate convergence rate [Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2022b], the resulting extra anchored gradient (EAG) algo- rithm achieves an O( 1 T) last-iterate convergence rate [Yoon and Ryu, 2021, Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2022a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Cai and Zheng [2023] obtain a single-call algorithm – Accelerated Reflected Gradient (ARG) that also achieves the same optimal last-iterate convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' However, EAG is not suitable for multi- player games, as it could exhibit linear regret as we demonstrated in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' ARG requires evaluating the gradient at points outside of the feasible domain, thus it is also incompatible with multi-player games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Our analysis is based on a construction from [Golowich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2020a], where they show that EG has linear regret in multi-player games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='1 Accelerated Optimistic Gradient We propose the following algorithm – the accelerated optimistic gradient (AOG) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The central idea is to combine optimism with Halpern iteration: in round t, the algorithm chooses action xt+ 1 2 and updates as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' xt+ 1 2 = ΠΩ � xt − ηtgt− 1 2 + 1 t + 1(x1 − xt) � , xt+1 = ΠΩ � xt − ηtgt+ 1 2 + 1 t + 1(x1 − xt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' (AOG) 7 Double Optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Our main result is that (AOG) is a doubly optimal online algorithm: with ηt = Θ( 1 √ t), (AOG) achieves optimal O( √ T) regret in adversarial setting (Theorem 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' when all players employ (AOG) with constant step size in a monotone game, their trajectory of play enjoys optimal O( 1 T) last-iterate convergence rate (Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Step-Size Adaptation We also present an implementation of (AOG) in Algorithm 1 with a step- size adaptation procedure (Line 7-11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' This procedure uses the player’s own second-order gradient variation St+1 = ∑t s=2 ∥gs+ 1 2 − gs− 1 2 ∥2 as a proxy for the environment and adapts the step-size ac- cordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The high level idea is that if all players use Algorithm 1 in a smooth monotone game, then each player’s second-order gradient variation remains to be bounded by a constant that only depends on L and D (Theorem 4), so the algorithm will keep a constant learning rate and achieve an O( 1 T) last-iterate convergence (Theorem 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' if the player’s second-order gradient variation ex- ceeds a certain constant threshold, then Algorithm 1 decreases the learning rate according to the second-order gradient variation, and by the standard argument of ”regret is bounded by stability”, we can essentially bound the player’s regret by the the second-order gradient variation, which is at most O( √ T) even in the adversarial setting (Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Algorithm 1 AOG with step-size adaptation 1: Input: L, D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 2: Initialize g 1 2 =⃗0, η1 = η = 1 3L, and choose an arbitrary x1 ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 3: for t = 1, 2, · · · do 4: xt+ 1 2 = ΠΩ[xt − ηtgt− 1 2 + 1 t+1(x1 − xt)] 5: Play xt+ 1 2 and receive feedback gt+ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 6: xt+1 = ΠΩ[xt − ηtgt+ 1 2 + 1 t+1(x1 − xt)] 7: if St+1 := ∑t s=2 ∥gs+ 1 2 − gs− 1 2 ∥2 > 4500πD2L2 then 8: ηt+1 = 1 √1+St+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 9: else 10: ηt+1 = ηt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 11: end if 12: end for Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In the adversarial setting, L and D can be any positive real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' If all players use Al- gorithm 1, L should be an upper bound of the Lipschitz constant of the game, and D should be an upper bound of the diameter ∥x − x′∥ ≤ D for x, x′ ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In other words, the players do not need to know exactly the environment that they are interacting with to carefully pick the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' As long as they know an upper bound for the Lipschitz constant and the diameter of all games that they could potentially participate in, Algorithm 1 will successfully choose the appropriate learning rate for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 8 4 Worst-Case Regret in the Adversarial Environment In this section, we view Algorithm 1 as a single-agent online learning algorithm in the adversarial setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' One could also interpret the result in the game setting, where we make no assumption on how the other players choose their actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We show in Theorem 1 that Algorithm 1 achieves min-max optimal O( √ T) regret when the gradient feedback is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' It shows that AOG is an optimal no-regret algorithm in the adversarial setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Theorem 1 (Regret Bound of Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Let G = maxt ∥gt+ 1 2 ∥2 and suppose the action set Ω is bounded by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The regret of Algorithm 1 is bounded by O(D2G √ T + G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We first establish a single-step regret inequality in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Lemma 2 (Single-Step Regret Inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Suppose the action set Ω is bounded by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' For all t ≥ 1 and any x′ ∈ X , the iterates of AOG satisfies � xt+ 1 2 − x′, gt+ 1 2 � ≤ 1 2ηt ���x′ − xt ��2 − ��x′ − xt+1 ��2� + ηt ���gt+ 1 2 − gt− 1 2 ��� 2 + D2 ηt(t + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The main idea behind Lemma 2 is to view the update rule of AOG as a standard update rule of OG with modified gradients gt− 1 2 − 1 ηt(t+1)(x1 − xt) and gt+ 1 2 − 1 ηt(t+1)(x1 − xt), which allows us to apply the classical analysis of OG [Rakhlin and Sridharan, 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Equipped with Lemma 2, we can bound the regret of Algorithm 1 even with adaptive size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We defer the proofs of Lemma 2 and Theorem 1 to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 5 Last-Iterate Convergence Rate to a Nash Equilibrium in Monotone Games In this section, we consider a multi-player learning setting where each player follows AOG with constant step size in smooth monotone games: each player i plays xi t+ 1 2 , receives gradient Vi(xt+ 1 2 ), and updates xi t+ 1 2 = ΠX i � xi t − ηVi(xt− 1 2 ) + 1 t + 1(xi 1 − xi t) � , xi t+1 = ΠX i � xi t − ηVi(xt+ 1 2 ) + 1 t + 1(xi 1 − xi t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We show in Theorem 2 that the trajectory of the action profile (xt+ 1 2 )t∈[T] converges to Nash equilibrium in last-iterate with an O( 1 T) rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Our convergence rate result matches the Ω( 1 T) lower bound by Yoon and Ryu [2021] and thus establishes that AOG is doubly optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 9 Theorem 2 (Optimal Last-Iterate Convergence Rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Let G = {N, (X i)i∈[N], (ℓi)i∈[N]} be a L-smooth monotone game, where the diameter of X = ∏i∈[N] X i is bounded by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' When all players employ AOG with a constant step size η ≤ 1 √ 6L in G, then for any T ≥ 2, we have rtan(xT+ 1 2 ) ≤ 55D ηT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' TGAP(xT+ 1 2 ) ≤ GAP(xT+ 1 2 ) ≤ 55D2 ηT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' A Sketch of the Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' First, recall that the tangent residual provides upper bounds for both the gap function and the total gap function due to Lemma 1, so it suffices to prove a last-iterate convergence rate with respect to the tangent residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' For x ∈ X , its tangent residual is defined as rtan(x) = minc∈NX (x) ∥V(x) + c∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The definition itself contains an optimization problem, thus is not explicit and difficult to directly work with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We relax the tangent residual by choosing an explicit c ∈ NX (x) as follows: for each player i ∈ [N] and iteration t ≥ 2, we define ci t = xi t−1 − ηVi(xt− 1 2 ) + 1 t (xi 1 − xi t−1) − xi t η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' According to the update rule of AOG, ci t ∈ NX i(xi t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Define ct = (c1 t , c2 t , · · · , cN t ) and we have ct ∈ NX (xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Thus rtan(xt) = minc∈NX (xt) ∥V(xt) + c∥ ≤ ∥V(xt) + ct∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Using ∥V(xt) + ct∥ as a proxy of the tangent residual rtan(xt), we construct a potential function of Pt in the order of Θ(t2 · ∥V(xt) + ct∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Although the potential function might increase between consecutive iterates, we manage prove that in Lemma 3 that the increment is sufficiently small: Pt+1 ≤ Pt + O(∥V(xt+1) + ct+1∥2) for any t ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Using the approximate monotonicity of Pt, we derive the following inequality for the sequence (∥V(xt) + ct∥2)t≥2 Θ(t2 · ∥V(xt) + ct∥2) ≤ O(1) + O( t−1 ∑ s=2 ∥V(xs) + cs∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Based on the above inequality, we show in Lemma 4 that ∥V(xt) + ct∥2 = O( 1 t2 ) for any t ≥ 2, which implies O( 1 T) last-iterate convergence rate for xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The final step is to relate the convergence on xt to the convergence of the action profile xt+ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='1 Proof of Theorem 2 Some of the proofs are postponed to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We also defer some auxiliary propositions to Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Potential Function We first formally define our potential function Pt: for t ≥ 2, let Pt be t(t + 1) 2 � ∥ηV(xt) + ηct∥2 + ���ηV(xt) − ηV(xt− 1 2 ) ��� 2� + t⟨ηV(xt) + ηct, xt − x1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 10 We first provide an upper bound on P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In the same setup of Theorem 2, P2 ≤ 9D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Now we present the main technical lemma of this section, where we show the potential func- tion Pt is approximately non-increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In the same setup of Theorem 2, if we choose η = √q L for any q ∈ (0, 1 4), then for all t ≥ 2, Pt+1 ≤ Pt + 3q 2(1 − 4q)∥ηV(xt+1) + ηct+1∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We show Pt − Pt+1 minus a few non-negative terms is at least − 3q 2(1−4q)∥ηV(xt+1) + ηct+1∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Here we present the list of non-negative terms that we use in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Non-Negative Terms Since the game is monotone, we have ⟨ηV(xt+1) − ηV(xt), xt+1 − xt⟩ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' (1) Using the L-Lipschitzness of V and the fact that (ηL)2 ≤ q, we have q ���xt+1 − xt+ 1 2 ��� 2 − ���ηV(xt+1) − ηV(xt+ 1 2 ) ��� 2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' (2) Since ct lies in the normal cone NX (xt) and ct+1 lies in the normal cone NX (xt+1), by the definition of normal cone we have ⟨ηct+1, xt+1 − xt⟩ ≥ 0 (3) � ηct, xt − xt+ 1 2 � ≥ 0 (4) ⟨ηct, xt − xt+1⟩ ≥ 0 (5) As xt − ηV(xt− 1 2 ) + 1 t+1(x1 − xt) − xt+ 1 2 lies in the normal cone NX (xt+ 1 2 ), we also have � xt − ηV(xt− 1 2 ) + x1 − xt t + 1 − xt+ 1 2 , xt+ 1 2 − xt+1 � ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' (6) 11 Descent Identity For convenience, we denote LHSI as “left-hand side of inequality”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We have the following identity by Proposition 3: Pt − Pt+1 − t(t + 1) · LHSI (1) − t(t + 1) 4q LHSI (2) − t(t + 1) · LHSI (3) − t(t + 1) 2 (LHSI (4) + LHSI (5) + LHSI (6)) = t(t + 1) 2 ���� xt+ 1 2 − xt+1 2 + ηV(xt) − ηV(xt+ 1 2 ) ���� 2 + t(t + 1) 2 ���� xt+ 1 2 + xt+1 2 − xt + ηV(xt) + ct − x1 − xt t + 1 ���� 2 + (1 − 4q)t − 4q 4q (t + 1) ���ηV(xt+ 1 2 ) − ηV(xt+1) ��� 2 � �� � I + (t + 1) · � ηV(xt+ 1 2 ) − ηV(xt+1), ηV(xt+1) + ηct+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' � �� � II Further using identity ∥a∥2 + ⟨a, b⟩ = ∥a + b 2∥ 2 − 1 4∥b∥2, we can simplify the last two terms: I + II = ���A(ηV(xt+ 1 2 ) − ηV(xt+1)) + B(ηV(xt+1) + ηct+1) ��� 2 − q(t + 1) (1 − 4q)t − 4q∥ηV(xt+1) + ct+1∥2 ≥ − 3q 2(1 − 4q)∥ηV(xt+1) + ct+1∥2, where A = � (1−4q)t−4q 4q (t + 1) , B = � q (1−4q)t−4q(t + 1), and we use the fact that t+1 t ≤ 3 2 for t ≥ 2 in the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Combining the above two inequalities and the fact that we only add non-positive terms to Pt − Pt+1, we conclude that Pt+1 ≤ Pt + 3q 2(1−4q)∥ηV(xt+1) + ct+1∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Using the fact that the potential function Pt is approximately non-increasing, we are able to use induction to show last-iterate convergence rate of the sequence (xt)t≥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' If X is bounded by D and η ∈ (0, 1 √ 6L), then we have for all T ≥ 2, ∥V(xT) + cT∥ ≤ 13D ηT and ���V(xT) − V(xT− 1 2 ) ��� ≤ 13D ηT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 12 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Let x⋆ be a Nash equilibrium of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' For any t ≥ 2, we have Pt = t(t + 1) 2 � ∥ηV(xt) + ηct∥2 + ���ηV(xt) − ηV(xt− 1 2 ) ��� 2� + t⟨ηV(xt) + ηct, x⋆ − x1⟩ + t⟨ηV(xt) + ηct, xt − x⋆⟩ ≥ t(t + 1) 2 � ∥ηV(xt) + ηct∥2 + ���ηV(xt) − ηV(xt− 1 2 ) ��� 2� + t⟨ηV(xt) + ηct, x⋆ − x1⟩ ≥ t(t + 1) 4 � ∥ηV(xt) + ηct∥2 + 2 ���ηV(xt) − ηV(xt− 1 2 ) ��� 2� − t t + 1∥x⋆ − x1∥2 ≥ t(t + 1) 4 � ∥ηV(xt) + ηct∥2 + 2 ���ηV(xt) − ηV(xt− 1 2 ) ��� 2� − ∥x⋆ − x1∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In the first inequality, we drop a positive term where ⟨V(xt), xt − x⋆⟩ ≥ ⟨V(x⋆), xt − x⋆⟩ ≥ 0 since x⋆ is Nash equilibrium, and ⟨ct, xt − x⋆⟩ ≥ 0 as ct ∈ NX (xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In the second inequality, we apply inequality ⟨a, b⟩ ≥ − α 4∥a∥2 − 1 α∥b∥2 with a = √ tη(V(xt) + ct), b = x⋆ − x1, and α = √ t + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' we use t t+1 ≤ 1 in the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Combing the above inequality with Lemma 3 and Proposition 1, we get for any t ≥ 2, t(t + 1) 4 � ∥ηV(xt) + ηct∥2 + 2 ���ηV(xt) − ηV(xt− 1 2 ) ��� 2� ≤ ∥x⋆ − x1∥2 + Pt ≤ ∥x⋆ − x1∥2 + P2 + 1 3 t−1 ∑ s=2 ∥ηV(xs) + ηcs∥2 ≤ 10D2 + 1 3 t−1 ∑ s=2 ∥ηV(xs) + ηcs∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' By Proposition 4, we can conclude that for any t ≥ 2, ∥ηV(xt) + ηct∥2 + 2 ���ηV(xt) − ηV(xt− 1 2 ) ��� 2 ≤ 160D2 t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' This completes the proof as 132 = 169 ≥ 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Using the last-iterate convergence rate on (xt)t≥2, we only need to bound the distance between xt and xt+ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In the same setup of Theorem 2, we have for any t ≥ 2, ���xt+ 1 2 − xt ��� ≤ 27D t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 13 Proof of Theorem 2 Given Lemma 4 that proves the last-iterate convergence rate on the sequence (xt)t≥2, and Lemma 5 that upper bounds the distance between xt and xt+ 1 2 , we are now ready to prove the last-iterate convergence rate for (xt+ 1 2 )t≥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Note that xt − ηV(xt− 1 2 ) + x1−xt t+1 − xt+ 1 2 ∈ NX (xt+ 1 2 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' thus we can upper bound the tangent residual at xt+ 1 2 by rtan(xt+ 1 2 ) = 1 η min c∈NX (xt+ 1 2 ) ���ηV(xt+ 1 2 ) + c ��� ≤ 1 η ����ηV(xt+ 1 2 ) + xt − ηV(xt− 1 2 ) + x1 − xt t + 1 − xt+ 1 2 ���� ≤ ���V(xt) − V(xt− 1 2 ) ��� + 1 + ηL η ���xt+ 1 2 − xt ��� + D η(t + 1) ≤ 13D ηt + 3 2 · 27D ηt + D η(t + 1) (Lemma 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 5 and ηL ≤ 1 2) ≤ 55D ηt ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' where we use the triangle inequality and the L-Lipschitzness of V in the second inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' This completes the first part of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The second part of Theorem 2 follows directly from the first part of Theorem 2 and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 6 Dynamic Regret and Second-Order Gradient Variation Recent works on no-regret learning in games have provided near-optimal bounds for players’ in- dividual external or swap regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In particular, Daskalakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' [2021], Anagnostides et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' [2022a,b] achieve logarithmic regret bounds for general-sum games, and the bound can be sharpen to O(1) if the games are monotone [Hsieh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' However, dynamic regret is a much stronger concept, which is impossible to achieve in the single-agent adversarial setting and tightly relates to the con- cept of last-iterate convergence in game settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' For example, the O( 1 √ T) last-iterate convergence rate of OG implies a O( √ T) individual dynamic regret bound in monotone games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' To the best of our knowledge, O( √ T) is the best bound for dynamic regret even in two-player zero-sum games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We significantly improve the bound and show that the individual dynamic regret is at most O(log T) if each player employs AOG in monotone games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' This is made possible by the fast O( 1 T) last-iterate convergence rate of AOG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Theorem 3 (Individual Dynamic Regret Bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In the same setup of Theorem 2, for any i ∈ [N] and T ≥ 2, DynamicRegi(T) ≤ O(log T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' By the definition of dynamic regret and total gap function, for any T ≥ 2, we have DynamicRegi(T) = T ∑ t=1 � ℓi(xt+ 1 2 ) − min x′∈X i ℓi(x′, x−i t+ 1 2 ) � ≤ O(1) + T ∑ t=2 TGAP(xt+ 1 2 ) ≤ T ∑ t=2 O(1 t ) = O(log T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Last-iterate convergence rate of AOG also implies each player’s bounded second-order gradi- ent variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We defer the proof of Theorem 4 to Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Theorem 4 (Bounded Second-Order Gradient Variation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In the same setup of Theorem 2 but with η = 1 3L, for any player i and time t ≥ 2, we have Si T ≤ 4500πD2L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Bounded second-order gradient variation guarantees when each player employs Algorithm 1 with the step-size adaptation procedure, they will always use constant step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Combining The- orem 1, Theorem 2, and Theorem 4, we conclude that Algorithm 1 is doubly optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Algorithm 1 automatically adapts to the environment and achieves O( √ T) regret in the adversarial setting and O( 1 T) last-iterate convergence rate in smooth monotone games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 7 Illustrative Experiments 100 101 102 103 104 105 Iteration 10 2 10 1 100 101 102 Tangent residual AOG OG 5000/(t+1) 0 20000 40000 60000 80000 100000 Iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='00 Dynamic regret 1e8 AOG OG Figure 1: Numerical Results of AOG and OG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In this section, we numerically verify our theoretical results through Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Let A ∈ Rn×n, b, h ∈ Rn, and X , Y ⊆ Rn, and f : X × Y → R be of the form f (x, y) = 1 2x⊤Hx − h⊤x − ⟨Ax − b, y⟩ [Ouyang and Xu, 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We consider a convex-concave min-max optimization prob- lem minx∈X maxy∈Y f (x, y), which is also a two-player zero-sum game G = ([2], (X , Y), ( f, − f )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Details of the choices of H, A, b, h, X , Y and step size η are deferred to Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 15 The numerical result is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We use z to denote (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' When players use AOG, the tangent residual of players’ action profile rtan(zt+ 1 2 ) decreases at a rate of O( 1 T), and corrobo- rates our theoretical results (Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Moreover, AOG significantly outperforms OG in terms of both the last-iterate convergence rate and the individual dynamic regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 8 Conclusion and Discussion In this paper, we propose the first doubly optimal online learning algorithm, the accelerated opti- mistic gradient (AOG) algorithm, which achieves optimal O( √ T) regret bound in the adversarial setting and optimal O( 1 T) last-iterate convergence rate in smooth monotone games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Extending our results in settings where players only receive noisy gradient or even bandit feedback is an interest- ing and challenging future direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Finally, We significantly improve the state-of-the-art upper bound of the individual dynamic regret from O( √ T) to O(log T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We believe that understanding the optimal individual dynamic regret is an interesting open question for learning in monotone games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Open Question: What is the optimal individual dynamic regret achievable in smooth monotone games using no-regret learning algorithms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' References Ioannis Anagnostides, Constantinos Daskalakis, Gabriele Farina, Maxwell Fishelson, Noah Golowich, and Tuomas Sandholm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Near-optimal no-regret learning for correlated equilibria in multi-player general-sum games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC), 2022a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Ioannis Anagnostides, Gabriele Farina, Christian Kroer, Chung-Wei Lee, Haipeng Luo, and Tuo- mas Sandholm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Uncoupled learning dynamics with o(log t) swap regret in multiplayer games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems (NeurIPS), 2022b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' LM Bregman and IN Fokin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Methods of Determining Equilibrium Situations in Zero-Sum Poly- matrix Games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Optimizatsia, 40(57):70–82, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Yang Cai and Constantinos Daskalakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' On minmax theorems for multiplayer games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In Proceed- ings of the twenty-second annual ACM-SIAM symposium on Discrete algorithms (SODA), 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Yang Cai and Weiqiang Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Accelerated single-call methods for constrained min-max opti- mization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' International Conference on Learning Representations (ICLR), 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' To appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Yang Cai, Ozan Candogan, Constantinos Daskalakis, and Christos Papadimitriou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Zero-Sum Poly- matrix Games: A Generalization of Minmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Mathematics of Operations Research, 41(2):648–655, May 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' ISSN 0364-765X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='1287/moor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='0745.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' URL https://pubsonline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='1287/moor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='0745.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Publisher: INFORMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 16 Yang Cai, Argyris Oikonomou, and Weiqiang Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Accelerated algorithms for monotone inclu- sion and constrained nonconvex-nonconcave min-max optimization.' 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30(1):687–716, January 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' ISSN 1052-6234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='1137/17M1134925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' URL https://epubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='siam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='org/doi/abs/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='1137/17M1134925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Publisher: Society for Indus- trial and Applied Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Martin Zinkevich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Online convex programming and generalized infinitesimal gradient ascent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In Proceedings of the 20th international conference on machine learning (ICML), 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 20 Contents 1 Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='1 Our Contributions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 6 3 No-Regret Learning Algorithms and Games 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='1 Accelerated Optimistic Gradient .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 7 4 Worst-Case Regret in the Adversarial Environment 9 5 Last-Iterate Convergence Rate to a Nash Equilibrium in Monotone Games 9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='1 Proof of Theorem 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 10 6 Dynamic Regret and Second-Order Gradient Variation 14 7 Illustrative Experiments 15 8 Conclusion and Discussion 16 A Related Works 21 B Missing proofs in Section 3 22 C Missing proofs in Section 5 24 D Linear Regret of EAG 25 E Proof of Theorem 4 26 F Details on Numerical Experiments 27 G Auxiliary Results 27 A Related Works Last-Iterate Convergence of No-regret learning in Games There is a vast literature on no-regret learning in games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' For strongly monotone games, linear last-iterate convergence rate is known [Tseng, 1995, Liang and Stokes, 2019, Mokhtari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2020b, Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Even under bandit feed- back or noisy gradient feedback, optimal sub-linear last-iterate convergence rate is achieved by no-regret learning algorithms for strongly monotone games [Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2022, Jordan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Obtaining last-iterate convergence rate to Nash equilibria beyond strongly monotone games received extensive attention recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Daskalakis and Panageas [2018] proved asymptotic conver- gence of the optimistic gradient (OG) algorithm in zero-sum games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Asymptotic convergence was also achieved in variationally stable games [Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2017b,a, Mertikopoulos and Zhou, 2019, Hsieh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2021] even with noisy feedback [Hsieh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Finite time O( 1 √ T) convergence was shown for unconstrained cocoercive games [Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2020] and unconstrained monotone games [Golowich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2020a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' For bilinear games over polytopes, Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' [2021] show linear convergence rate of OG but this rate depends on a problem constant c which can be arbitrarily large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Recently, Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' [2022b] proved a tight O( 1 √ T) last-iterate convergence rate of OG and the extragradient (EG) algortihm for constrained monotone games, matching the lower bound of p- SCIL algorithms by Golowich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' [2020a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We remark that for general gradient-based algorithms, the lower bound is Ω( 1 T) [Ouyang and Xu, 2021, Yoon and Ryu, 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Regret Minimization in Games There is a large body of works on minimizing individual re- gret in games, from early results in two-player zero-sum games [Daskalakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2011, Kangar- shahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2018] to more recent works on general-sum games [Syrgkanis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2015, Chen and Peng, 2020, Daskalakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2021, Anagnostides et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2022a,b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Among them, Daskalakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' [2021], Anagnostides et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' [2022a,b] achieves O(log T) regret for general-sum games and Hsieh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' [2021] achieves O(1) regret for variationally stable games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Little is known, however, for the stronger notion of dynamic regret except for O( √ T) bound of OG in monotone games [Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=', 2022b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' B Missing proofs in Section 3 Proof of Lemma 2: Let us view the update rule of AOG as standard update rule of OG with mod- ified gradients gt− 1 2 − 1 ηt(t+1)(x1 − xt) and gt+ 1 2 − 1 ηt(t+1)(x1 − xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Thus by the standard analysis of OG (see Rakhlin and Sridharan [2013][Lemma 1]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' we have for any t ≥ 1 and any x′ ∈ X ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' � gt+ 1 2 − 1 ηt(t + 1)(x1 − xt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' xt+ 1 2 − x′ � ≤ 1 2ηt ���xt − x′��2 − ��xt+1 − x′��2� + ���gt+ 1 2 − gt− 1 2 ��� · ���xt+ 1 2 − xt+1 ��� ≤ 1 2ηt ���xt − x′��2 − ��xt+1 − x′��2� + ηt ���gt+ 1 2 − gt− 1 2 ��� 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' where in the second inequality we use the following inequality: ���xt+ 1 2 − xt+1 ��� ≤ ����ΠX � xt − ηtgt− 1 2 − 1 t + 1(x1 − xt) � − ΠX � xt − ηtgt+ 1 2 − 1 t + 1(x1 − xt) ����� ≤ ���gt− 1 2 − gt+ 1 2 ��� (ΠX is non-expansive) 22 Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' we can bound the single-step regret by � gt+ 1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' xt+ 1 2 − x′� ≤ 1 2ηt ���xt − x′��2 − ��xt+1 − x′��2� + ηt ���gt+ 1 2 − gt− 1 2 ��� 2 + � 1 ηt(t + 1)(x1 − xt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' xt+ 1 2 − x′ � ≤ 1 2ηt ���xt − x′��2 − ��xt+1 − x′��2� + ηt ���gt+ 1 2 − gt− 1 2 ��� 2 + D2 ηt(t + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' (Cauchy-Schwarz/ inequality and X is bounded by D) Thi completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' □ Proof of Theorem 1: Let T1 ≥ 2 be the last time the player uses constant step size η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' By line 7 of Algorithm 1, we know the the second-order gradient variation ST1+1 ≤ ST1 + 2G2 is upper bounded by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' By telescoping the inequality from Lemma 2, we know that the player’s regret up to time T1 is at most T1 ∑ t=1 � gt+ 1 2 , xt+ 1 2 − x′� ≤ ∥x1 − x′∥2 2η + ηST1+1 + G2 + T1 ∑ t=1 D2 η(t + 1) ≤ O(G2 + log T1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Now we consider t ≥ T1 + 1 when the player switches to an adaptive step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Using Lemma 2, for any T ≥ T1 + 1, we have T ∑ t=T1+1 � gt+ 1 2 , xt+ 1 2 − x′� ≤ T ∑ t=T1+1 1 ηt (∥xt − x∗∥2 − ∥xt+1 − x∗∥2) � �� � I + T ∑ t=T1+1 ηt ���gt+ 1 2 − gt− 1 2 ��� 2 � �� � II + T ∑ t=T1+1 D2 ηt(t + 1) � �� � III .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Since for any t ≥ 1, ∥gt+ 1 2 − gt− 1 2 ∥2 ≤ 2∥gt+ 1 2 ∥2 + 2∥gt− 1 2 ∥2 ≤ 4G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We have St ≤ 4G2t and ηt = 1 √1+St ≥ 1 2G √ t for any t ≥ T1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We now proceed to bound each terms as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' I ≤ D2 ηT1+1 + T ∑ t=T1+2 D2 � 1 ηt − 1 ηt−1 � ≤ D2 ηT ≤ O(D2G √ T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 23 II = T ∑ t=T1+1 (ηt+1 + ηt − ηt+1) ���gt+ 1 2 − gt− 1 2 ��� 2 ≤ T ∑ t=T1+1 � �∥gt+ 1 2 − gt− 1 2 ∥2 √1 + St+1 + 4G2(ηt − ηt+1) � � ≤ T ∑ t=T1+1 (√1 + St+1 − √1 + St)(√1 + St+1 + √1 + St) √1 + St+1 + 4G2 ≤ T ∑ t=T1+1 2( � 1 + St+1 − � 1 + St) + 4G2 ≤ 2 � � � �1 + T ∑ t=1 ���gt+ 1 2 − gt− 1 2 ��� 2 + 4G2 = O(G √ T + G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' III ≤ D2 t ∑ i=1 √1 + St t + 1 ≤ D2 T ∑ t=1 O( G √ t) = O(D2G √ T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Combing the above inequalities, we get the regret between T1 and T is at most O(D2G √ T + G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' □ C Missing proofs in Section 5 Proof of Proposition 1: Note that x3/2 = x1 and ηc2 = x1 − ηV(x1) − x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Thus ∥ηV(x2) + ηc2∥ = ∥ηV(x2) + x1 − ηV(x1) − x2∥ ≤ η∥V(x2) − V(x1)∥ + ∥x1 − x2∥ ≤ (1 + ηL)∥x1 − x2∥ (V is L-Lipschitz) ≤ 3D 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' (ηL ≤ 1 2) Using the above inequality, we can bound P2 as follows: P2 = 3 � ∥ηV(x2) + ηc2∥2 + ∥ηV(x2) − ηV(x1)∥2� + 2⟨ηV(x2) + ηc2, x2 − x1⟩ ≤ 3 � ∥ηV(x2) + ηc2∥2 + ηL∥x2 − x1∥2� + 2∥ηV(x2) + ηc2∥∥x2 − x1∥ ≤ 3 �9D2 4 + D2 4 � + 3D2 (ηL ≤ 1 2) = 33D2 4 ≤ 9D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 24 This completes the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' □ Proof of Lemma 5: Fix any t ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Using triangle inequality, we have ���xt+ 1 2 − xt ��� ≤ ���xt+ 1 2 − ΠX [xt − ηV(xt)] ��� + ∥ΠX [xt − ηV(xt)] − xt∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We can bound the first term as follows: ���xt+ 1 2 − ΠX [xt − ηV(xt)] ��� = ����ΠX � xt − ηV(xt− 1 2 ) + 1 t + 1(x1 − xt) � − ΠX [xt − ηV(xt)] ���� ≤ ����ηV(xt) − ηV(xt− 1 2 ) + 1 t + 1(x1 − xt) ���� (ΠX is non-expansive) ≤ ���ηV(xt) − ηV(xt− 1 2 ) ��� + ∥x1 − xt∥ t + 1 ≤ 14D t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' (Lemma 4) Since ct ∈ NX (xt), we have xt = ΠX [xt + ηct].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Using this fact we can bound the second term: ∥ΠX [xt − ηV(xt)] − xt∥ = ∥ΠX [xt − ηV(xt)] − ΠX [xt + ηct]∥ ≤ ∥ηV(xt) + ηct∥ (ΠX is non-expansive) ≤ 13D t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' (Lemma 4) Combing the above inequalities, we have ∥xt+ 1 2 − xt∥ ≤ 27D t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' This completes the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' □ D Linear Regret of EAG In this section, we review the definition of the Extra Anchored Gradient (EAG) algorithm and show that it is not a no-regret algorithm when implemented it in the online learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The proof is similar to the linear regret proof of EG Golowich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' [2020a] and we include it for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Given a game G with gradient operator V, initial point x1 ∈ X , the Extra Anchored Gradient algorithm updates as follows: xt+ 1 2 = ΠX � xt − ηV(xt) + 1 t + 1(x1 − xt) � , xt+1 = ΠX � xt − ηV(xt+ 1 2 ) + 1 t + 1(x1 − xt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' (EAG) The key difference of EAG compared to AOG is that in one iteration, the update of EAG requires two gradients V(xt) and V(xt+ 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Since in online learning setting, players only see the gradients corresponding to the action they play, players must play both xt and xt+ 1 2 using EAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Thus to implement EAG in standard online learning setting, we need two iterations for each iteration 25 of EAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Specifically, each player i plays yi t for t ≥ 1, while yi 2t−1 = xi t and yi 2t = xi t+ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The corresponding update is for t ≥ 1, yi 2t = ΠX i � yi 2t−1 − ηVi(y2t−1) + 1 t + 1(yi 1 − yi 2t−1) � , (7) yi 2t+1 = ΠX i � yi 2t−1 − ηVi(y2t) + 1 t + 1(yi 1 − yi 2t−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' (8) We will show when the other players’ action y−i t is adversarial, EAG has linear regret and is not no-regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' There exits a two-player zero-sum 1-smooth game G = ([2], {X1, X2}, ( f, − f )), such that for an adversarial choice of (y2 t )t∈[T], the EAG updates (7) and (8) for the first player has Ω(T) regret for any T ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We use exactly the same construction as Golowich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' [2020a][Proposition 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We take X 1 = X 2 = [−1, 1] and f : X → R to be f (y1, y2) = y1 · y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Player 2 play the following sequence of actions: y2 t = � 1 t is odd 0 t is even Then for any t ≥ 1, we have V1(y2t−1) = y2 2t−1 = 1, V1(y2t) = y2 2t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Suppose y1 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Then we have y1 2t−1 = 0 and y1 2t = max{−η, −1} for any t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Thus the accumulative loss for player 1 until T ≥ 1 round is ∑T t=1 f (y1 t , y2 t ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' However, the accumulative loss of action y1 = −1 is only ∑T t=1 f (−1, y2 t ) ≤ − T 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Thus the regret is at least T 2 = Ω(T) E Proof of Theorem 4 Proof of Theorem 4: In the game setting, player i’s second-order gradient variation is Si T = ∑T t=2 ∥Vi(xt+ 1 2 ) − Vi(xt− 1 2 )∥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Using Lemma 4 and Lemma 5, we have ���Vi(xt+ 1 2 ) − Vi(xt− 1 2 ) ��� 2 ≤ ���V(xt+ 1 2 ) − V(xt− 1 2 ) ��� 2 ≤ 2L2���xt+ 1 2 − xt ��� 2 + 2 ���V(xt) − V(xt− 1 2 ) ��� 2 (L-Lipschitzness of V) ≤ 2L2 · 272D2 t2 + 2 · 132D2 η2t2 = (1458L2 + 338 η2 )D2 t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 26 For a choice of η = 1 3L, we have ���Vi(xt+ 1 2 ) − Vi(xt− 1 2 ) ��� 2 ≤ 4500D2L2 t2 and Si T ≤ 4500πD2L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' □ F Details on Numerical Experiments We choose A = 1 4 � ����� −1 1 · · · · −1 1 −1 1 1 � ����� ∈ Rn×n, b = 1 4 � ����� 1 1 · · 1 1 � ����� ∈ Rn, h = 1 4 � ����� 0 0 · · 0 1 � ����� ∈ Rn, and H = 2A⊤A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' As shown in [Ouyang and Xu, 2021], ∥A∥ ≤ 1 2 and ∥H∥ ≤ 1 2 which implies f = 1 2x⊤Hx − h⊤x − ⟨Ax − b, y⟩ is 1-smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We choose n = 100, X = Y = [−200, 200]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We run both AOG and OG with step size η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='3 and initial points x1 = y1 = 1 n1 for 105 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The code can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='com/weiqiangzheng1999/Doubly-Optimal-No-Regret-Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' G Auxiliary Results Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' In the setup of Lemma 3, the following identity holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Pt − Pt+1 − t(t + 1) · LHSI (1) − t(t + 1) 4q LHSI (2) − t(t + 1) · LHSI (3) − t(t + 1) 2 (LHSI (4) + LHSI (5) + LHSI (6)) = t(t + 1) 2 ���� xt+ 1 2 − xt 2 + ηV(xt) − ηV(xt+ 1 2 ) ���� 2 + (1 − 4q)t − 4q 4q (t + 1) ���ηV(xt+ 1 2 ) − ηV(xt+1) ��� 2 + (t + 1) · � ηV(xt+ 1 2 ) − ηV(xt+1), ηV(xt+1) + ηct+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' We use MATLAB to verify the following inequality, which implies the claim by suitable change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' For any vectors a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' a3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' a4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' b1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' b2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' b3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' b4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' u4 ∈ Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' any real numbers t ≥ 1 and q > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' if a4 = a2 − b3 + 1 t + 1(a0 − a2) − u4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 27 then the following identity holds t(t + 1) 2 � ∥a2 + u2∥2 + ∥b2 − b1∥2� + t⟨b2 + u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' a2 − a0⟩ − (t + 1)(t + 2) 2 � ∥a4 + u4∥2 + ∥b4 − b3∥2� + t⟨b4 + u4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' a4 − a0⟩ − t(t + 1)⟨b4 − b2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' a4 − a2⟩ − t(t + 1) 4q � q∥a4 − a3∥2 − ∥b4 − b3∥2� − t(t + 1)⟨u4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' a4 − a2⟩ − t(t + 1) 2 ⟨u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' a2 − a3⟩ − t(t + 1) 2 ⟨u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' a2 − a4⟩ − t(t + 1) 2 � a2 − b1 + 1 t + 1(a0 − a2) − a3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' a3 − a4 � =t(t + 1) 2 ���� a3 − a4 2 + b1 − b2 ���� 2 + t(t + 1) 2 ���� a3 + a4 2 − a2 + b2 + u2 − a0 − a2 t + 1 ���� 2 + (1 − 4q)t − 4q 4q (t + 1)∥b3 − b4∥2 + (t + 1)⟨b3 − b4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' b4 + u4⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' The MATLAB code for verification of the above identity is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content='com/ weiqiangzheng1999/Doubly-Optimal-No-Regret-Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' To see how the above identity im- plies the claimed identity, we replace a0 with x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' replace ak with xt−1+ k 2 for k ∈ [4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' replace bk with ηV(xt−1+ k 2 ) for k ∈ [4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' replace u2 with ηct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' replace u4 with ηct+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' and note that by the definition of ct+1, we have xt+1 = xt − ηV(xt+ 1 2 ) + 1 t + 1(x1 − xt) − ηct+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Proposition 4 (Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' [2022a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Let {ak ∈ R+}k≥2 be a sequence of real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' Let C1 ≥ 0 and p ∈ (0, 1 3) be two real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' If the following condition holds for every k ≥ 2, k2 4 · ak ≤ C1 + p 1 − p · k−1 ∑ t=2 at, then for each k ≥ 2 we have ak ≤ 4 · C1 1 − 3p · 1 k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} +page_content=' 28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFPT4oBgHgl3EQflDVE/content/2301.13120v1.pdf'} diff --git a/NdE0T4oBgHgl3EQfjQHt/content/tmp_files/2301.02458v1.pdf.txt b/NdE0T4oBgHgl3EQfjQHt/content/tmp_files/2301.02458v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b8ec0f2ba593cde1a8353a1a8d50b59bec628e1e --- /dev/null +++ b/NdE0T4oBgHgl3EQfjQHt/content/tmp_files/2301.02458v1.pdf.txt @@ -0,0 +1,1491 @@ +Topics as Entity Clusters: Entity-based Topics from Language Models and +Graph Neural Networks +Manuel V. Loureiro and Steven Derby and Tri Kurniawan Wijaya +Huawei Ireland Research Centre +Georges Court, Townsend St, +Dublin 2, D02 R156, Ireland +{manuel.loureiro, tri.kurniawan.wijaya}@huawei.com +steven.derby@huawei-partners.com +Abstract +Topic models aim to reveal the latent structure +behind a corpus, typically conducted over a +bag-of-words representation of documents. In +the context of topic modeling, most vocabu- +lary is either irrelevant for uncovering underly- +ing topics or contains strong relationships with +relevant concepts, impacting the interpretabil- +ity of these topics. Furthermore, their limited +expressiveness and dependency on language +demand considerable computation resources. +Hence, we propose a novel approach for +cluster-based topic modeling that employs con- +ceptual entities. Entities are language-agnostic +representations of real-world concepts rich in +relational information. +To this end, we ex- +tract vector representations of entities from +(i) an encyclopedic corpus using a language +model; and (ii) a knowledge base using a graph +neural network. We demonstrate that our ap- +proach consistently outperforms other state-of- +the-art topic models across coherency metrics +and find that the explicit knowledge encoded +in the graph-based embeddings provides more +coherent topics than the implicit knowledge +encoded with the contextualized embeddings +of language models. +1 +Introduction +Following the seminal work of Blei et al. (2003), +topic models have since become the de facto +method for extracting and elucidating prominent +themes from corpora. Traditionally, the semantic +content of a document is composed of document- +term frequencies or latently through a mixture of +distributions of topics, common with probabilistic +generative models such as Latent Dirichlet Alloca- +tion (LDA). Here, individual topics are represented +by salient lexical constituents such as words that +depict some subjects of the corpora (Blei et al., +2003; Blei and Lafferty, 2006; Li and McCallum, +2006; Teh et al., 2006; Crain et al., 2012). In recent +years, the field of Natural Language Processing +(NLP) has seen a trend toward continuous vector +representations of words, which look to capture +the paradigmatic relationship between concepts by +learning distributional co-occurrence patterns in +text. For example, large-scale language models +such as BERT (Devlin et al., 2018) have explored +robust contextualized representations that can ex- +plain an array of linguistic phenomena and implicit +real-world knowledge (Peters et al., 2018; Tenney +et al., 2019a,b; Petroni et al., 2019; Rogers et al., +2020), making them highly advantageous for topic +modeling (Sia et al., 2020; Bianchi et al., 2021). +Despite their successes, it becomes evident that +certain limitations emerge from conventional topic +modeling due to the superfluous nature and lim- +ited expressiveness of word-level tokens. These +methods rely on data-driven techniques — while +ignoring real-world knowledge — to uncover statis- +tical patterns and infer relevant lexical items, which +results in topics with limited guarantees of inter- +pretability. Furthermore, in a multilingual setting, +these models require expansive, resource-intensive +lexicons that may not produce a desirable set of +shared, language-free universal topics (Ni et al., +2009; Boyd-Graber and Blei, 2009). +To overcome these challenges, in this paper +we focus on entities; they are distinct, free +form human-derived concepts that are represented +through encyclopedic-based definitions and a num- +ber of key relational attributes, which offer a bet- +ter alternative for topic modeling (Chemudugunta +et al., 2008; Andrzejewski et al., 2009, 2011; Al- +lahyari and Kochut, 2016). We supersede word- +level topic modeling with real-world entities, as +these are both rich in conceptual information and +language-agnostic. We demonstrate that by consid- +ering purely entity-level units in the text, it is possi- +ble to construct topics that are both interpretable to +humans and founded on a rich set of prior knowl- +edge. We pursue this approach using two sources to +represent entities: (1) contextualized text represen- +arXiv:2301.02458v1 [cs.CL] 6 Jan 2023 + +tations constructed from entity definitions and (2) +structured graph data extracted from a knowledge +base that we use to train a graph neural network +to learn node embeddings. Furthermore, we pro- +pose Topics as Entity Clusters (TEC), a novel topic +modeling algorithm that can discover meaningful +and highly informative topics by clustering either +type of entity vectors or a combination of both. We +successfully verify that, through the experimental +procedure, our approach outperforms a number of +state-of-the-art topic models on a range of metrics +across numerous datasets. +2 +Literature Review +Previous research has attempted to represent topic +models using entities. +For instance, Newman +et al. (2006) proposed representing documents +with salient entities obtained using Named Entity +Recognition (NER) instead of using the words di- +rectly. Others have attempted to capture the pat- +terns among words, entities, and topics, either by +expanding LDA (Blei et al., 2003) or more complex +Bayesian topic models — see Alghamdi and Al- +falqi (2015), Chauhan and Shah (2021) and Vayan- +sky and Kumar (2020) for a general overview — +by describing entities using words (Newman et al., +2006; Kim et al., 2012; Hu et al., 2013). +2.1 +Word embeddings +Researchers have also found success by capitaliz- +ing on contemporary work in distributional seman- +tics, integrating embedding lookup tables into their +frameworks to represent words and documents. For +instance, lda2vec (Moody, 2016) combines em- +beddings with topic models by embedding word, +document, and topic vectors into a common rep- +resentation space. Concurrently, Van Gysel et al. +(2016) introduce an unsupervised model that learns +unidirectional mappings between latent vector rep- +resentations of words and entities. Using a shared +embedding space for words and topics, Dieng et al. +(2020) instead present the Embedded Topic Model +(ETM), which merges traditional topic models with +the neural-based word embeddings of Mikolov et al. +(2013)1. +2.2 +Neural topic models +In recent years, researchers have also looked to +incorporate modern deep learning techniques that +1We do not compare our model against ETM because to +do so requires us to pick the entity embeddings to be used in +place of word embeddings. +utilize contextualized representations in contrast to +more traditional static embeddings (Zhao et al., +2021). +Srivastava and Sutton (2016) propose +ProdLDA, a neural variational inference method +for LDA that explicitly approximates the Dirich- +let prior. Other models, however, such as Neu- +ral Variational Document Model (NVDM) (Miao +et al., 2017), employ a multinomial factor model +of documents that uses amortized variational infer- +ence to learn document representations. Bianchi +et al. (2021) expand on ProdLDA presenting Com- +binedTM, which improves the model with contex- +tualized embeddings. +2.3 +Knowledge extraction +More related to our work, Piccardi and West +(2021) — leveraging the self-referencing nature of +Wikipedia — define a cross-lingual topic model in +which documents are represented by extracted and +densified bags-of-links. The adoption of large-scale +lexical resources has recently gained popularity in +NLP as a way to directly inject knowledge into the +model (Gillick et al., 2019; Sun et al., 2020; Liu +et al., 2020), which further motivates our research. +2.4 +Clustering +Clustering techniques have also proved effective for +topic modeling. For instance, Sia et al. (2020) intro- +duce clustering to generate coherent topic models +from word embeddings, lowering complexity and +producing better runtimes compared to traditional +topic modeling approaches. Thompson and Mimno +(2020) experiment with different pretrained con- +textualized embeddings and demonstrate that clus- +tering contextualized representations at the token +level is indistinguishable from a Gibbs sampling +state for LDA. These findings were also recently +corroborated by Zhang et al. (2022) who cluster +sentence embeddings and extract top topic words +using TF-IDF to produce more coherent and di- +verse topics than neural topic models. In contrast +to our work, none of these works considers the +expressiveness of entities. +3 +Topics as Entity Clusters +In this section, we describe the steps necessary to +perform topic modeling with entities and the novel +approach for extracting salient entities to represent +topics. We present an overview of the model in +Figure 1. + +Entity representation +Topic inference +Entity extraction +Entity clustering +Text +document +Pattern +matching +Neural +Language +Model +Clustering +Document +Embedding +Graph +Embeddings +Language Model +Embeddings +Graph +Neural +Network +Concatenated +Embeddings +Entity +representation +Reranking top entities +Averaging +Top topics +per +document +Concatenated +embeddings +Text pattterns +Top topics for the +whole corpus +Top entities +per topic +Knowledge +Base +Disambiguation +Measuring +distance and +interpolating +Algorithm 1 +Topic +centroids +Next sequential step +Preprocessed data +Entity +descriptions +Entity +triplets +List of +entities +Figure 1: Overview of Topics as Entity Clusters (TEC). The top half illustrates the processing of entity embeddings, +topic centroids and top entities per topic, while the bottom half inferencing the top topics per document. +3.1 +Entity representation +We explore methods to encode entities for cluster- +based topic modeling. Broadly, we construct ex- +pressive entity representations from two sources of +information: Implicit Knowledge from a large pre- +trained language model and Explicit Knowledge +extracted directly from a knowledge graph. +Language embeddings. Language models are +used to construct document representations and de- +pict knowledge obtained implicitly through a con- +siderable amount of unsupervised learning (Petroni +et al., 2019). To encode the entities, we first ex- +tract their definitions from an encyclopedic corpus +before using these descriptions to build sentence +embeddings to represent each entity — for exam- +ple, utilizing Reimers and Gurevych (2019). Using +the text description of the entity rather than the +entity alone as a query for these unsupervised mod- +els elicits a stronger response due to their highly +contextualized nature (Ethayarajh, 2019). +Graph embeddings. Another advantage of us- +ing entities from lexical resources such as a knowl- +edge base is that they provide a systematic frame- +work for organizing and describing curated rela- +tionships between concepts. Similar to a semantic +network, these entities exhibit a complex structure +that provides meaningful information about their +content, provided in the form of a directed graph. +For instance, the triplet contains intricate encyclopedic knowl- +edge about the city of Petra that can be difficult to +learn with less specialized corpora. Language mod- +els may fail to adequately capture this relationship +due to the abstract notion of the concept. Hence, +to effectively capture these human-curated and re- +fined factual relationships, we employ the graph +neural network node2vec (Grover and Leskovec, +2016) to encode information about the sophisti- +cated semantic structure between these entities. +Combining Approaches. In this work, we bal- +ance the contribution of the language model and our +graph neural network. For some normalized lan- +guage and graph embeddings ˆELM ∈ RdLM and +ˆEG ∈ RdG, respectively, we weight their contribu- +tions using the following concatenation function, +ˆE = +�� +1 +1 + α · ˆET +LM, +� +α +1 + α · ˆET +G +�T +(1) +where α ∈ R is the scalar ratio of embedding +weights and ˆE ∈ RdLM+dG is our final embedding +used in entity clustering. We take the square root +to guarantee that the final embedding is normalized +similarly to the input embeddings. +3.2 +Entity clustering +Independent of the specific method, we represent +entities in an embedding space. Clustering allow +us to define centroids which we interpret as topic +centroids. Therefore, we model topics to have rep- +resentations in a shared embedding space with enti- +ties. To this effect, we apply K-Means to the set of +entities contained in a corpus, using the implemen- +tation available in FAISS (Johnson et al., 2021). + +3.3 +Entity extraction +We adopt a two-stage approach to extract entities, +which allows us to represent text as a language- +agnostic collection of entity identifiers arranged in +order of appearance. +Pattern matching. We first extract candidate en- +tities by finding language-specific text patterns in +the original text. Inspired by Mendes et al. (2011) +and Daiber et al. (2013), we use the deterministic +Aho-Corasick algorithm (Aho and Corasick, 1975) +due to its speed and effectiveness in extracting text +patterns. The only language-specific components +are the preprocessing components, such as lemma- +tizers, that increase the number of relevant entity +matches. These preprocessing components are in- +dependent of each other. Consequently, we can +expand the model to additional languages without +compromising the performance of the others. +Disambiguation. Since text patterns could rep- +resent multiple entities — for example, acronyms +of organizations or people sharing the same name +— we perform disambiguation and entity filtering. +For each textual pattern and its corresponding set of +entities, we choose the entity that best fits the text. +We embed the text using using the same model +used to derive the language embeddings and cal- +culate their cosine similarity. We choose the best +candidate based on the highest score if it is above a +set similarity threshold. Otherwise, we discard it. +3.4 +Topic inference +Topic inference requires the representation of doc- +uments in the same embedding space as entities +and topic centroids. To accomplish this, we ex- +tract entities as described in Section 3.3. We then +obtain the document representation by calculat- +ing the weighted average of those entity embed- +dings. With K representing the number of top- +ics, we can now measure the Euclidean distances +d = [d1, d2, ..., dK]T of the document to the topic +centroids. Documents are assumed to contain a +share of all topics. We infer the topic weight con- +tribution w = [w1, w2, ..., wK]T to the document +using the inverse distance squared weighted inter- +polation2 (Shepard, 1968): +wi = +d−2 +i +�K +j=1 d−2 +j +, ∀ i ∈ {1, ... , K} . +(2) +2If we consider the embedding of a document as an inter- +polation of topic centroids, squaring the distances yields more +weight to the closest topic centroids. +3.5 +Reranking top entities +A list of highly descriptive entities, weighted by +their importance, can be used to express the theme +of a topic. However, the closest entities to topic +centroids are not necessarily the most descriptive +as that does not consider entity co-occurrences +in the corpus. +In Algorithm 1, we propose a +novel inference-based method to rerank top entities, +which assigns the entity frequency of a document +to the top topic centroid, as measured by w. +We start by assigning entities to topics based +on their distances weighted by a small value, ϵ +(Lines 1-3). This ensures all topics have top enti- +ties. We follow by inferring the top topic for each +document and updating the top entities in that topic +using the document entity frequency. The update +is proportional to the inference score, max (w), as +it represents the degree of confidence in the infer- +ence. (Lines 4-10). To increase topic diversity, we +only update the top topic. Lastly, we calculate the +relative frequencies to obtain the top entities per +topic (Lines 11-13). +Algorithm 1: Reranking top entities +Input: Number of topics K, number of top entities +per topic N, small initialization weight ϵ, +documents Docs, all entity identifiers in the +corpus entities, entity embeddings ˆE +Output: Lists of top entities per topic topEntities, +each element is a list of pairs +(entityId, frequency) +1 for topicId ∈ {1, ..., K} do +2 +topEntities[topicId] ← +ClosestEntities(topicId, ˆE, N, ϵ) +3 end +4 for doc ∈ Docs do +5 +w, entityFrequency ← TopicInference(doc) +// Section 3.4 +6 +topTopic ← argmax(w) +7 +for entityId ∈ entities do +8 +topEntities [topTopic] += +max (w) · entityFrequency[entityId] +9 +end +10 end +11 for topicId ∈ [1, ..., K] do +12 +topEntities[topicId] ← +RelativeFrequency(topEntities[topicId]) +13 end +4 +Experiments +We study the performance of TEC and qualitatively +compare it to other state-of-the-art topic models +using a set of corpora preprocessed into lists of en- +tity identifiers. By contrasting the top entities and +measuring results across several coherency metrics, +we can infer the quality of each topic model. + +Table 1: Statistics of the corpora. +Corpus +Vocabulary +Documents +Avg. Entities +per Document +WIKIPEDIA +359,507 +359,507 +44.62 +CC-NEWS +94,936 +412,731 +13.97 +MLSUM +89,383 +661,422 +11.71 +In summary, we find that TEC produces signifi- +cantly more coherent topics. These gains are more +pronounced when using graph embeddings. +4.1 +Entity extractor +We build the entity extractor using Wikidata3 as the +source of our knowledge base and Wikipedia4 as +the encyclopedic corpus. Wikidata currently has +more than 97 million entities, most of which would +be a long tail of entities in a topic model therefore +we restrict the entity extractor to only include the +top one million entities, as ranked by QRank5 – a +public domain project that ranks page views across +Wikimedia projects. Out of these entities, we se- +lect those matching at least one predicate-object +pair from lists of preselected objects for predicates +"instance of", "subclass of", and "facet of". We gen- +erate the entity embeddings used in disambiguation +using SBERT6. Entities are matched to Wikipedia +articles using Wikidata identifiers. +4.2 +Corpora +We evaluate all models on various corpora: +Wikipedia, CC-News7, and MLSUM (Scialom et al., +2020); Table 1 contains a statistics summary. The +Wikipedia corpus consists of a sample of prepro- +cessed documents, each matching an entity in the +vocabulary. +CC-News consists of monolingual +news articles written in English. MLSUM is a col- +lection of news articles written in German, Spanish, +French, Russian, and Turkish. +We preprocess the documents according to +Section 3.3. The language-specific components +for documents in English, German, Spanish and +French are spaCy lemmatizers (Honnibal and Mon- +tani, 2017), for documents in Russian we use py- +morphy2 (Korobov, 2015), and for documents in +Turkish we use zeyrek8. +3Wikidata JSON dump downloaded on March 24th, 2022. +4Collected with Beautiful Soup on March 28th, 2022. +5QRank downloaded on March 24th, 2022. +6We use paraphrase-multilingual-mpnet-base-v2. +7CC-News available at Hugging Face. +8zeyrek available on GitHub. +4.3 +Models +We start by comparing our approach with LDA +(Blei et al., 2003) due to its pervasiveness in topic +model literature. Specifically, we use the Mallet +implementation of LDA (McCallum, 2002). On +top of that, we compare using other state-of-the-art +topic models from the literature. +NVDM-GSM. Neural Variational Document +Model (NVDM) is a neural network-based topic +model that discovers topics through variational in- +ference training, proposing a number of ways to +construct topic distributions, such as a Gaussian +Softmax (GSM) function (Miao et al., 2017). +ProdLDA. Similar to NVDM-GSM, this model +is an autoencoder trained to reconstruct the input +embeddings with variational inference-based train- +ing (Srivastava and Sutton, 2016). +CombinedTM. This model is a direct exten- +sion to ProdLDA that includes pre-trained contex- +tualized embeddings from a pretrained language +model (Bianchi et al., 2021). In this case, the au- +thors extract contextual vectors for documents us- +ing SBERT. +WikiPDA. We also consider the Wikipedia- +based Polyglot Dirichlet Allocation model, an LDA +model trained on entities extracted from Wikipedia +(Piccardi and West, 2021). WikiPDA has its own +preprocessing method. +4.4 +Metrics +Topic models produce subjective results, so we +calculate different measures to understand model +performance. We use topic coherence measures to +estimate the relationship between top entities of a +topic (Röder et al., 2015). +Cft = 1 +T +� +t∈{1..T} +� +��� +2 +N(N − 1) +� +i∈{1..N} +j∈{1..i−1} +ft(wi, wj) +� +��� +(3) +All coherence metrics are calculated using Eq. 3, +as implemented in gensim (Rehurek and Sojka, +2011), over the top most relevant N entities for +all topics t ∈ {1..T}, with N = 10. The specific +element ft changes for each measure. +Coherence UCI. Newman et al. (2010) present +a coherence measure that averages the Pointwise +Mutual Information (PMI, Eq. 4) of all entity pairs +in a topic using a sliding window of entities: +PMI(wi, wj) = log +� p(wi, wj) +p(wi)p(wj) +� +. +(4) + +Model +Sample Topic +LDA +United Nations (Q1065) | Teenage Mutant Ninja Turtles (Q12296099) | Miles Davis (Q93341) +| Star Trek (Q1092) | United Nations Security Council (Q37470) | United Nations Relief and +Works Agency for Palestine Refugees in the Near East (Q846656) | public health (Q189603) | +Dizzy Gillespie (Q49575) | Greenpeace (Q81307) | John Coltrane (Q7346) +NVDM-GSM +bitcoin (Q131723) | Apple Inc. (Q312) | Halloween [film franchise] (Q1364022) | Fisker Inc. +[automaker] (Q1420893) | IBM (Q37156) | Michael Myers (Q1426891) | Yakuza [video game +series] (Q2594935) | Facebook (Q355) | cryptocurrency (Q13479982) | Vancouver (Q234053) +ProdLDA +Paul McCartney (Q2599) | Maxim Gorky (Q12706) | Lucy-Jo Hudson (Q1394969) | Bob +Dylan (Q392) | sport utility vehicle (Q192152) | FIFA World Cup (Q19317) | sedan (Q190578) +| American football (Q41323) | concept car (Q850270) | racing automobile (Q673687) +CombinedTM +vocalist (Q2643890) | United States of America (Q30) | music interpreter (Q3153559) | England +(Q21) | Ryuichi Sakamoto (Q345494) | human rights (Q8458) | David Tennant (Q214601) | +Harry Potter (Q76164749) | Comedian (Q2591461) | Aoni Production (Q1359479) +WikiPDA +a cappella (Q185298) | X-Men (Q128452) | Marvel Comics (Q173496) | To Be [music album] +(Q17025795) | The Allman Brothers Band (Q507327) | proton–proton chain reaction (Q223073) +| features of the Marvel Universe (Q5439694) | Features of the Marvel Cinematic Universe +(Q107088537) | Uncanny X-Men (Q1399747) | member of parliament (Q486839) +TEC ELM (α = 0) +Google (Q95) | Amazon (Q3884) | Microsoft (Q2283) | open source (Q39162) | Apple +Inc. (Q312) | Facebook (Q355) | Meta Platforms (Q380) | Cisco Systems (Q173395) | +Salesforce.com (Q941127) | Citrix Systems (Q916196) +TEC EG (α = ∞) +Mike Tyson (Q79031) | World Boxing Organization (Q830940) | International Boxing Fed- +eration (Q742944) | Floyd Mayweather (Q318204) | World Boxing Association (Q725676) | +Tyson Fury (Q1000592) | Manny Pacquiao (Q486359) | World Boxing Council (Q724450) | +Evander Holyfield (Q313451) | Joe Frazier (Q102301) +Table 2: Example topics using WIKIPEDIA corpus for models trained with 300 topics. Each topic is represented +by its top 10 entities. +WIKIPEDIA +Model +CNPMI +CUCI +UMass +TD +TQ +Number of Topics ×100 +LDA +−0.05 (0.01) +−4.72 (0.21) +−11.23 (0.23) +0.98 (0.00) +−0.05 (0.01) +NVDM-GSM +0.06 (0.02) +−2.66 (0.36) +−9.17 (0.31) +0.87 (0.02) +0.05 (0.02) +ProdLDA +−0.16 (0.03) +−6.55 (0.44) +−12.91 (0.52) +0.62 (0.16) +−0.10 (0.03) +CombinedTM +−0.10 (0.02) +−5.94 (0.35) +−11.54 (0.49) +0.22 (0.03) +−0.02 (0.00) +WikiPDA +0.08 (0.01) +0.37 (0.14) +-3.60 (0.11) +0.73 (0.01) +0.06 (0.00) +TEC ELM (α = 0) +0.18 (0.01) +0.66 (0.18) +−5.91 (0.35) +0.95 (0.00) +0.17 (0.01) +TEC α = 1/2 +0.21 (0.01) +1.10 (0.21) +−4.79 (0.27) +0.95 (0.01) +0.20 (0.01) +TEC α = 1 +0.21 (0.01) +1.10 (0.17) +−4.82 (0.21) +0.96 (0.01) +0.21 (0.01) +TEC α = 2 +0.22 (0.01) +1.26 (0.17) +−4.67 (0.23) +0.97 (0.00) +0.22 (0.01) +TEC EG (α = ∞) +0.24 (0.01) +1.67 (0.15) +−4.94 (0.26) +0.97 (0.00) +0.23 (0.01) +Number of Topics ×300 +LDA +0.09 (0.01) +−2.00 (0.19) +−9.42 (0.22) +0.97 (0.00) +0.09 (0.01) +NVDM-GSM +0.06 (0.02) +−2.31 (0.37) +−9.23 (0.39) +0.69 (0.03) +0.04 (0.02) +ProdLDA +−0.14 (0.02) +−5.82 (0.37) +−13.28 (0.36) +0.44 (0.17) +−0.06 (0.03) +CombinedTM +−0.13 (0.03) +−6.20 (0.47) +−13.01 (0.38) +0.15 (0.03) +−0.02 (0.00) +WikiPDA +0.06 (0.01) +−0.30 (0.13) +−5.72 (0.14) +0.84 (0.01) +0.05 (0.00) +TEC ELM (α = 0) +0.25 (0.01) +2.03 (0.12) +−6.31 (0.20) +0.95 (0.00) +0.24 (0.01) +TEC α = 1/2 +0.29 (0.01) +2.51 (0.09) +−4.89 (0.15) +0.95 (0.00) +0.28 (0.01) +TEC α = 1 +0.30 (0.01) +2.62 (0.09) +−4.63 (0.12) +0.96 (0.00) +0.29 (0.01) +TEC α = 2 +0.31 (0.01) +2.70 (0.10) +-4.50 (0.15) +0.96 (0.00) +0.30 (0.01) +TEC EG (α = ∞) +0.31 (0.01) +2.88 (0.08) +−5.08 (0.18) +0.96 (0.00) +0.30 (0.01) +Table 3: Results on WIKIPEDIA corpus for all topic models. We record the results on five metrics, including +CNPMI : Normalized pointwise mutual information, more correlated with humans, UMass : How often a word +appears with another against how often it appears on its own TD : (Topic Diversity) the ratio of unique entities +to total entities and TQ : (Topic Quality) Topic Diversity × CNPMI. The results are reported as averages (95% +confidence interval) based on 10 random experimental runs. Our model outperforms all baselines across all metrics +except for TD and CUCI at 100 topics. + +CC − NEWS +Model +CNPMI +CUCI +UMass +TD +TQ +Number of Topics ×100 +LDA +−0.13 (0.01) +−6.10 (0.13) +−12.69 (0.14) +0.97 (0.00) +−0.13 (0.01) +NVDM-GSM +−0.03 (0.02) +−3.53 (0.48) +−9.68 (0.89) +0.61 (0.14) +−0.02 (0.01) +ProdLDA +−0.30 (0.01) +−8.69 (0.13) +−14.18 (0.24) +0.23 (0.01) +−0.07 (0.00) +CombinedTM +−0.32 (0.01) +−9.34 (0.32) +−15.07 (1.04) +0.37 (0.21) +−0.12 (0.07) +TEC ELM (α = 0) +0.11 (0.01) +−0.97 (0.15) +−7.54 (0.21) +0.79 (0.01) +0.08 (0.01) +TEC α = 1/2 +0.17 (0.01) +0.16 (0.24) +−6.36 (0.29) +0.81 (0.01) +0.14 (0.01) +TEC α = 1 +0.19 (0.02) +0.39 (0.29) +−6.18 (0.30) +0.82 (0.01) +0.15 (0.02) +TEC α = 2 +0.19 (0.01) +0.39 (0.21) +−6.28 (0.26) +0.83 (0.01) +0.16 (0.01) +TEC EG (α = ∞) +0.20 (0.02) +0.50 (0.28) +-6.17 (0.31) +0.83 (0.01) +0.16 (0.02) +Number of Topics ×300 +LDA +−0.07 (0.01) +−5.08 (0.11) +−12.94 (0.11) +0.91 (0.00) +−0.07 (0.01) +NVDM-GSM +0.04 (0.01) +−2.54 (0.17) +−9.30 (0.26) +0.52 (0.05) +0.02 (0.01) +ProdLDA +−0.21 (0.01) +−6.92 (0.27) +−13.11 (0.29) +0.16 (0.01) +−0.03 (0.00) +CombinedTM +−0.32 (0.01) +−9.45 (0.09) +−16.49 (1.03) +0.41 (0.17) +−0.13 (0.05) +TEC ELM (α = 0) +0.11 (0.01) +−1.06 (0.14) +−8.74 (0.15) +0.72 (0.00) +0.08 (0.01) +TEC α = 1/2 +0.18 (0.01) +0.17 (0.12) +−7.46 (0.16) +0.75 (0.01) +0.13 (0.01) +TEC α = 1 +0.18 (0.01) +0.26 (0.14) +-7.34 (0.21) +0.75 (0.00) +0.14 (0.01) +TEC α = 2 +0.18 (0.01) +0.27 (0.09) +−7.35 (0.14) +0.76 (0.01) +0.14 (0.00) +TEC EG (α = ∞) +0.18 (0.01) +0.28 (0.15) +−7.39 (0.19) +0.76 (0.01) +0.14 (0.01) +Table 4: Results on CC-NEWS corpus for all topic models. We record the results on five metrics, including +CNPMI : Normalized pointwise mutual information, more correlated with humans, UMass : How often a word +appears with another against how often it appears on its own TD : (Topic Diversity) the ratio of unique entities +to total entities and TQ : (Topic Quality) Topic Diversity × CNPMI. The results are reported as averages (95% +confidence interval) based on 10 random experimental runs. Our model outperforms all baselines across all metrics +except for TD. +MLSUM +Model +CNPMI +CUCI +UMass +TD +TQ +Number of Topics ×100 +LDA +−0.02 (0.01) +−3.89 (0.26) +−10.49 (0.28) +0.96 (0.00) +−0.02 (0.01) +NVDM-GSM +0.08 (0.01) +−1.19 (0.32) +−7.44 (0.64) +0.59 (0.09) +0.04 (0.01) +ProdLDA +−0.21 (0.02) +−6.79 (0.42) +−12.95 (0.42) +0.36 (0.04) +−0.08 (0.01) +CombinedTM +−0.25 (0.01) +−7.54 (0.24) +−12.67 (0.56) +0.25 (0.09) +−0.06 (0.02) +TEC ELM (α = 0) +0.16 (0.01) +0.27 (0.25) +−6.80 (0.23) +0.79 (0.01) +0.13 (0.01) +TEC α = 1/2 +0.24 (0.01) +1.48 (0.16) +−5.49 (0.16) +0.82 (0.01) +0.19 (0.01) +TEC α = 1 +0.24 (0.01) +1.45 (0.17) +−5.58 (0.15) +0.82 (0.01) +0.19 (0.01) +TEC α = 2 +0.24 (0.01) +1.53 (0.15) +-5.45 (0.19) +0.82 (0.01) +0.20 (0.01) +TEC EG (α = ∞) +0.24 (0.01) +1.46 (0.20) +−5.62 (0.19) +0.83 (0.01) +0.20 (0.01) +Number of Topics ×300 +LDA +0.03 (0.01) +−3.06 (0.17) +−10.79 (0.18) +0.88 (0.00) +0.02 (0.01) +NVDM-GSM +0.13 (0.01) +−0.45 (0.21) +−7.00 (0.28) +0.42 (0.04) +0.06 (0.01) +ProdLDA +−0.16 (0.01) +−5.54 (0.13) +−12.01 (0.16) +0.17 (0.01) +−0.03 (0.00) +CombinedTM +−0.20 (0.01) +−6.44 (0.15) +−12.13 (0.17) +0.12 (0.01) +−0.02 (0.00) +TEC ELM (α = 0) +0.14 (0.01) +−0.24 (0.14) +−8.37 (0.14) +0.74 (0.01) +0.10 (0.01) +TEC α = 1/2 +0.22 (0.01) +1.17 (0.12) +−6.84 (0.14) +0.76 (0.00) +0.16 (0.01) +TEC α = 1 +0.22 (0.01) +1.22 (0.13) +−6.79 (0.12) +0.76 (0.01) +0.17 (0.01) +TEC α = 2 +0.22 (0.01) +1.27 (0.08) +-6.75 (0.12) +0.76 (0.01) +0.17 (0.01) +TEC EG (α = ∞) +0.23 (0.01) +1.26 (0.09) +−6.77 (0.13) +0.76 (0.01) +0.17 (0.00) +Table 5: Results on MLSUM corpus for all topic models. We record the results on five metrics, including CNPMI : +Normalized pointwise mutual information, more correlated with humans, UMass : How often a word appears with +another against how often it appears on its own TD : (Topic Diversity) the ratio of unique entities to total entities +and TQ : (Topic Quality) Topic Diversity × CNPMI. The results are reported as averages (95% confidence +interval) based on 10 random experimental runs. Our model outperforms all baselines across all metrics except for +TD. + +Coherence NPMI. Bouma (2009) proposes an +alternative coherence measure, where the above el- +ements are substituted by Normalized PMI (NPMI, +Eq. (5)) as it was found that these have higher cor- +relation to human topic coherence ratings: +NPMI(wi, wj) = +PMI(wi, wj) +− log (p(wi, wj)). +(5) +Coherence UMass. Mimno et al. (2011) sug- +gests the asymmetrical coherence measure UMass +(Eq. 6), which is also calculated based on intrin- +sic entity co-occurrences conditioned to top entity +occurrences: +UMass(wi, wj) = log +�p(wi, wj) +p(wj) +� +. +(6) +Topic diversity and quality. Topic diversity +(TD) is the ratio between the number of unique +entities and the total number of entities, consider- +ing the top 25 entities per topic (Dieng et al., 2020). +Topic quality (TQ) is the product of topic coher- +ence, as measured by CNPMI, and topic diversity. +4.5 +Experiments specifications +For each combination of model, corpus, and the +number of topics — 100 and 300 —, we compute +metrics over 10 runs and present both the averages +and 95% confidence interval range in Tables 3, 4 +and 5. We use sequential seeds for the sake of +reproducibility. We use implementation defaults +for all models, including TEC, with the exceptions +of NVDM-GSM, where we run 100 epochs, and +ProdLDA and CombinedTM, that we run each for +250 epochs. We report metrics for the epoch with +higher CNPMI. +We run the experiments in a shared Linux ma- +chine with 72 CPU cores, 256GB RAM and use a +Tesla V100-SXM2-16GB GPU. +4.6 +Qualitative results +We present exemplar topics for the different models +in Table 2. Using visual inspection, we find cases +where some top entities do not match the general +topic theme. These must be attributed to limitations +in the model as they all share the same preprocessed +corpora, with the exception of WikiPDA. Overall +these issues seem less prevalent with TEC. Particu- +larly for ProdLDA and CombinedTM, we also find +unrelated entities that linger across many topics, +with the lingering entities varying between runs. +We also find topics covering multiple themes, such +as the ones resulting from LDA and WikiPDA. +4.7 +Quantitative results +For all combinations of corpora and number of top- +ics, TEC achieves better CNPMI, CUCI, UMass +and TQ when compared to the other models. +UMass has a single exception where WikiPDA per- +forms better for 100 topics. +As opposed to word-based preprocessing, entity +extraction results in sparser representations of cor- +pora, and, for that reason, we observe significantly +worse results to those presented in the original topic +model papers. +Documents generally assume the reader has +background knowledge on the subject. Models +like LDA, NVDM-GSM, ProdLDA and WikiPDA +learn based on entity co-occurrences. Relation- +ships that are not explicit are neglected, justifying +their lack of performance in comparison to TEC. +WikiPDA considers the relationship across entities +during its preprocessing however training is based +on LDA so the same limitations apply to it. Com- +binedTM uses implicit knowledge, but much like +ProdLDA, seems to be affected by component col- +lapse as can be verified by their low TD scores +— a state where variational autoencoders can get +stuck in a poor local optimum, due to the choice +in objective function, that results in topics being +similar (Masada, 2022). +The results suggest that both embedding types +are valuable sources of knowledge to use with topic +models. Models using graph-based embeddings +perform significantly better than models using em- +beddings obtained with language models, and we +only find a few circumstances where some combi- +nation of both embeddings produces better results +than graph-based embeddings alone. +5 +Conclusions +We explore entity-based topic models based on +the clustering of vector representations of enti- +ties. TEC internally represents documents using +language-agnostic entity identifiers, which results +in a single set of topics shared across languages and +allows it to extend to new languages without sacri- +ficing the performance of the existing languages. +Our results suggest that the implicit knowledge +provided by language models is superior to the +state-of-the-art in terms of coherence and quality. +Nevertheless, these results are surpassed by the +explicit knowledge encoded in graph-based embed- +dings, using human contributed Wikidata knowl- +edge base as a source. + +6 +Limitations +TEC assumes that documents contain entities, yet +this is not necessarily the case. The proposed model +is specifically valuable for entity-rich applications +such as news articles. A potential solution we are +interested in exploring in the future is to train a self- +supervised model to generate word embeddings us- +ing the bag-of-words as input and the document em- +bedding as the target. It results in word embeddings +having a representation in the shared embedding +space. +We produce graph-based embeddings using +node2vec — a shallow neural network that may +be unable to learn deeper, more complex relation- +ships between entities. We believe that our results +can improve if we obtain embeddings using a multi- +layer graph neural network with unsupervised train- +ing. Furthermore, while our approach outperforms +other models on a range of metrics, they still lag +behind when it comes to topic diversity. Finding +a way to improve the diversity of the topics while +preserving their intrinsic performance could make +for important future work. +Lastly, updating the knowledge base will force +the retraining of the model, which does not cur- +rently guarantee a direct relationship between for- +mer and new topics. It requires additional research +as this can be a hindrance for some applications. +References +Alfred V. Aho and Margaret J. 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Survey Track. + diff --git a/NdE0T4oBgHgl3EQfjQHt/content/tmp_files/load_file.txt b/NdE0T4oBgHgl3EQfjQHt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7dfebaabc0b569011bb0c0756c9f82b3143799c6 --- /dev/null +++ b/NdE0T4oBgHgl3EQfjQHt/content/tmp_files/load_file.txt @@ -0,0 +1,1226 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf,len=1225 +page_content='Topics as Entity Clusters: Entity-based Topics from Language Models and Graph Neural Networks Manuel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Loureiro and Steven Derby and Tri Kurniawan Wijaya Huawei Ireland Research Centre Georges Court, Townsend St, Dublin 2, D02 R156, Ireland {manuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='loureiro, tri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='kurniawan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='wijaya}@huawei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='com steven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='derby@huawei-partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='com Abstract Topic models aim to reveal the latent structure behind a corpus, typically conducted over a bag-of-words representation of documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' In the context of topic modeling, most vocabu- lary is either irrelevant for uncovering underly- ing topics or contains strong relationships with relevant concepts, impacting the interpretabil- ity of these topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Furthermore, their limited expressiveness and dependency on language demand considerable computation resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Hence, we propose a novel approach for cluster-based topic modeling that employs con- ceptual entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Entities are language-agnostic representations of real-world concepts rich in relational information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' To this end, we ex- tract vector representations of entities from (i) an encyclopedic corpus using a language model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' and (ii) a knowledge base using a graph neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We demonstrate that our ap- proach consistently outperforms other state-of- the-art topic models across coherency metrics and find that the explicit knowledge encoded in the graph-based embeddings provides more coherent topics than the implicit knowledge encoded with the contextualized embeddings of language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 1 Introduction Following the seminal work of Blei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (2003), topic models have since become the de facto method for extracting and elucidating prominent themes from corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Traditionally, the semantic content of a document is composed of document- term frequencies or latently through a mixture of distributions of topics, common with probabilistic generative models such as Latent Dirichlet Alloca- tion (LDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Here, individual topics are represented by salient lexical constituents such as words that depict some subjects of the corpora (Blei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Blei and Lafferty, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Li and McCallum, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Teh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Crain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' In recent years, the field of Natural Language Processing (NLP) has seen a trend toward continuous vector representations of words, which look to capture the paradigmatic relationship between concepts by learning distributional co-occurrence patterns in text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' For example, large-scale language models such as BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2018) have explored robust contextualized representations that can ex- plain an array of linguistic phenomena and implicit real-world knowledge (Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Tenney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2019a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Petroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2020), making them highly advantageous for topic modeling (Sia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Despite their successes, it becomes evident that certain limitations emerge from conventional topic modeling due to the superfluous nature and lim- ited expressiveness of word-level tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' These methods rely on data-driven techniques — while ignoring real-world knowledge — to uncover statis- tical patterns and infer relevant lexical items, which results in topics with limited guarantees of inter- pretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Furthermore, in a multilingual setting, these models require expansive, resource-intensive lexicons that may not produce a desirable set of shared, language-free universal topics (Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Boyd-Graber and Blei, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' To overcome these challenges, in this paper we focus on entities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' they are distinct, free form human-derived concepts that are represented through encyclopedic-based definitions and a num- ber of key relational attributes, which offer a bet- ter alternative for topic modeling (Chemudugunta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Andrzejewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2009, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Al- lahyari and Kochut, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We supersede word- level topic modeling with real-world entities, as these are both rich in conceptual information and language-agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We demonstrate that by consid- ering purely entity-level units in the text, it is possi- ble to construct topics that are both interpretable to humans and founded on a rich set of prior knowl- edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We pursue this approach using two sources to represent entities: (1) contextualized text represen- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02458v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='CL] 6 Jan 2023 tations constructed from entity definitions and (2) structured graph data extracted from a knowledge base that we use to train a graph neural network to learn node embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Furthermore, we pro- pose Topics as Entity Clusters (TEC), a novel topic modeling algorithm that can discover meaningful and highly informative topics by clustering either type of entity vectors or a combination of both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We successfully verify that, through the experimental procedure, our approach outperforms a number of state-of-the-art topic models on a range of metrics across numerous datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 2 Literature Review Previous research has attempted to represent topic models using entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' For instance, Newman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (2006) proposed representing documents with salient entities obtained using Named Entity Recognition (NER) instead of using the words di- rectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Others have attempted to capture the pat- terns among words, entities, and topics, either by expanding LDA (Blei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2003) or more complex Bayesian topic models — see Alghamdi and Al- falqi (2015), Chauhan and Shah (2021) and Vayan- sky and Kumar (2020) for a general overview — by describing entities using words (Newman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='1 Word embeddings Researchers have also found success by capitaliz- ing on contemporary work in distributional seman- tics, integrating embedding lookup tables into their frameworks to represent words and documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' For instance, lda2vec (Moody, 2016) combines em- beddings with topic models by embedding word, document, and topic vectors into a common rep- resentation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Concurrently, Van Gysel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (2016) introduce an unsupervised model that learns unidirectional mappings between latent vector rep- resentations of words and entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Using a shared embedding space for words and topics, Dieng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (2020) instead present the Embedded Topic Model (ETM), which merges traditional topic models with the neural-based word embeddings of Mikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (2013)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='2 Neural topic models In recent years, researchers have also looked to incorporate modern deep learning techniques that 1We do not compare our model against ETM because to do so requires us to pick the entity embeddings to be used in place of word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' utilize contextualized representations in contrast to more traditional static embeddings (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Srivastava and Sutton (2016) propose ProdLDA, a neural variational inference method for LDA that explicitly approximates the Dirich- let prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Other models, however, such as Neu- ral Variational Document Model (NVDM) (Miao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2017), employ a multinomial factor model of documents that uses amortized variational infer- ence to learn document representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (2021) expand on ProdLDA presenting Com- binedTM, which improves the model with contex- tualized embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='3 Knowledge extraction More related to our work, Piccardi and West (2021) — leveraging the self-referencing nature of Wikipedia — define a cross-lingual topic model in which documents are represented by extracted and densified bags-of-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' The adoption of large-scale lexical resources has recently gained popularity in NLP as a way to directly inject knowledge into the model (Gillick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2020), which further motivates our research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='4 Clustering Clustering techniques have also proved effective for topic modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' For instance, Sia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (2020) intro- duce clustering to generate coherent topic models from word embeddings, lowering complexity and producing better runtimes compared to traditional topic modeling approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Thompson and Mimno (2020) experiment with different pretrained con- textualized embeddings and demonstrate that clus- tering contextualized representations at the token level is indistinguishable from a Gibbs sampling state for LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' These findings were also recently corroborated by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (2022) who cluster sentence embeddings and extract top topic words using TF-IDF to produce more coherent and di- verse topics than neural topic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' In contrast to our work, none of these works considers the expressiveness of entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 3 Topics as Entity Clusters In this section, we describe the steps necessary to perform topic modeling with entities and the novel approach for extracting salient entities to represent topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We present an overview of the model in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='Entity representation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='Topic inference ' metadata={'source': 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entities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='per topic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='Knowledge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='Base ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='Disambiguation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='Measuring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='distance and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='interpolating ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='Algorithm 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='Topic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='centroids ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='Next sequential step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='Preprocessed data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='Entity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='descriptions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='Entity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='triplets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='List of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='entities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='Figure 1: Overview of Topics as Entity Clusters (TEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' The top half illustrates the processing of entity embeddings, topic centroids and top entities per topic, while the bottom half inferencing the top topics per document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='1 Entity representation We explore methods to encode entities for cluster- based topic modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Broadly, we construct ex- pressive entity representations from two sources of information: Implicit Knowledge from a large pre- trained language model and Explicit Knowledge extracted directly from a knowledge graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Language embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Language models are used to construct document representations and de- pict knowledge obtained implicitly through a con- siderable amount of unsupervised learning (Petroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' To encode the entities, we first ex- tract their definitions from an encyclopedic corpus before using these descriptions to build sentence embeddings to represent each entity — for exam- ple, utilizing Reimers and Gurevych (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Using the text description of the entity rather than the entity alone as a query for these unsupervised mod- els elicits a stronger response due to their highly contextualized nature (Ethayarajh, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Graph embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Another advantage of us- ing entities from lexical resources such as a knowl- edge base is that they provide a systematic frame- work for organizing and describing curated rela- tionships between concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Similar to a semantic network, these entities exhibit a complex structure that provides meaningful information about their content, provided in the form of a directed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' For instance, the triplet contains intricate encyclopedic knowl- edge about the city of Petra that can be difficult to learn with less specialized corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Language mod- els may fail to adequately capture this relationship due to the abstract notion of the concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Hence, to effectively capture these human-curated and re- fined factual relationships, we employ the graph neural network node2vec (Grover and Leskovec, 2016) to encode information about the sophisti- cated semantic structure between these entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Combining Approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' In this work, we bal- ance the contribution of the language model and our graph neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' For some normalized lan- guage and graph embeddings ˆELM ∈ RdLM and ˆEG ∈ RdG, respectively, we weight their contribu- tions using the following concatenation function, ˆE = �� 1 1 + α · ˆET LM, � α 1 + α · ˆET G �T (1) where α ∈ R is the scalar ratio of embedding weights and ˆE ∈ RdLM+dG is our final embedding used in entity clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We take the square root to guarantee that the final embedding is normalized similarly to the input embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='2 Entity clustering Independent of the specific method, we represent entities in an embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Clustering allow us to define centroids which we interpret as topic centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Therefore, we model topics to have rep- resentations in a shared embedding space with enti- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' To this effect, we apply K-Means to the set of entities contained in a corpus, using the implemen- tation available in FAISS (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='3 Entity extraction We adopt a two-stage approach to extract entities, which allows us to represent text as a language- agnostic collection of entity identifiers arranged in order of appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Pattern matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We first extract candidate en- tities by finding language-specific text patterns in the original text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Inspired by Mendes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (2011) and Daiber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (2013), we use the deterministic Aho-Corasick algorithm (Aho and Corasick, 1975) due to its speed and effectiveness in extracting text patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' The only language-specific components are the preprocessing components, such as lemma- tizers, that increase the number of relevant entity matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' These preprocessing components are in- dependent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Consequently, we can expand the model to additional languages without compromising the performance of the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Disambiguation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Since text patterns could rep- resent multiple entities — for example, acronyms of organizations or people sharing the same name — we perform disambiguation and entity filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' For each textual pattern and its corresponding set of entities, we choose the entity that best fits the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We embed the text using using the same model used to derive the language embeddings and cal- culate their cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We choose the best candidate based on the highest score if it is above a set similarity threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Otherwise, we discard it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='4 Topic inference Topic inference requires the representation of doc- uments in the same embedding space as entities and topic centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' To accomplish this, we ex- tract entities as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We then obtain the document representation by calculat- ing the weighted average of those entity embed- dings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' With K representing the number of top- ics, we can now measure the Euclidean distances d = [d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', dK]T of the document to the topic centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Documents are assumed to contain a share of all topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We infer the topic weight con- tribution w = [w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', wK]T to the document using the inverse distance squared weighted inter- polation2 (Shepard, 1968): wi = d−2 i �K j=1 d−2 j , ∀ i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' , K} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (2) 2If we consider the embedding of a document as an inter- polation of topic centroids, squaring the distances yields more weight to the closest topic centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='5 Reranking top entities A list of highly descriptive entities, weighted by their importance, can be used to express the theme of a topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' However, the closest entities to topic centroids are not necessarily the most descriptive as that does not consider entity co-occurrences in the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' In Algorithm 1, we propose a novel inference-based method to rerank top entities, which assigns the entity frequency of a document to the top topic centroid, as measured by w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We start by assigning entities to topics based on their distances weighted by a small value, ϵ (Lines 1-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' This ensures all topics have top enti- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We follow by inferring the top topic for each document and updating the top entities in that topic using the document entity frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' The update is proportional to the inference score, max (w), as it represents the degree of confidence in the infer- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (Lines 4-10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' To increase topic diversity, we only update the top topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Lastly, we calculate the relative frequencies to obtain the top entities per topic (Lines 11-13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Algorithm 1: Reranking top entities Input: Number of topics K, number of top entities per topic N, small initialization weight ϵ, documents Docs, all entity identifiers in the corpus entities, entity embeddings ˆE Output: Lists of top entities per topic topEntities, each element is a list of pairs (entityId, frequency) 1 for topicId ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', K} do 2 topEntities[topicId] ← ClosestEntities(topicId, ˆE, N, ϵ) 3 end 4 for doc ∈ Docs do 5 w, entityFrequency ← TopicInference(doc) // Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='4 6 topTopic ← argmax(w) 7 for entityId ∈ entities do 8 topEntities [topTopic] += max (w) · entityFrequency[entityId] 9 end 10 end 11 for topicId ∈ [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', K] do 12 topEntities[topicId] ← RelativeFrequency(topEntities[topicId]) 13 end 4 Experiments We study the performance of TEC and qualitatively compare it to other state-of-the-art topic models using a set of corpora preprocessed into lists of en- tity identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' By contrasting the top entities and measuring results across several coherency metrics, we can infer the quality of each topic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Table 1: Statistics of the corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Corpus Vocabulary Documents Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Entities per Document WIKIPEDIA 359,507 359,507 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='62 CC-NEWS 94,936 412,731 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='97 MLSUM 89,383 661,422 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='71 In summary, we find that TEC produces signifi- cantly more coherent topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' These gains are more pronounced when using graph embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='1 Entity extractor We build the entity extractor using Wikidata3 as the source of our knowledge base and Wikipedia4 as the encyclopedic corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Wikidata currently has more than 97 million entities, most of which would be a long tail of entities in a topic model therefore we restrict the entity extractor to only include the top one million entities, as ranked by QRank5 – a public domain project that ranks page views across Wikimedia projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Out of these entities, we se- lect those matching at least one predicate-object pair from lists of preselected objects for predicates "instance of", "subclass of", and "facet of".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We gen- erate the entity embeddings used in disambiguation using SBERT6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Entities are matched to Wikipedia articles using Wikidata identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='2 Corpora We evaluate all models on various corpora: Wikipedia, CC-News7, and MLSUM (Scialom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Table 1 contains a statistics summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' The Wikipedia corpus consists of a sample of prepro- cessed documents, each matching an entity in the vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' CC-News consists of monolingual news articles written in English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' MLSUM is a col- lection of news articles written in German, Spanish, French, Russian, and Turkish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We preprocess the documents according to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' The language-specific components for documents in English, German, Spanish and French are spaCy lemmatizers (Honnibal and Mon- tani, 2017), for documents in Russian we use py- morphy2 (Korobov, 2015), and for documents in Turkish we use zeyrek8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 3Wikidata JSON dump downloaded on March 24th, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 4Collected with Beautiful Soup on March 28th, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 5QRank downloaded on March 24th, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 6We use paraphrase-multilingual-mpnet-base-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 7CC-News available at Hugging Face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 8zeyrek available on GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='3 Models We start by comparing our approach with LDA (Blei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2003) due to its pervasiveness in topic model literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Specifically, we use the Mallet implementation of LDA (McCallum, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' On top of that, we compare using other state-of-the-art topic models from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' NVDM-GSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Neural Variational Document Model (NVDM) is a neural network-based topic model that discovers topics through variational in- ference training, proposing a number of ways to construct topic distributions, such as a Gaussian Softmax (GSM) function (Miao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' ProdLDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Similar to NVDM-GSM, this model is an autoencoder trained to reconstruct the input embeddings with variational inference-based train- ing (Srivastava and Sutton, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' CombinedTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' This model is a direct exten- sion to ProdLDA that includes pre-trained contex- tualized embeddings from a pretrained language model (Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' In this case, the au- thors extract contextual vectors for documents us- ing SBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' WikiPDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We also consider the Wikipedia- based Polyglot Dirichlet Allocation model, an LDA model trained on entities extracted from Wikipedia (Piccardi and West, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' WikiPDA has its own preprocessing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='4 Metrics Topic models produce subjective results, so we calculate different measures to understand model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We use topic coherence measures to estimate the relationship between top entities of a topic (Röder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Cft = 1 T � t∈{1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='.T} � ��� 2 N(N − 1) � i∈{1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='.N} j∈{1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='.i−1} ft(wi, wj) � ��� (3) All coherence metrics are calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 3, as implemented in gensim (Rehurek and Sojka, 2011), over the top most relevant N entities for all topics t ∈ {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='.T}, with N = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' The specific element ft changes for each measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Coherence UCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Newman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (2010) present a coherence measure that averages the Pointwise Mutual Information (PMI, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 4) of all entity pairs in a topic using a sliding window of entities: PMI(wi, wj) = log � p(wi, wj) p(wi)p(wj) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (4) Model Sample Topic LDA United Nations (Q1065) | Teenage Mutant Ninja Turtles (Q12296099) | Miles Davis (Q93341) | Star Trek (Q1092) | United Nations Security Council (Q37470) | United Nations Relief and Works Agency for Palestine Refugees in the Near East (Q846656) | public health (Q189603) | Dizzy Gillespie (Q49575) | Greenpeace (Q81307) | John Coltrane (Q7346) NVDM-GSM bitcoin (Q131723) | Apple Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (Q312) | Halloween [film franchise] (Q1364022) | Fisker Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='[automaker] (Q1420893) | IBM (Q37156) | Michael Myers (Q1426891) | Yakuza [video game ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='series] (Q2594935) | Facebook (Q355) | cryptocurrency (Q13479982) | Vancouver (Q234053) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='ProdLDA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='Paul McCartney (Q2599) | Maxim Gorky (Q12706) | Lucy-Jo Hudson (Q1394969) | Bob ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='Dylan (Q392) | sport utility vehicle (Q192152) | FIFA World Cup (Q19317) | sedan (Q190578) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='| American football (Q41323) | concept car (Q850270) | racing automobile (Q673687) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='CombinedTM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='vocalist (Q2643890) | United States of America (Q30) | music interpreter (Q3153559) | England ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='(Q21) | Ryuichi Sakamoto (Q345494) | human rights (Q8458) | David Tennant (Q214601) | ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='Harry Potter (Q76164749) | Comedian (Q2591461) | Aoni Production (Q1359479) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='WikiPDA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='a cappella (Q185298) | X-Men (Q128452) | Marvel Comics (Q173496) | To Be [music album] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='(Q17025795) | The Allman Brothers Band (Q507327) | proton–proton chain reaction (Q223073) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='| features of the Marvel Universe (Q5439694) | Features of the Marvel Cinematic Universe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='(Q107088537) | Uncanny X-Men (Q1399747) | member of parliament (Q486839) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='TEC ELM (α = 0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='Google (Q95) | Amazon (Q3884) | Microsoft (Q2283) | open source (Q39162) | Apple ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (Q312) | Facebook (Q355) | Meta Platforms (Q380) | Cisco Systems (Q173395) | Salesforce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='com (Q941127) | Citrix Systems (Q916196) TEC EG (α = ∞) Mike Tyson (Q79031) | World Boxing Organization (Q830940) | International Boxing Fed- eration (Q742944) | Floyd Mayweather (Q318204) | World Boxing Association (Q725676) | Tyson Fury (Q1000592) | Manny Pacquiao (Q486359) | World Boxing Council (Q724450) | Evander Holyfield (Q313451) | Joe Frazier (Q102301) Table 2: Example topics using WIKIPEDIA corpus for models trained with 300 topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Each topic is represented by its top 10 entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' WIKIPEDIA Model CNPMI CUCI UMass TD TQ Number of Topics ×100 LDA −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='05 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='72 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='21) −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='23 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='23) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='98 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='05 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) NVDM-GSM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='06 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='66 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='36) −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} 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+page_content='55 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='44) −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='91 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='52) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='62 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='16) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='10 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='03) CombinedTM −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='10 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02) −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='94 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='35) −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='54 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='49) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='22 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='03) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) WikiPDA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='08 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='37 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='14) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='60 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='11) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='73 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='06 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) TEC ELM (α = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='18 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='66 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='18) −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='91 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='35) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='95 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC α = 1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='21 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='10 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='21) −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='79 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='27) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='95 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='20 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC α = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='21 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='10 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='17) −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='82 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='21) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='96 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='21 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC α = 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='22 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='26 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='17) −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='67 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='23) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='97 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='22 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC EG (α = ∞) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='24 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='67 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='15) −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='94 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='26) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='97 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='23 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) Number of Topics ×300 LDA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='09 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='19) −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='42 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='22) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='97 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='09 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) NVDM-GSM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='06 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='31 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='37) −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='23 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='39) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='69 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='04 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02) ProdLDA −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='14 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02) −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='82 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='37) −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='28 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='36) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='44 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='17) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='06 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='03) CombinedTM −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='13 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='03) −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='20 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='47) −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='38) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='15 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='03) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) WikiPDA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='06 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='30 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='13) −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='72 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='14) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='84 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='05 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) TEC ELM (α = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='25 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='03 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='12) −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='31 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='20) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='95 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='24 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC α = 1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='29 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='51 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='09) −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='89 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='15) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='95 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='28 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC α = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='30 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='62 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='09) −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='63 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='12) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='96 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='29 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC α = 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='31 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='70 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='10) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='50 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='15) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='96 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='30 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC EG (α = ∞) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='31 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='88 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='08) −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='08 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='18) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='96 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='30 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) Table 3: Results on WIKIPEDIA corpus for all topic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We record the results on five metrics, including CNPMI : Normalized pointwise mutual information, more correlated with humans, UMass : How often a word appears with another against how often it appears on its own TD : (Topic Diversity) the ratio of unique entities to total entities and TQ : (Topic Quality) Topic Diversity × CNPMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' The results are reported as averages (95% confidence interval) based on 10 random experimental runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Our model outperforms all baselines across all metrics except for TD and CUCI at 100 topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' CC − NEWS Model CNPMI CUCI UMass TD TQ Number of Topics ×100 LDA −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='13 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='10 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='13) −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='69 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='14) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='97 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='13 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) NVDM-GSM −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='03 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02) −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='53 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='48) −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='68 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='89) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='61 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='14) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) ProdLDA −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='30 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='69 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='13) −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='18 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='24) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='23 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='07 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) CombinedTM −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='32 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='34 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='32) −15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='07 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='04) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='37 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='21) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='12 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='07) TEC ELM (α = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='11 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='97 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='15) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='54 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='21) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='79 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='08 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC α = 1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='16 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='24) −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='36 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='29) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='81 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='14 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC α = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='19 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='39 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='29) −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='18 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='30) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='82 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='15 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02) TEC α = 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='19 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='39 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='21) −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='28 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='26) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='83 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='16 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC EG (α = ∞) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='20 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='50 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='28) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='31) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='83 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='16 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02) Number of Topics ×300 LDA −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='07 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='08 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='11) −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='94 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='11) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='91 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='07 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) NVDM-GSM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='04 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='54 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='17) −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='30 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='26) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='52 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='05) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) ProdLDA −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='21 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='92 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='27) −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='11 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='29) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='16 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='03 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) CombinedTM −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='32 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='45 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='09) −16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='49 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='41 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='17) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='13 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='05) TEC ELM (α = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='11 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='06 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='14) −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='74 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='15) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='72 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='08 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC α = 1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='18 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='12) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='46 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='16) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='75 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='13 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC α = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='18 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='26 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='14) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='34 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='21) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='75 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='14 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC α = 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='18 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='27 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='09) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='35 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='14) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='76 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='14 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) TEC EG (α = ∞) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='18 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='28 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='15) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='39 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='19) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='76 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='14 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) Table 4: Results on CC-NEWS corpus for all topic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We record the results on five metrics, including CNPMI : Normalized pointwise mutual information, more correlated with humans, UMass : How often a word appears with another against how often it appears on its own TD : (Topic Diversity) the ratio of unique entities to total entities and TQ : (Topic Quality) Topic Diversity × CNPMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' The results are reported as averages (95% confidence interval) based on 10 random experimental runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Our model outperforms all baselines across all metrics except for TD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' MLSUM Model CNPMI CUCI UMass TD TQ Number of Topics ×100 LDA −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='89 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='26) −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='28) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='96 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) NVDM-GSM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='08 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='19 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='32) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='44 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='64) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='59 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='09) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='04 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) ProdLDA −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='21 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02) −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='79 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='42) −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='95 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='42) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='36 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='04) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='08 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) CombinedTM −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='25 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='54 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='24) −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='67 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='56) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='25 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='09) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='06 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02) TEC ELM (α = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='16 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='27 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='25) −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='80 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='23) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='79 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='13 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC α = 1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='24 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='48 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='16) −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='16) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='82 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='19 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC α = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='24 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='45 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='17) −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='58 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='15) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='82 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='19 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC α = 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='24 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='53 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='15) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='45 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='19) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='82 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='20 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC EG (α = ∞) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='24 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='46 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='20) −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='62 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='19) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='83 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='20 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) Number of Topics ×300 LDA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='03 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='06 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='17) −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='79 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='18) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='88 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) NVDM-GSM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='13 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='45 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='21) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='28) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='42 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='04) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='06 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) ProdLDA −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='16 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='54 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='13) −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='16) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='03 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) CombinedTM −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='20 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='44 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='15) −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='13 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='17) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='12 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='02 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) TEC ELM (α = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='14 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='24 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='14) −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='37 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='14) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='74 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='10 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC α = 1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='22 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='12) −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='84 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='14) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='76 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='16 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC α = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='22 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='22 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='13) −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='79 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='12) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='76 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC α = 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='22 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='27 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='08) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='75 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='12) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='76 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) TEC EG (α = ∞) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='23 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='26 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='09) −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='77 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='13) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='76 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='00) Table 5: Results on MLSUM corpus for all topic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We record the results on five metrics, including CNPMI : Normalized pointwise mutual information, more correlated with humans, UMass : How often a word appears with another against how often it appears on its own TD : (Topic Diversity) the ratio of unique entities to total entities and TQ : (Topic Quality) Topic Diversity × CNPMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' The results are reported as averages (95% confidence interval) based on 10 random experimental runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Our model outperforms all baselines across all metrics except for TD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Coherence NPMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Bouma (2009) proposes an alternative coherence measure, where the above el- ements are substituted by Normalized PMI (NPMI, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (5)) as it was found that these have higher cor- relation to human topic coherence ratings: NPMI(wi, wj) = PMI(wi, wj) − log (p(wi, wj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (5) Coherence UMass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Mimno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (2011) sug- gests the asymmetrical coherence measure UMass (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 6), which is also calculated based on intrin- sic entity co-occurrences conditioned to top entity occurrences: UMass(wi, wj) = log �p(wi, wj) p(wj) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' (6) Topic diversity and quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Topic diversity (TD) is the ratio between the number of unique entities and the total number of entities, consider- ing the top 25 entities per topic (Dieng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Topic quality (TQ) is the product of topic coher- ence, as measured by CNPMI, and topic diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='5 Experiments specifications For each combination of model, corpus, and the number of topics — 100 and 300 —, we compute metrics over 10 runs and present both the averages and 95% confidence interval range in Tables 3, 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We use sequential seeds for the sake of reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We use implementation defaults for all models, including TEC, with the exceptions of NVDM-GSM, where we run 100 epochs, and ProdLDA and CombinedTM, that we run each for 250 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We report metrics for the epoch with higher CNPMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We run the experiments in a shared Linux ma- chine with 72 CPU cores, 256GB RAM and use a Tesla V100-SXM2-16GB GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='6 Qualitative results We present exemplar topics for the different models in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Using visual inspection, we find cases where some top entities do not match the general topic theme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' These must be attributed to limitations in the model as they all share the same preprocessed corpora, with the exception of WikiPDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Overall these issues seem less prevalent with TEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Particu- larly for ProdLDA and CombinedTM, we also find unrelated entities that linger across many topics, with the lingering entities varying between runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We also find topics covering multiple themes, such as the ones resulting from LDA and WikiPDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content='7 Quantitative results For all combinations of corpora and number of top- ics, TEC achieves better CNPMI, CUCI, UMass and TQ when compared to the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' UMass has a single exception where WikiPDA per- forms better for 100 topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' As opposed to word-based preprocessing, entity extraction results in sparser representations of cor- pora, and, for that reason, we observe significantly worse results to those presented in the original topic model papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Documents generally assume the reader has background knowledge on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Models like LDA, NVDM-GSM, ProdLDA and WikiPDA learn based on entity co-occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Relation- ships that are not explicit are neglected, justifying their lack of performance in comparison to TEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' WikiPDA considers the relationship across entities during its preprocessing however training is based on LDA so the same limitations apply to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Com- binedTM uses implicit knowledge, but much like ProdLDA, seems to be affected by component col- lapse as can be verified by their low TD scores — a state where variational autoencoders can get stuck in a poor local optimum, due to the choice in objective function, that results in topics being similar (Masada, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' The results suggest that both embedding types are valuable sources of knowledge to use with topic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Models using graph-based embeddings perform significantly better than models using em- beddings obtained with language models, and we only find a few circumstances where some combi- nation of both embeddings produces better results than graph-based embeddings alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 5 Conclusions We explore entity-based topic models based on the clustering of vector representations of enti- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' TEC internally represents documents using language-agnostic entity identifiers, which results in a single set of topics shared across languages and allows it to extend to new languages without sacri- ficing the performance of the existing languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Our results suggest that the implicit knowledge provided by language models is superior to the state-of-the-art in terms of coherence and quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Nevertheless, these results are surpassed by the explicit knowledge encoded in graph-based embed- dings, using human contributed Wikidata knowl- edge base as a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' 6 Limitations TEC assumes that documents contain entities, yet this is not necessarily the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' The proposed model is specifically valuable for entity-rich applications such as news articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' A potential solution we are interested in exploring in the future is to train a self- supervised model to generate word embeddings us- ing the bag-of-words as input and the document em- bedding as the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' It results in word embeddings having a representation in the shared embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We produce graph-based embeddings using node2vec — a shallow neural network that may be unable to learn deeper, more complex relation- ships between entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' We believe that our results can improve if we obtain embeddings using a multi- layer graph neural network with unsupervised train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Furthermore, while our approach outperforms other models on a range of metrics, they still lag behind when it comes to topic diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Finding a way to improve the diversity of the topics while preserving their intrinsic performance could make for important future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Lastly, updating the knowledge base will force the retraining of the model, which does not cur- rently guarantee a direct relationship between for- mer and new topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' It requires additional research as this can be a hindrance for some applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' References Alfred V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Aho and Margaret J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Corasick.' metadata={'source': 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Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} +page_content=' Survey Track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQfjQHt/content/2301.02458v1.pdf'} diff --git a/NtE4T4oBgHgl3EQfjg2I/content/tmp_files/2301.05143v1.pdf.txt b/NtE4T4oBgHgl3EQfjg2I/content/tmp_files/2301.05143v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3e6fae2a8049a12308e75a19bbdffae850747800 --- /dev/null +++ b/NtE4T4oBgHgl3EQfjg2I/content/tmp_files/2301.05143v1.pdf.txt @@ -0,0 +1,966 @@ +Impacts of Distribution Network Reconfiguration on +Aggregated DER Flexibility +Andrey Churkin1, Miguel Sanchez-Lopez1,2, Mohammad Iman Alizadeh3, Florin Capitanescu3, +Eduardo A. Mart´ınez Cese˜na1,4, Pierluigi Mancarella1,5 +1Department of Electrical and Electronic Engineering, the University of Manchester, UK +2Departamento de Ingenieria Electrica, Universidad de Chile, Chile +3Luxembourg Institute of Science and Technology (LIST), Luxembourg +4Tyndall Centre for Climate Change Research, UK +5Department of Electrical and Electronic Engineering, the University of Melbourne, Australia +{andrey.churkin; alex.martinezcesena; p.mancarella}@manchester.ac.uk, {mohammad.alizadeh; florin.capitanescu}@list.lu +Abstract—The ongoing integration of controllable distributed +energy resources (DER) makes distribution networks capable +of aggregating flexible power and providing flexibility services +at both transmission and distribution levels. The aggregated +flexibility of an active distribution network (ADN) can be +represented as its feasible operating area in the P-Q space. +The limits of this area are pivotal for arranging flexibility +markets and coordinating transmission and distribution system +operators (TSOs and DSOs). However, motivated by the current +technical limitations of distribution networks (e.g., protection +schemes), existing literature on ADN flexibility and TSO-DSO +coordination mostly focuses on radial networks, overlooking +the potential benefits of network reconfiguration. This paper, +using a realistic meshed distribution system from the UK and +the exact ACOPF model for flexibility estimation, demonstrates +that network reconfiguration can increase the limits of ADN +aggregated flexibility and improve the economic efficiency of +flexibility markets. +Index Terms—Active distribution network (ADN), distributed +energy resources (DER), flexibility services, network reconfigu- +ration, TSO-DSO coordination. +I. INTRODUCTION +Modern +distribution +networks +incorporate +increasing +amounts +of +distributed +energy +resources +(DER) +and +continuously improve their observability and controllability +[1]. +With +these +controllable +flexible +resources, +active +distribution networks (ADNs) now offer attractive means +to aggregate flexible power and provide flexibility services +at both distribution and transmission levels. To enable +coordination between transmission and distribution system +operators (TSOs and DSOs) and arrange TSO-DSO flexibility +markets, it is necessary to assess the limits of aggregated +flexibility that ADNs can deliver [2]–[4]. This has motivated +research on flexibility areas in the P-Q space to capture +the aggregated ADN flexibility at the primary substation or +TSO/DSO interface [5]–[10]. However, despite the rapidly +evolving research on ADN flexibility, existing studies mostly +focus +on +radial +distribution +networks, +overlooking +the +potential benefits of network reconfiguration. There have been +This work was carried out as a part of the ATTEST project (the Horizon +2020 research and innovation programme, grant agreement No 864298). +some attempts to quantify the value of ADN reconfiguration, +e.g., in [11]. Yet, the impacts of network reconfiguration +on aggregated DER flexibility and TSO-DSO coordination +remain largely unexplored. +Distribution network reconfiguration is well known in power +systems research as a tool for improving system security +and optimising its operation, e.g., for loss reduction and +voltage control [12], [13]. But, historically speaking, active +network reconfiguration and meshed operation have not been +economically attractive at the lowest voltage levels of the +distribution networks, being typically considered only during +emergency conditions [14], [15]. As a result, existing assets +(e.g., protections) are generally only suitable for radial opera- +tion, and existing literature on ADN flexibility has focused on +radial networks [4], [7], [8]. However, network reconfigura- +tion, including meshed options, can become more attractive +with increasing DER integration and the rising interest in +reducing power losses and environmental concerns [16]. +This paper demonstrates the benefits that network reconfig- +uration can bring in terms of increased ADN flexibility and +improved economic efficiency of flexibility markets. For this +purpose, a mixed integer quadratically constrained program- +ming (MIQCP) model is developed to estimate the limits of +flexibility P-Q areas and the cost of ADN aggregated flexibility +under different network configurations. The impacts of the +reconfiguration are illustrated using a meshed UK distribution +network. The rest of the paper is structured as follows. +Section II presents the proposed MIQCP flexibility estimation +model. The case study, which highlights how ADN flexibility +changes subject to different configurations, is presented in +Section III. Finally, section IV concludes this work. +II. MODELLING FRAMEWORK +In this section, a mathematical programming model is +proposed to analyse the effects of different network config- +urations on aggregated network flexibility (e.g., at the primary +substation or TSO/DSO interface). The model presented in +(1a)-(1p) is a single-stage (single-period) network flexibility +estimation problem. This formulation is based on an exact AC +arXiv:2301.05143v1 [eess.SY] 12 Jan 2023 + +optimal power flow model in rectangular voltage coordinates +(ACROPF) [7]. Variables ek and fk are the real and imaginary +rectangular voltage components at bus k (i.e., uk = ek +jfk), +pij and qij stand for active and reactive power flows between +buses i and j, and pk,g and qk,g denote the active and reactive +power of generators located in the network. The power of each +flexible unit f ∈ F is given by its available P-Q upward and +downward regulation capacities indicated by the corresponding +arrows. It is worth noting that, in practice, the available upward +and downward flexibility capacities would vary based on the +initial operation of the flexible units before a flexibility service +is requested. Finally, xij are binary variables representing the +status of lines (if lines are switched on or off). +The model uses two sets of objective functions and con- +straints designed for two purposes: map the flexibility area +limits or deploy the cheapest flexible units. Objective function +(1a) minimises or maximises network power consumption at a +selected reference bus (e.g., primary substation or TSO/DSO +interface). Coefficients wp +k and wq +k are introduced to control +the optimisation directions and can be used to iteratively +reconstruct the feasible P-Q flexibility area. Objective function +(1b) minimises the total cost of all flexible power regulations. +It is used to find the optimal dispatch decisions and analyse +the efficiency of the flexibility market for a given operat- +ing point. The first set of constraints, (1c)-(1n), defines the +technical constraints of the network and flexible units. Active +and reactive power flows are determined with (1c) and (1d), +where Gij and Bij are the conductance and susceptance of +lines, respectively. Equations (1e) and (1f) represent active +and reactive power balance for each node, where pk,d and +qk,d are the loads at bus k. Line capacity limits and nodal +voltage magnitude limits are imposed in (1g) and (1h). The +output of generators and flexible units is constrained in (1i)- +(1j) and (1k)-(1n). This set of constraints, (1c)-(1n), together +with the objective function (1a), enables estimating the limits +of the aggregated network flexibility and approximating it as +the network feasibility boundary. When minimising the total +flexibility cost with the objective function (1b), additional +constraints (1o)-(1p) must be introduced to specify the selected +feasible operating point.1 +The presented ACROPF flexibility estimation formulation is +a MIQCP problem. This problem is nonlinear and nonconvex, +which imposes requirements on the solver: it must combine +algorithms for both combinatorial optimisation (e.g., branch +and bound) and nonlinear optimisation. Solving the MIQCP +flexibility estimation problem can be intractable for an arbi- +trarily complex distribution network with an arbitrarily large +number of flexible units. However, realistic networks can have +a limited number of possible configurations. For example, as +will be demonstrated in Section III, some distribution networks +in the UK (typically 6.6 kV and 11 kV networks) have only +one loop, a normally open point (NOP) connecting adjacent +feeders. For such cases, the MIQCP flexibility estimation prob- +1Note that the cost-minimising model does not consider intertemporal +constraints and switching costs. The inclusion of additional constraints and +costs is the subject of future research. +MODEL ACROPF for flexibility estimation +[MIQCP] +Variables: +ek, fk +∀k ∈ K +pij, qij +∀(i, j) ∈ L +pk,g, qk,g +∀k ∈ K, ∀g ∈ G +p↑ +k,f, p↓ +k,f, q↑ +k,f, q↓ +k,f +∀k ∈ K, ∀f ∈ F +xij ∈ +� +{0, 1}, +if reconfiguration line, +1 +other in-service lines. +Objective I: +min wp +kpk,g + wq +kqk,g +k = kref +(1a) +Objective II: +min +� +k∈K +� +f∈F +Ck,f(p↑ +k,f, p↓ +k,f, q↑ +k,f, q↓ +k,f) +(1b) +Constraints I: +pij = (ei +2 + fi +2)Gij − (eiej + fifj)Gij +(1c) +−(fiej − eifj)Bij +∀(i, j) ∈ L +qij = −(ei +2 + fi +2)Bij + (eiej + fifj)Bij +(1d) +−(fiej − eifj)Gij +∀(i, j) ∈ L +pk,g − pk,d + p↑ +k,f − p↓ +k,f +(1e) +− +� +(k,j)∈L +pkjxkj = 0 +∀k ∈ K +qk,g − qk,d + q↑ +k,f − q↓ +k,f +(1f) +− +� +(k,j)∈L +qkjxkj = 0 +∀k ∈ K +pij +2 + qij +2 ≤ Smax +ij +2 +∀(i, j) ∈ L +(1g) +vmin +k +2 ≤ ek +2 + fk +2 ≤ vmax +k +2 +∀k ∈ K +(1h) +pk,g +min ≤ pk,g ≤ pk,g +max +∀k, ∀g +(1i) +qk,g +min ≤ qk,g ≤ qk,g +max +∀k, ∀g +(1j) +0 ≤ p↑ +k,f ≤ p↑max +k,f +∀k, ∀f +(1k) +0 ≤ p↓ +k,f ≤ p↓max +k,f +∀k, ∀f +(1l) +0 ≤ q↑ +k,f ≤ q↑max +k,f +∀k, ∀f +(1m) +0 ≤ q↓ +k,f ≤ q↓max +k,f +∀k, ∀f +(1n) +Constraints II: +pk,g = pk,g +′ +k = kref +(1o) +qk,g = qk,g +′ +k = kref +(1p) +lem can be decomposed into continuous QCP subproblems +corresponding to possible network configurations and solved +with software for nonlinear continuous systems, such as Ipopt. +In the next section, the flexibility estimation model (1a)-(1p) +will be applied to analyse the effects of network reconfigu- +ration on the aggregated network flexibility. Changes in both + +the technical limits of network flexibility and the economics of +flexibility provision will be traced and discussed. Additionally, +the application of the exact ACROPF model to a realistic +distribution network will demonstrate the nonlinearities and +nonconvexities of ADN operation and highlight potential co- +ordination issues of flexible units. +III. RESULTS AND DISCUSSION +A. Case Study: 38-bus Synthetic Distribution Network +The impacts of network reconfiguration on the aggregated +DER flexibility are illustrated with a 38-bus 6.6 kV UK +distribution network [17], which has been anonymised. The +network is displayed in Fig. 1 as a graph, where the sizes of the +nodes represent power demand or generation at each bus, and +lengths of the edges are proportional to the impedance between +buses. Flexible operation of this network is challenging due +to potential voltage problems and congestion. For example, +feeder 2 has buses with high demand, which causes a signif- +icant voltage drop and restricts flexible power consumption. +Conversely, feeder 1 has uncontrollable generation at buses +11, 17, 18, 19, and 20, which increases the voltage profile +and dictates the power flows in the feeder. Four flexible units +(denoted as FU in Fig. 1) are placed in the network. Their +characteristics are listed in Table I. It is assumed that units can +produce and consume power, which is consistent with DER +technologies such as battery storage and flexible loads. +The network includes two radial feeders that can be inter- +connected through the NOP (typically for customer restoration +purposes). That is, there are two sectionalising switches and +one tie switch, which enables dynamic network reconfigura- +tion. Excluding cases where some customers are isolated from +the system, there are four possible network configurations +to consider: (i) normal operation with NOP open (radial +network), (ii) normal operation with NOP closed (meshed +network), (iii) contingency power supply via feeder 1 (radial +network, NOP closed, line 8-1 open), and (iv) contingency +power supply via feeder 2 (radial network, NOP closed, line 8- +7 open). The limited number of possible configurations makes +it possible to decompose the flexibility estimation problem +(1a)-(1p) into four subproblems, which will be explored in +the following subsections. In this work, the ACROPF model +was programmed in JuMP 1.4.0 for Julia 1.6.1 programming +language, then decomposed into continuous QCP subproblems +for the four configurations, and solved with Ipopt 3.14.4 solver. +TABLE I +PARAMETERS OF FLEXIBLE UNITS PLACED IN THE NETWORK +Parameter +Flexible unit +A +B +C +D +Bus # +24 +17 +12 +36 +P regulation limits (MW) +±1.0 +±1.0 +±1.0 +±1.0 +Q regulation limits (MVAr) +±1.0 +±1.0 +±1.0 +±1.0 +P cost ($/MWh) +375.0 +350.0 +325.0 +300.0 +Q cost ($/MVArh) +187.5 +175.0 +162.5 +150.0 +1 +2 +3 +4 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +25 +26 +27 +28 +29 +30 +31 +32 +33 +34 +35 +36 +37 +38 +5 +primary +substation +0 MVA +0.683 MVA +Reconfiguration lines: +switch closed +switch open +Power demand/generation: +FU +FUD +FUC +FUB +FUA +Nodes with flexible units: +feeder 1 +feeder 2 +NOP +Fig. 1. +Case study: 38-bus distribution network with two feeders and four +flexible units. +B. Aggregated Flexibility Limits +For each of the four considered network configurations, +model (1a), (1c)-(1n) was solved iteratively 200 times (for +200 extreme operating points, with a step of 0.08 MVA) +to approximate the boundary of the network feasibility area +at the primary substation. The flexibility areas estimated by +model (1a), (1c)-(1n) for the 38-bus system, subject to the four +selected configurations, are shown in Fig. 2. The coordinates +represent the total active and reactive power consumption of +the network, measured at the primary substation, and the +cross marker corresponds to the initial operating point with +all flexible units switched off. Note that the slightly curved +horizontal and vertical lines correspond to unconstrained net- +work operation where flexible units can fully deploy their P- +Q regulation capabilities. Other inclined lines reflect the con- +strained network operation where flexible units cannot be fully +activated. Considering normal network operation under radial +and meshed topologies, the flexibility P-Q areas are similar +for most operating points, except for the upper right part of +the areas (increased active and reactive power consumption). +When the NOP is open (radial case), the voltage profile of +feeder 2 has a significant drop and is close to the lower voltage +limit of 0.94 p.u. Therefore, in this case, flexible units A +and D cannot fully increase their power consumption due to +voltage constraints. When the NOP is closed (meshed case), +voltages in feeder 2 increase which expands the limits of the +aggregated flexibility. In addition to improving the voltage +profile, meshing radial networks increases the capacity for +transferring flexible power, making the operation of distant +flexible units less constrained. The difference in the feasibility + +−2 +0 +2 +4 +6 +P, MW +−4 +−2 +0 +2 +4 +Q, MVAr +normal operation: radial network (NOP open) +normal operation: meshed network (NOP closed) +contingency: supply via feeder 1 (line 8-1 open) +contingency: supply via feeder 2 (line 8-7 open) +initial operating point +N-1 secure flexibility provision +Fig. 2. Boundaries of the feasibility areas for different network configurations. +areas demonstrates that network reconfiguration (meshing in +this case), besides offering options to manage power losses +and voltages, can be used to maximise aggregated flexibility. +The results show that the flexibility area greatly decreases in +the last two cases (contingencies when the network is supplied +only with feeder 1 or 2). This happens due to voltage problems +at busses 18 and 33 and congestion of lines 7-14 and 3- +32. The “N-1 secure” flexibility area can be estimated as the +intersection of areas under different network configurations, as +shown in Fig. 2 with a green polygon. Considering the effects +of network reconfiguration, DSO can impose restrictions on +some flexibility services to guarantee the security of flexible +power provision and ADN operation against contingencies. +C. Economics of Flexibility Provision +Different network configurations can affect the economics +of flexible power provision. To explore these economic effects, +model (1b)-(1p) was solved for 25,000 feasible operating +points (with a step of 0.05 MVA). The simulations were +performed for both the radial and meshed cases. Only normal +network operation was considered in the economic analysis +of flexibility provision, while configurations corresponding to +contingencies were omitted. +The optimal (least-cost) operation of the flexible units under +the radial network topology (NOP open) is visualised in +Fig. 3 as P-Q maps of flexible power regulations. For each +operating point, the maps specify which units should produce +or consume flexible power to minimise the total cost of +flexibility service, subject to the relevant voltage and thermal +network constraints. For example, as expected, the cheapest +unit, unit D, is activated for most flexibility service requests +(e.g., a large portion of the flexibility area is highlighted in +dark red and blue colours). However, this is not the case +when the aggregated flexibility requests correspond to high +power consumption (the top right areas highlighted in white +or light red colours). At such points, the voltage level at bus +33 drops down to 0.94 p.u., which makes further flexible +power consumption of unit D infeasible. Under such stressed +conditions, other (more expensive) units have to consume +flexible power, while unit D decreases its consumption or +even starts producing power to alleviate voltage problems. This +complex behavior is displayed in Fig. 3 as nonlinear changes +in the power output of the units.2 +The nonlinear, sometimes abrupt, changes in the optimal +power output of the flexible units pose two key challenges to +the cost-effective operation of ADN and flexibility markets. +Firstly, in practice, some units have ramp constraints that +would prevent deploying the optimal portfolio of units, and +would encourage the use of security margins to prevent +infeasible operation (e.g., exceeding voltage or thermal lim- +its). Secondly, information exchanges and strong coordination +between units would be required to facilitate the complex unit +coordination required to provide some flexibility services, es- +pecially under stressed network conditions (e.g., in the exam- +ple above, when unit D is used to manage voltage constraints). +Without such coordination, for instance, in decentralised peer- +to-peer flexibility markets, DSOs would provide much more +conservative flexible P-Q support for transmission systems. +The P-Q maps of the optimal flexible power regulations +for the meshed network topology (NOP closed) are presented +in Fig. 4. Compared to the results for the radial network +configuration, the cheapest unit D can now fully perform +flexible power regulation for most feasible operating points +(e.g., a larger portion of the area, compared with the radial +case, is highlighted in dark red and blue colours). The voltage +problems constraining the operation of unit D are now alle- +viated by network reconfiguration, making flexibility services +cheaper and the flexibility market more efficient. The nonlinear +rapid changes in the unit dispatch observed in Fig. 3 are +now smoothed out and shifted to more expensive units. It +follows that network reconfiguration can be used to reduce +the cost of flexibility services and ease network operation +and coordination between flexible units. A comprehensive +cost analysis of aggregated flexibility under different network +configurations is given in Fig. 5, where costs for feasible +operating points are displayed as surfaces. The flexibility cost +under the radial network operation (grey surface) increases +rapidly and unevenly for operating points with high power +consumption. Conversely, the cost function corresponding to +the meshed network operation (blue surface) grows slower and +2The observed complex nonlinear behavior of the flexible units is not the +product of numerical instability or solver convergence. The results have been +verified by other OPF formulations and solvers, e.g., for the radial network +configuration, the same results were obtained with the DistFlow OPF model +solved with Gurobi 10.0. + +Flexible unit C +(bus 12) +Flexible active power management, MW: +−2 +0 +2 +4 +6 +P, MW +−4 +−2 +0 +2 +4 +Q, MVAr +−2 +0 +2 +4 +6 +P, MW +−4 +−2 +0 +2 +4 +Q, MVAr +−2 +0 +2 +4 +6 +P, MW +−4 +−2 +0 +2 +4 +Q, MVAr +Flexible unit B +(bus 17) +−2 +0 +2 +4 +6 +P, MW +−4 +−2 +0 +2 +4 +Q, MVAr +−2 +0 +2 +4 +6 +P, MW +−4 +−2 +0 +2 +4 +Q, MVAr +Flexible unit A +(bus 24) +−2 +0 +2 +4 +6 +P, MW +−4 +−2 +0 +2 +4 +Q, MVAr +−2 +0 +2 +4 +6 +P, MW +−4 +−2 +0 +2 +4 +Q, MVAr +Flexible unit D +(bus 36) +−2 +0 +2 +4 +6 +P, MW +−4 +−2 +0 +2 +4 +Q, MVAr +Flexible reactive power management, MVAr: +-1.00 -0.50 +0 +0.50 +1.00 +Flexible power of units, MW or MVAr: +(consumption) +(production) +Network feasibility set (NOP open) +Initial operating point +Fig. 3. P-Q maps of the optimal flexible power regulations for the radial network topology (NOP open). The red-blue color scheme indicates operating points +where flexible units consume or produce active and reactive power, in MW and MVAr. +does not have cost spikes. The difference between the two +surfaces illustrates the potential improvement in the flexibility +market efficiency due to network reconfiguration. +D. Discussion: Computational Aspects and Applicability +In the above simulations, the computational challenges of +the aggregated flexibility estimation problem were overcome +by decomposing it into a small number of subproblems and +fixing corresponding binary reconfiguration variables. How- +ever, the proposed ACROPF model (1a)-(1p) is generally +hard to solve for meshed ADNs with multiple binary recon- +figuration variables and numerous flexible units [18]. Such +MIQCP models pose several challenges for solvers, such +as difficulties in finding a feasible solution (an incumbent +solution), convergence issues, and the inability to guarantee +the global optimum. For example, for the considered 38-bus +weakly meshed network, Gurobi 10.0 is able to solve the +cost-minimising problem correctly for some operating points, +but cannot correctly estimate the limits of the aggregated +flexibility (the feasibility area boundary). Larger systems with +more reconfiguration variables can create intractable problems. +Therefore, there is a need for testing different optimisation +algorithms, ACOPF formulations and their approximations for +meshed ADNs with flexible units. A discussion of flexibility +models programming and numerical issues can be found in +[9]–[11], [18], [19]. Note that existing ACOPF approximations +and relaxations should be used with caution, as they can lead +to inaccurate solutions and overestimation of ADN aggre- +gated flexibility. Future research can focus on developing a +tractable and accurate tool for estimating aggregated flexibility +in ADNs, considering the effects of network reconfiguration. +Additionally, the effects of including intertemporal constraints +and switching costs should be explored. +Deploying the proposed flexibility estimation framework in +real distribution systems and running the flexibility market +will require overcoming several practical challenges. First, +the security of flexibility provision and ADN operation must +be guaranteed. DSO cannot commit to flexibility services +that put a distribution network in a highly-stressed unreliable +operation. Reliability analysis must be performed to estimate +the impacts of potential contingencies or flexible units not +delivering the right amount of flexible power. Second, it is +necessary to introduce an information exchange system to +coordinate the actions of flexible units. As demonstrated by the +simulations, some units might need to perform fast regulations +or produce flexible power to alleviate network constraints. +Lack of unit coordination can result in an infeasible network + +Flexible unit C +(bus 12) +Flexible active power management, MW: +Flexible unit B +(bus 17) +P, MW +Flexible unit A +(bus 24) +Flexible unit D +(bus 36) +Flexible reactive power management, MVAr: +-1.00 -0.50 +0 +0.50 +1.00 +Flexible power of units, MW or MVAr: +(consumption) +(production) +Network feasibility set (NOP closed) +Initial operating point +−4 +−2 +0 +2 +4 +Q, MVAr +−2 +0 +2 +4 +6 +P, MW +−4 +−2 +0 +2 +4 +Q, MVAr +−2 +0 +2 +4 +6 +P, MW +−4 +−2 +0 +2 +4 +Q, MVAr +−2 +0 +2 +4 +6 +P, MW +−4 +−2 +0 +2 +4 +Q, MVAr +−2 +0 +2 +4 +6 +−4 +−2 +0 +2 +4 +Q, MVAr +−2 +0 +2 +4 +6 +P, MW +−4 +−2 +0 +2 +4 +Q, MVAr +−2 +0 +2 +4 +6 +P, MW +−4 +−2 +0 +2 +4 +Q, MVAr +−2 +0 +2 +4 +6 +P, MW +−4 +−2 +0 +2 +4 +Q, MVAr +−2 +0 +2 +4 +6 +P, MW +Fig. 4. P-Q maps of the optimal flexible power regulations for the meshed network topology (NOP closed). The red-blue color scheme indicates operating +points where flexible units consume or produce active and reactive power, in MW and MVAr. +normal operation: radial network (NOP open) +normal operation: meshed network (NOP closed) +initial operating point +500 +1000 +1500 +2000 +P, MW +Q, MVAr +0 +Cost, $/h +-2 +0 +2 +4 +6 +-4 +-2 +0 +2 +4 +Fig. 5. Total flexibility costs for different network configurations, $/h. +operation and power outage. Finally, relay protection schemes +must be upgraded to enable the operation of multiple flexible +units in meshed ADNs. +IV. CONCLUSION +This paper explores the operation of ADNs with flexible +resources and demonstrates the effects of different network +configurations on the aggregated DER flexibility. An accu- +rate ACROPF flexibility estimation model, formulated as a +MIQCP problem, is applied to a weakly meshed distribution +network with two feeders in the UK. Extensive numerical +simulations are performed for different network configurations +to compare the limits of the aggregated flexibility and the eco- +nomic efficiency of flexibility provision. The results illustrate +that changing network topology (in this case from radial to +meshed) enables increasing aggregated DER flexibility and +reducing the cost of flexible power. Therefore, DSOs can +use network reconfiguration as a tool to actively manage +and optimise available flexible resources. However, to deploy +this approach, multiple challenges have to be solved in the +future, such as the security of flexible power provision under +different network configurations, information exchange and +coordination between relay protection and flexible units, and +computational issues of flexibility estimation models. + +REFERENCES +[1] C. Eid, P. Codani, Y. Perez, J. Reneses, and R. Hakvoort, “Managing +electric flexibility from Distributed Energy Resources: A review of +incentives for market design,” Renewable Sustain. Energy Rev., vol. 64, +2016. +[2] T. Schittekatte and L. Meeus, “Flexibility markets: Q&A with project +pioneers,” Utilities Policy, vol. 63, 2020. +[3] A. G. Givisiez, K. Petrou, and L. F. Ochoa, “A Review on TSO-DSO +Coordination Models and Solution Techniques,” Electr. Power Syst. Res., +vol. 189, 2020. +[4] A. Sanjab, H. Le Cadre, and Y. 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Available: https://doi.org/10.25747/RE7B-ES77 +[18] F. Capitanescu, “A relax and reduce sequential decomposition rolling +horizon algorithm to value dynamic network reconfiguration in smart +distribution grid,” in 2017 IEEE PES Innovative Smart Grid Technolo- +gies Conference Europe (ISGT-Europe), 2017. +[19] L. Lopez, A. Gonzalez-Castellanos, D. Pozo, M. Roozbehani, and +M. Dahleh, “QuickFlex: a Fast Algorithm for Flexible Region Construc- +tion for the TSO-DSO Coordination,” in 2021 International Conference +on Smart Energy Systems and Technologies (SEST), 2021. + diff --git a/NtE4T4oBgHgl3EQfjg2I/content/tmp_files/load_file.txt b/NtE4T4oBgHgl3EQfjg2I/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ec0f54fb4349357c9c67a063f19814c0b39f5e5d --- /dev/null +++ b/NtE4T4oBgHgl3EQfjg2I/content/tmp_files/load_file.txt @@ -0,0 +1,468 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf,len=467 +page_content='Impacts of Distribution Network Reconfiguration on Aggregated DER Flexibility Andrey Churkin1, Miguel Sanchez-Lopez1,2, Mohammad Iman Alizadeh3, Florin Capitanescu3, Eduardo A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Mart´ınez Cese˜na1,4, Pierluigi Mancarella1,5 1Department of Electrical and Electronic Engineering, the University of Manchester, UK 2Departamento de Ingenieria Electrica, Universidad de Chile, Chile 3Luxembourg Institute of Science and Technology (LIST), Luxembourg 4Tyndall Centre for Climate Change Research, UK 5Department of Electrical and Electronic Engineering, the University of Melbourne, Australia {andrey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='churkin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' alex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='martinezcesena;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='mancarella}@manchester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='uk, {mohammad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='alizadeh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' florin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='capitanescu}@list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='lu Abstract—The ongoing integration of controllable distributed energy resources (DER) makes distribution networks capable of aggregating flexible power and providing flexibility services at both transmission and distribution levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The aggregated flexibility of an active distribution network (ADN) can be represented as its feasible operating area in the P-Q space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The limits of this area are pivotal for arranging flexibility markets and coordinating transmission and distribution system operators (TSOs and DSOs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' However, motivated by the current technical limitations of distribution networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=', protection schemes), existing literature on ADN flexibility and TSO-DSO coordination mostly focuses on radial networks, overlooking the potential benefits of network reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' This paper, using a realistic meshed distribution system from the UK and the exact ACOPF model for flexibility estimation, demonstrates that network reconfiguration can increase the limits of ADN aggregated flexibility and improve the economic efficiency of flexibility markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Index Terms—Active distribution network (ADN), distributed energy resources (DER), flexibility services, network reconfigu- ration, TSO-DSO coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' INTRODUCTION Modern distribution networks incorporate increasing amounts of distributed energy resources (DER) and continuously improve their observability and controllability [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' With these controllable flexible resources, active distribution networks (ADNs) now offer attractive means to aggregate flexible power and provide flexibility services at both distribution and transmission levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' To enable coordination between transmission and distribution system operators (TSOs and DSOs) and arrange TSO-DSO flexibility markets, it is necessary to assess the limits of aggregated flexibility that ADNs can deliver [2]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' This has motivated research on flexibility areas in the P-Q space to capture the aggregated ADN flexibility at the primary substation or TSO/DSO interface [5]–[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' However, despite the rapidly evolving research on ADN flexibility, existing studies mostly focus on radial distribution networks, overlooking the potential benefits of network reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' There have been This work was carried out as a part of the ATTEST project (the Horizon 2020 research and innovation programme, grant agreement No 864298).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' some attempts to quantify the value of ADN reconfiguration, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=', in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Yet, the impacts of network reconfiguration on aggregated DER flexibility and TSO-DSO coordination remain largely unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Distribution network reconfiguration is well known in power systems research as a tool for improving system security and optimising its operation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=', for loss reduction and voltage control [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' But, historically speaking, active network reconfiguration and meshed operation have not been economically attractive at the lowest voltage levels of the distribution networks, being typically considered only during emergency conditions [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' As a result, existing assets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=', protections) are generally only suitable for radial opera- tion, and existing literature on ADN flexibility has focused on radial networks [4], [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' However, network reconfigura- tion, including meshed options, can become more attractive with increasing DER integration and the rising interest in reducing power losses and environmental concerns [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' This paper demonstrates the benefits that network reconfig- uration can bring in terms of increased ADN flexibility and improved economic efficiency of flexibility markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' For this purpose, a mixed integer quadratically constrained program- ming (MIQCP) model is developed to estimate the limits of flexibility P-Q areas and the cost of ADN aggregated flexibility under different network configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The impacts of the reconfiguration are illustrated using a meshed UK distribution network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The rest of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Section II presents the proposed MIQCP flexibility estimation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The case study, which highlights how ADN flexibility changes subject to different configurations, is presented in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Finally, section IV concludes this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MODELLING FRAMEWORK In this section, a mathematical programming model is proposed to analyse the effects of different network config- urations on aggregated network flexibility (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=', at the primary substation or TSO/DSO interface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The model presented in (1a)-(1p) is a single-stage (single-period) network flexibility estimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' This formulation is based on an exact AC arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='05143v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='SY] 12 Jan 2023 optimal power flow model in rectangular voltage coordinates (ACROPF) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Variables ek and fk are the real and imaginary rectangular voltage components at bus k (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=', uk = ek +jfk), pij and qij stand for active and reactive power flows between buses i and j, and pk,g and qk,g denote the active and reactive power of generators located in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The power of each flexible unit f ∈ F is given by its available P-Q upward and downward regulation capacities indicated by the corresponding arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' It is worth noting that, in practice, the available upward and downward flexibility capacities would vary based on the initial operation of the flexible units before a flexibility service is requested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Finally, xij are binary variables representing the status of lines (if lines are switched on or off).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The model uses two sets of objective functions and con- straints designed for two purposes: map the flexibility area limits or deploy the cheapest flexible units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Objective function (1a) minimises or maximises network power consumption at a selected reference bus (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=', primary substation or TSO/DSO interface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Coefficients wp k and wq k are introduced to control the optimisation directions and can be used to iteratively reconstruct the feasible P-Q flexibility area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Objective function (1b) minimises the total cost of all flexible power regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' It is used to find the optimal dispatch decisions and analyse the efficiency of the flexibility market for a given operat- ing point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The first set of constraints, (1c)-(1n), defines the technical constraints of the network and flexible units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Active and reactive power flows are determined with (1c) and (1d), where Gij and Bij are the conductance and susceptance of lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Equations (1e) and (1f) represent active and reactive power balance for each node, where pk,d and qk,d are the loads at bus k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Line capacity limits and nodal voltage magnitude limits are imposed in (1g) and (1h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The output of generators and flexible units is constrained in (1i)- (1j) and (1k)-(1n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' This set of constraints, (1c)-(1n), together with the objective function (1a), enables estimating the limits of the aggregated network flexibility and approximating it as the network feasibility boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' When minimising the total flexibility cost with the objective function (1b), additional constraints (1o)-(1p) must be introduced to specify the selected feasible operating point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='1 The presented ACROPF flexibility estimation formulation is a MIQCP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' This problem is nonlinear and nonconvex, which imposes requirements on the solver: it must combine algorithms for both combinatorial optimisation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=', branch and bound) and nonlinear optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Solving the MIQCP flexibility estimation problem can be intractable for an arbi- trarily complex distribution network with an arbitrarily large number of flexible units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' However, realistic networks can have a limited number of possible configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' For example, as will be demonstrated in Section III, some distribution networks in the UK (typically 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='6 kV and 11 kV networks) have only one loop, a normally open point (NOP) connecting adjacent feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' For such cases, the MIQCP flexibility estimation prob- 1Note that the cost-minimising model does not consider intertemporal constraints and switching costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The inclusion of additional constraints and costs is the subject of future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MODEL ACROPF for flexibility estimation [MIQCP] Variables: ek, fk ∀k ∈ K pij, qij ∀(i, j) ∈ L pk,g, qk,g ∀k ∈ K, ∀g ∈ G p↑ k,f, p↓ k,f, q↑ k,f, q↓ k,f ∀k ∈ K, ∀f ∈ F xij ∈ � {0, 1}, if reconfiguration line, 1 other in-service lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Objective I: min wp kpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g + wq kqk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g k = kref (1a) Objective II: min � k∈K � f∈F Ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='f(p↑ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' p↓ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' q↑ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' q↓ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='f) (1b) Constraints I: pij = (ei 2 + fi 2)Gij − (eiej + fifj)Gij (1c) −(fiej − eifj)Bij ∀(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' j) ∈ L qij = −(ei 2 + fi 2)Bij + (eiej + fifj)Bij (1d) −(fiej − eifj)Gij ∀(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' j) ∈ L pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g − pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='d + p↑ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='f − p↓ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='f (1e) − � (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='j)∈L pkjxkj = 0 ∀k ∈ K qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g − qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='d + q↑ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='f − q↓ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='f (1f) − � (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='j)∈L qkjxkj = 0 ∀k ∈ K pij 2 + qij 2 ≤ Smax ij 2 ∀(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' j) ∈ L (1g) vmin k 2 ≤ ek 2 + fk 2 ≤ vmax k 2 ∀k ∈ K (1h) pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g min ≤ pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g ≤ pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g max ∀k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' ∀g (1i) qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g min ≤ qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g ≤ qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g max ∀k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' ∀g (1j) 0 ≤ p↑ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='f ≤ p↑max k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='f ∀k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' ∀f (1k) 0 ≤ p↓ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='f ≤ p↓max k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='f ∀k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' ∀f (1l) 0 ≤ q↑ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='f ≤ q↑max k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='f ∀k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' ∀f (1m) 0 ≤ q↓ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='f ≤ q↓max k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='f ∀k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' ∀f (1n) Constraints II: pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g = pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g ′ k = kref (1o) qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g = qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g ′ k = kref (1p) lem can be decomposed into continuous QCP subproblems corresponding to possible network configurations and solved with software for nonlinear continuous systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' such as Ipopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' In the next section, the flexibility estimation model (1a)-(1p) will be applied to analyse the effects of network reconfigu- ration on the aggregated network flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Changes in both the technical limits of network flexibility and the economics of flexibility provision will be traced and discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Additionally, the application of the exact ACROPF model to a realistic distribution network will demonstrate the nonlinearities and nonconvexities of ADN operation and highlight potential co- ordination issues of flexible units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' RESULTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Case Study: 38-bus Synthetic Distribution Network The impacts of network reconfiguration on the aggregated DER flexibility are illustrated with a 38-bus 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='6 kV UK distribution network [17], which has been anonymised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The network is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' 1 as a graph, where the sizes of the nodes represent power demand or generation at each bus, and lengths of the edges are proportional to the impedance between buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Flexible operation of this network is challenging due to potential voltage problems and congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' For example, feeder 2 has buses with high demand, which causes a signif- icant voltage drop and restricts flexible power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Conversely, feeder 1 has uncontrollable generation at buses 11, 17, 18, 19, and 20, which increases the voltage profile and dictates the power flows in the feeder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Four flexible units (denoted as FU in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' 1) are placed in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Their characteristics are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' It is assumed that units can produce and consume power, which is consistent with DER technologies such as battery storage and flexible loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The network includes two radial feeders that can be inter- connected through the NOP (typically for customer restoration purposes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' That is, there are two sectionalising switches and one tie switch, which enables dynamic network reconfigura- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Excluding cases where some customers are isolated from the system, there are four possible network configurations to consider: (i) normal operation with NOP open (radial network), (ii) normal operation with NOP closed (meshed network), (iii) contingency power supply via feeder 1 (radial network, NOP closed, line 8-1 open), and (iv) contingency power supply via feeder 2 (radial network, NOP closed, line 8- 7 open).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The limited number of possible configurations makes it possible to decompose the flexibility estimation problem (1a)-(1p) into four subproblems, which will be explored in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' In this work, the ACROPF model was programmed in JuMP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='0 for Julia 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='1 programming language, then decomposed into continuous QCP subproblems for the four configurations, and solved with Ipopt 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='4 solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' TABLE I PARAMETERS OF FLEXIBLE UNITS PLACED IN THE NETWORK Parameter Flexible unit A B C D Bus # 24 17 12 36 P regulation limits (MW) ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='0 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='0 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='0 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='0 Q regulation limits (MVAr) ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='0 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='0 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='0 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='0 P cost ($/MWh) 375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='0 350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='0 325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='0 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='0 Q cost ($/MVArh) 187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='5 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='0 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='5 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='0 1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 5 primary substation 0 MVA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='683 MVA Reconfiguration lines: switch closed switch open Power demand/generation: FU FUD FUC FUB FUA Nodes with flexible units: feeder 1 feeder 2 NOP Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Case study: 38-bus distribution network with two feeders and four flexible units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Aggregated Flexibility Limits For each of the four considered network configurations, model (1a), (1c)-(1n) was solved iteratively 200 times (for 200 extreme operating points, with a step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='08 MVA) to approximate the boundary of the network feasibility area at the primary substation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The flexibility areas estimated by model (1a), (1c)-(1n) for the 38-bus system, subject to the four selected configurations, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The coordinates represent the total active and reactive power consumption of the network, measured at the primary substation, and the cross marker corresponds to the initial operating point with all flexible units switched off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Note that the slightly curved horizontal and vertical lines correspond to unconstrained net- work operation where flexible units can fully deploy their P- Q regulation capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Other inclined lines reflect the con- strained network operation where flexible units cannot be fully activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Considering normal network operation under radial and meshed topologies, the flexibility P-Q areas are similar for most operating points, except for the upper right part of the areas (increased active and reactive power consumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' When the NOP is open (radial case), the voltage profile of feeder 2 has a significant drop and is close to the lower voltage limit of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='94 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Therefore, in this case, flexible units A and D cannot fully increase their power consumption due to voltage constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' When the NOP is closed (meshed case), voltages in feeder 2 increase which expands the limits of the aggregated flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' In addition to improving the voltage profile, meshing radial networks increases the capacity for transferring flexible power, making the operation of distant flexible units less constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The difference in the feasibility −2 0 2 4 6 P, MW −4 −2 0 2 4 Q, MVAr normal operation: radial network (NOP open) normal operation: meshed network (NOP closed) contingency: supply via feeder 1 (line 8-1 open) contingency: supply via feeder 2 (line 8-7 open) initial operating point N-1 secure flexibility provision Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Boundaries of the feasibility areas for different network configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' areas demonstrates that network reconfiguration (meshing in this case), besides offering options to manage power losses and voltages, can be used to maximise aggregated flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The results show that the flexibility area greatly decreases in the last two cases (contingencies when the network is supplied only with feeder 1 or 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' This happens due to voltage problems at busses 18 and 33 and congestion of lines 7-14 and 3- 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The “N-1 secure” flexibility area can be estimated as the intersection of areas under different network configurations, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' 2 with a green polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Considering the effects of network reconfiguration, DSO can impose restrictions on some flexibility services to guarantee the security of flexible power provision and ADN operation against contingencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Economics of Flexibility Provision Different network configurations can affect the economics of flexible power provision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' To explore these economic effects, model (1b)-(1p) was solved for 25,000 feasible operating points (with a step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='05 MVA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The simulations were performed for both the radial and meshed cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Only normal network operation was considered in the economic analysis of flexibility provision, while configurations corresponding to contingencies were omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The optimal (least-cost) operation of the flexible units under the radial network topology (NOP open) is visualised in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' 3 as P-Q maps of flexible power regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' For each operating point, the maps specify which units should produce or consume flexible power to minimise the total cost of flexibility service, subject to the relevant voltage and thermal network constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' For example, as expected, the cheapest unit, unit D, is activated for most flexibility service requests (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=', a large portion of the flexibility area is highlighted in dark red and blue colours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' However, this is not the case when the aggregated flexibility requests correspond to high power consumption (the top right areas highlighted in white or light red colours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' At such points, the voltage level at bus 33 drops down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='94 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=', which makes further flexible power consumption of unit D infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Under such stressed conditions, other (more expensive) units have to consume flexible power, while unit D decreases its consumption or even starts producing power to alleviate voltage problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' This complex behavior is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' 3 as nonlinear changes in the power output of the units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='2 The nonlinear, sometimes abrupt, changes in the optimal power output of the flexible units pose two key challenges to the cost-effective operation of ADN and flexibility markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Firstly, in practice, some units have ramp constraints that would prevent deploying the optimal portfolio of units, and would encourage the use of security margins to prevent infeasible operation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=', exceeding voltage or thermal lim- its).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Secondly, information exchanges and strong coordination between units would be required to facilitate the complex unit coordination required to provide some flexibility services, es- pecially under stressed network conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=', in the exam- ple above, when unit D is used to manage voltage constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Without such coordination, for instance, in decentralised peer- to-peer flexibility markets, DSOs would provide much more conservative flexible P-Q support for transmission systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The P-Q maps of the optimal flexible power regulations for the meshed network topology (NOP closed) are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Compared to the results for the radial network configuration, the cheapest unit D can now fully perform flexible power regulation for most feasible operating points (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=', a larger portion of the area, compared with the radial case, is highlighted in dark red and blue colours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The voltage problems constraining the operation of unit D are now alle- viated by network reconfiguration, making flexibility services cheaper and the flexibility market more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The nonlinear rapid changes in the unit dispatch observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' 3 are now smoothed out and shifted to more expensive units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' It follows that network reconfiguration can be used to reduce the cost of flexibility services and ease network operation and coordination between flexible units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' A comprehensive cost analysis of aggregated flexibility under different network configurations is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' 5, where costs for feasible operating points are displayed as surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The flexibility cost under the radial network operation (grey surface) increases rapidly and unevenly for operating points with high power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Conversely, the cost function corresponding to the meshed network operation (blue surface) grows slower and 2The observed complex nonlinear behavior of the flexible units is not the product of numerical instability or solver convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The results have been verified by other OPF formulations and solvers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=', for the radial network configuration, the same results were obtained with the DistFlow OPF model solved with Gurobi 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Flexible unit C (bus 12) Flexible active power management,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MW: −2 0 2 4 6 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MW −4 −2 0 2 4 Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MVAr −2 0 2 4 6 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MW −4 −2 0 2 4 Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MVAr −2 0 2 4 6 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MW −4 −2 0 2 4 Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MVAr Flexible unit B (bus 17) −2 0 2 4 6 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MW −4 −2 0 2 4 Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MVAr −2 0 2 4 6 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MW −4 −2 0 2 4 Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MVAr Flexible unit A (bus 24) −2 0 2 4 6 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MW −4 −2 0 2 4 Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MVAr −2 0 2 4 6 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MW −4 −2 0 2 4 Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MVAr Flexible unit D (bus 36) −2 0 2 4 6 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MW −4 −2 0 2 4 Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MVAr Flexible reactive power management,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MVAr: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='00 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='00 Flexible power of units, MW or MVAr: (consumption) (production) Network feasibility set (NOP open) Initial operating point Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' P-Q maps of the optimal flexible power regulations for the radial network topology (NOP open).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The red-blue color scheme indicates operating points where flexible units consume or produce active and reactive power, in MW and MVAr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' does not have cost spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The difference between the two surfaces illustrates the potential improvement in the flexibility market efficiency due to network reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Discussion: Computational Aspects and Applicability In the above simulations, the computational challenges of the aggregated flexibility estimation problem were overcome by decomposing it into a small number of subproblems and fixing corresponding binary reconfiguration variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' How- ever, the proposed ACROPF model (1a)-(1p) is generally hard to solve for meshed ADNs with multiple binary recon- figuration variables and numerous flexible units [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Such MIQCP models pose several challenges for solvers, such as difficulties in finding a feasible solution (an incumbent solution), convergence issues, and the inability to guarantee the global optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' For example, for the considered 38-bus weakly meshed network, Gurobi 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='0 is able to solve the cost-minimising problem correctly for some operating points, but cannot correctly estimate the limits of the aggregated flexibility (the feasibility area boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Larger systems with more reconfiguration variables can create intractable problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Therefore, there is a need for testing different optimisation algorithms, ACOPF formulations and their approximations for meshed ADNs with flexible units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' A discussion of flexibility models programming and numerical issues can be found in [9]–[11], [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Note that existing ACOPF approximations and relaxations should be used with caution, as they can lead to inaccurate solutions and overestimation of ADN aggre- gated flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Future research can focus on developing a tractable and accurate tool for estimating aggregated flexibility in ADNs, considering the effects of network reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Additionally, the effects of including intertemporal constraints and switching costs should be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Deploying the proposed flexibility estimation framework in real distribution systems and running the flexibility market will require overcoming several practical challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' First, the security of flexibility provision and ADN operation must be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' DSO cannot commit to flexibility services that put a distribution network in a highly-stressed unreliable operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Reliability analysis must be performed to estimate the impacts of potential contingencies or flexible units not delivering the right amount of flexible power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Second, it is necessary to introduce an information exchange system to coordinate the actions of flexible units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' As demonstrated by the simulations, some units might need to perform fast regulations or produce flexible power to alleviate network constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Lack of unit coordination can result in an infeasible network Flexible unit C (bus 12) Flexible active power management, MW: Flexible unit B (bus 17) P, MW Flexible unit A (bus 24) Flexible unit D (bus 36) Flexible reactive power management, MVAr: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='00 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content='00 Flexible power of units,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MW or MVAr: (consumption) (production) Network feasibility set (NOP closed) Initial operating point −4 −2 0 2 4 Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MVAr −2 0 2 4 6 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MW −4 −2 0 2 4 Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MVAr −2 0 2 4 6 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MW −4 −2 0 2 4 Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MVAr −2 0 2 4 6 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MW −4 −2 0 2 4 Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MVAr −2 0 2 4 6 −4 −2 0 2 4 Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MVAr −2 0 2 4 6 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MW −4 −2 0 2 4 Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MVAr −2 0 2 4 6 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MW −4 −2 0 2 4 Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MVAr −2 0 2 4 6 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MW −4 −2 0 2 4 Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MVAr −2 0 2 4 6 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' MW Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' P-Q maps of the optimal flexible power regulations for the meshed network topology (NOP closed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The red-blue color scheme indicates operating points where flexible units consume or produce active and reactive power, in MW and MVAr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' normal operation: radial network (NOP open) normal operation: meshed network (NOP closed) initial operating point 500 1000 1500 2000 P, MW Q, MVAr 0 Cost, $/h 2 0 2 4 6 4 2 0 2 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Total flexibility costs for different network configurations, $/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' operation and power outage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Finally, relay protection schemes must be upgraded to enable the operation of multiple flexible units in meshed ADNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' CONCLUSION This paper explores the operation of ADNs with flexible resources and demonstrates the effects of different network configurations on the aggregated DER flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' An accu- rate ACROPF flexibility estimation model, formulated as a MIQCP problem, is applied to a weakly meshed distribution network with two feeders in the UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Extensive numerical simulations are performed for different network configurations to compare the limits of the aggregated flexibility and the eco- nomic efficiency of flexibility provision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' The results illustrate that changing network topology (in this case from radial to meshed) enables increasing aggregated DER flexibility and reducing the cost of flexible power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} +page_content=' Therefore, DSOs can use network reconfiguration as a tool to actively manage and optimise available flexible resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE4T4oBgHgl3EQfjg2I/content/2301.05143v1.pdf'} 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Reference Editor for Kannada +(Based on OPOK! and OHOK! principles, and Domain Knowledge) +Vishweshwar V. Dixit +KRIA, Kannada Research Institute of America +714-322-9748 namovish@kannadakali.com; namovish@gmail.com +Abstract +Kannudi is a reference editor for Kannada based on +OPOK! +and +OHOK! +principles, +and +domain +knowledge. It introduces a method of input for +Kannada, called OHOK!, that is, Ottu Hāku Ottu Koḍu! +(apply pressure and give ottu). This is especially suited +for pressure sensitive input devices, though the current +online implementation uses the regular mechanical +keyboard. OHOK! has three possible modes, namely, +sva-ottu (self-conjunct), kandante (as you see), and +andante (as you say). It may be noted that kandante +mode does not follow the phonetic order. However, this +mode may work well for those who are inclined to +visualize as they type rather than vocalizing the sounds. +Kannudi also demonstrates how domain knowledge can +be effectively used to potentially increase speed, +accuracy, and user friendliness. For example, selection +of a default vowel, automatic shunyification, and +arkification. Also implemented are four types Deletes +that are necessary for phono-syllabic languages like +Kannada. +Kannudi can be accessed at +https://kannadakali.com/kannudi/kannudi.html +1 Introduction +Many tools are available for digital inputting of +Kannada and other Indian language text, such as Input +Method Editors (IME), and real time transliteration +tools, free and commercial, online as well as offline. +Most of these are generic in the sense that they are +designed to address all Indian languages. That makes +sense as the scripts for these languages, as they all +descended from the same Brahmi script, share many +common features such as being alpha-syllabic. +However, there are subtle differences in the writing +styles of these languages. These language specifics can +be used to “optimize” and make the editors and IMEs +more efficient and user friendly. An earliest +implementation of one such editor attempting to use the +‘domain knowledge’ was described by Dixit [1] [2] [3]. +It was an ambitious effort as it aimed for a universal +framework. It identified notions such as Śūn'yīfication, +Non-initial Vowel, and hinted at a-rule for voice input. +However, the implementation was limited to Kannada +and limited in scope of the features. This was a DOS +based editor and no versions were released later on +Windows or other platforms. Another notable DOS +editor was developed in Visual Basic by Rangachar [4]. +Described in this article is a reference implementation +introducing a new method of input for ottaksharas +(conjuncts) while incorporating previous ideas in +Chitragupta and Unived [1] [2]. +2 Input Methods +Keyboarding, being the norm, is a required method of +input. A phonetic input method generally assigns a +single key to a single phoneme and the keys are typed +in the order of pronunciation. A strict one phoneme - +one key (OPOK!) mapping of keys to phonemes may +not desirable for the sake of convenience. Some may +assign multiple combinations of 1-3 keys to a single +phoneme for convenience to resemble common writing +using Latin script. For example, one may assign B, bh, + +2 + +Bh, or BH for ಭ (mahāprāṇa b) for the sake of +convenience. +A mapping of letters to (ASCII) keys was developed by +Kannada Ganaka Parishattu (KGP) and is used in its +Nudi editor [5]. This has been designated as the official +standard by Government of Karnataka [6]. +A context sensitive dynamic keyboarding has been +described by Joshi et.al. [7] +Handwriting, using stylus on a tablet or mobile screen, +is another method of input. The complexity of graphics +processing makes it slow. Variations in individual +writing styles also introduces errors. Correcting these +errors as one goes takes time and interrupts the thought +process of the user which reduces the speed. +Voice input is a promising method. However, Voice +recognition is not perfect. This is especially problematic +in Kannada where regional and other variations in +pronunciations +abound. +Additionally, +current +implementations are mostly dictionary based. They +suffer from the same difficulties with all Indian +languages, namely, variations in pronunciation, +dictionary limitations, and indetermination between +writing as pronounced and the dictionary entries. +Current voice input implementations may make it more +difficult to type. One needs to keep looking constantly +at the words being entered, select among the choices, or +correct manually. A user seems to spend significant +time in ‘correcting’ the dictionary words and ultimately, +being frustrated, ends up using or taking help of a basic +keyboard layout such as Inscript. Thus, voice input, +though convenient, current implementations fail the +user in both accuracy and speed. +3 Improvement Opportunities +Three main considerations in any input method are +accuracy, speed, and convenience. Appropriate tradeoff +among the three is also of concern. Elimination or +minimization of corrections (backspace and deletes) +and minimization of required keystrokes become +important parameters. Certainly, there is a need and +room to improve the existing methods in these regards. +Current implementations aim and try to cater to all +Indian languages and therein implementing a set of +common minimum features. They become minimally +useful, as most users are not interested in 15+ +languages. Hence, an implementation must be +extensible so that domain knowledge specific to each +language and script can be incorporated +Mobile platforms present interesting opportunities for +novel methods using soft keyboards, dynamic context, +and new input mechanisms such as swipes. +4 Letter Frequencies +Whereas earlier studies have found 36% mūla akṣaras +(vowel a or C+a) and 14% conjuncts, using a sample of +articles from Kannada Wikipedia [8] and Kannada Kali +[9],we found the letter frequencies as shown in Table 1. +moola akshara +42.4% +Gunitakshara +39.3% +anusvāra / sonne / śūn'ya +5.6% +Ottu and end virāma +18.2% + Sajāti/Dvitva (self-conjunct) +8.7% + Dvitva post-vowel +1.1% + Vijāti (non-self-conjunct) +8.8% + End-virāma +0.7% +Table 1: Syllable Frequencies +5 Kannudi +Kannudi [10] is a reference editor introducing several +innovative and experimental features. Currently, an +online implementation invoked via a web browser, is +available. Some features in this implementation require +a keyboard. Kannudi input method follows the phonetic +order, i.e., phonemes are entered in the order of their +pronunciation. +Major principles in Kannudi are OPOK!, OHOK!, user +friendliness, prevention/minimization of errors and +illegal combinations and letter formations. + +3 + +5.1 OPOK! Principle +First major principle in Kannudi implementation of One +Phoneme One Key (OPOK!) Here, keys are assigned to +phonemes, not graphemes; and further, a one-one +correspondence exists between a key and a phoneme. +Key assignments adhere to the standard specified by +Government of Karnataka. + + ೧ + 1 ೨ + 2 ೩ + 3 ೪ + 4 +೫ + 5 +೬ + 6 +೭ + 7 +೮ + 8 +೯ + 9 +೦ + 0 + + ಟ + q ಡ + w ಎ + e +ರ + r +ತ + t +ಯ + y +ಉ + u +ಇ + i +ಒ + o ಪ + p + + +ಅ + a ಸ + s ದ + d ್ + f +ಗ + g +ಹ + h +ಜ + j +ಕ + k ಲ + l + +Shift ⇧ ಙ + z ಷ + x +ಚ + c +ವ + v +ಬ + b +ನ + n +ಮ + m +Figure 1. Keyboard Layout – non-Shift State + + + ಠ + Q ಢ + W ಏ + E ಋ + R +ಥ + T +ಐ + Y ಊ + U +ಈ + I +ಓ + O ಫ + P + + +ಆ + A +ಶ + S +ಧ + D + + F ಘ + G ್ + H +ಝ + J +ಖ + K ಳ + L + +Shift ⇧ ಞ + Z + + X ಛ + C +ಔ + V ಭ + B +ಣ + N +ಂ + M +Figure 2. Keyboard Layout – Shift State + +5.2 Default Vowel – null ಂ or a ಅ +A pure phonetic method of input would be to type every +vowel and consonant in the order pronounced. This +assumes no vowel inherently attached to a consonant in +the alphabet. Thus, with OPOK! in force, ಕ್ needs only +one keystroke but each kāguṇita ka kā ki... kau ಕ, ಕಾ, ಕಿ, +...ಕೌ requires 2 keystrokes. +As can be seen from the Table 1, majority of the +syllables are mūla akṣaras, i.e., independent vowels and +consonants with vowel a ಅ. Hence, in the alphabet, the +graphemes for consonants have been designed with an +assumed or default vowel (ūhita svara) a ಅ. +Ottu is the secondary form of a consonant that appears +in a conjunct (sanyuktākṣara). An otttu is produced +when there is no vowel between the two consonants, +indicated by null vowel or virama ್ . +If the default vowel is virama ್ , then one simply types +consonant keys in succession. However, if the default +vowel a ಅ, then the null vowel must be typed in with an +additional keystroke. For example, to type ಕತ three keys +need to be pressed as shown in Table 2. +It adds an “extra” key, thus significantly negating the +savings provided by the default vowel. Even then, +having a ಅ as the default vowel saves 24% of +keystrokes (42-18=24%) compared to having no default +vowel (or assuming null ್ as the default vowel). +However, one may find this somewhat unnatural and +not a pure phonetic method, and experience a loss in the +speed of typing. +Default +Vowel +a ಅ + +null ್ +Key pressed +k +f +t + +k +t +a +Result +ಕ ಕ್ ಕತ +ಕ್ ಕ್ತ ಕತ +Table 2: Typing an ottakshara conjunct +As such, Kannudi provides a choice of default vowels a +ಅ and null ್ . A user can choose one as a preference +or switch between the two as suitable. +5.3 OHOK! Principle +OHOK! uses pressure as input to produce secondary +forms of consonants (ottu). It is to apply pressure or +press and hold a key to produce an ottu. +In case of mechanical keyboards, which are not pressure +sensitive, OHOK!, which can be thought of as Otti Hidi +Ottu Kodu! ಒತ್ತತ ಹಿಡಿ ಒತತತ ಕ ೊಡತ!, is simply time based, +that is, dependent on how long the key is held pressed. +Though keyboard timings, touch, and pressure +sensitivities can be optimized to speed OHOK! we +understand that it may be beyond the normal +capabilities of a user. + +4 + +In case of pressure sensitive (mobile) devices, OHOK! +is called Ottu Hāku Ottu Koḍu! ಒತತತ ಹಾಕತ ಒತತತ ಕ ೊಡತ! +(apply pressure and give ottu). This offers greater +potential for savings in time as well as key strokes. +OHOK! can have three modes. Here OHOK! of a key, +namely applying pressure (or holding pressed) is +denoted by superscript + sign. +1. Sva-ottu Mode (SO): This is Self-Ottu, also +known as Dvitva where a consonant gets its +own secondary form (ottu) attached. For +example, k+ will produce ಕ್ಕ. And kn+w, +equivalent to typing knfnw, produces ಕನನಡ. +2. Kaṇḍante Ottu Mode (KO): This is visual +mode where the input follows the written order +– holding a consonant will add its ottu to +preceding consonant. For example, the key +sequence st+r+I (= sftfrI) produces ಸ್ತ್ರೀ. +3. Andante Ottu Mode (AO): This is “as you say” +or phonetic mode where the input follows the +order of pronunciation. Pressing and holding +(ಒತ್ತತ ಹಿಡಿ) of a consonant key will prepare it for +an ottu by adding the null vowel (virama). This +agrees with the order of pronunciation as the +ottu (accent) is on this consonant and the next +consonant is turned into an ottu. Here k+ is +equivalent to kf. To produce ಸ್ತ್ರೀ enter s+t+rI. In +essence this is an alternative to f key in normal +mode. + + + +Default Vowel = a ಅ +Syllable Type + +Mode +Normal +OHOK! SO +Dvitva +OHOK! KO +Kandante +OHOK! AO +Andante +Self-conjunct +(dvitva) +Keys Typed +k +f +k +k+ + + +k +k+ +k+ +k +Display +ಕ +ಕ್ +ಕಕ +ಕಕ + + +ಕ +ಕಕ +ಕ್ +ಕಕ +Non-self- +conjunct +Keys Typed +g +f +r +g +f +r +g +r+ +g+ +r +Display +ಗ +ಗ್ +ಗರ +ಗ +ಗ್ +ಗರ +ಗ +ಗರ +ಗ್ +ಗರ +End-Virama +Keys Typed +n +f + +n +f + +n +f +n+ + +Display +ನ +ನ್ + +ನ +ನ್ + +ನ +ನ್ +ನ್ + +V[MH]? + self- +conjunct +Keys Typed +ak +f +k +ak+ + + +ak + k+ +ak+ +k +Display +ಅಕ ಅಕ್ ಅಕಕ ಅಕಕ + + +ಅಕ +ಅಕಕ +ಅಕ್ +ಅಕಕ +Keys saved per 1000 syllables +0 +174 +196 +193 +Table 3: Key savings with a-default + + + + +5 + +Default Vowel = null ್ +Syllable Type + +Mode +Normal +OHOK! +Dvitva +OHOK! KO +Kandante +OHOK! AO +Andante +Self-conjunct +(dvitva) +Keys Typed +k +k +k+ + +k +k+ +k+ +k +Display +ಕ್ +ಕ್ಕ +ಕ್ಕ + +ಕ್ +ಕ್ಕ +ಕ್ +ಕ್ಕ +Non-self- +conjunct +Keys Typed +g +r +g +r +g +r+ +g+ +r +Display +ಗ್ +ಗ್ರ +ಗ್ +ಗ್ರ +ಗ್ +ಗ್ರ +ಗ್ +ಗ್ರ +Virama +Keys Typed +n + +n + + + +n+ + +Display +ನ್ + +ನ್ + + + +ನ್ + +V[MH]? + +self-conjunct +Keys Typed +ak +k +ak+ + +ak+ + +ak+ +k +Display +ಅಕ್ +ಅಕ್ಕ +ಅಕ್ಕ + +ಅಕ್ಕ + +ಅಕ್ +ಅಕ್ಕ +keys saved per 1000 syllables +0 +87 +11 +0 +Table 4: Key Savings with null-default +5.4 Key Savings +Table 3 shows the key savings, with a-default, for +various conjunct syllable types for the three modes of +OHOK!. For example, a dvitva requiring 3 keys +normally, can be produced with only 1 key in dvitva +mode (SO). +Similarly, Table 4 lists the key savings with null- +default. +These tables also show the keys saved per 1000 +syllables in a typical document, based on the frequency +data in Table 1. Thus, under a-default, key savings of +174 can be achieved in OHOK! Dvitva mode (SO). +It may be noted that, most key savings occur in a- +default mode while the differences among the three +OHOK! modes remain insignificant. +Another scheme: when a key is pressed and held next +key becomes an ottu. e.g press-holding a k then typing +r will produce ಕರ in a-default-mode. This scheme has +physical limitations due to positions of keys being fixed +on a keyboard. It becomes necessary to cross hands or +fingers or quickly decide which hand to use for a key. +This is contrary to the trained typist who expects to +blindly use the same finger at the same physical location +for a given key. It can be physically impossible or +totally +confusing. +Hence +this +scheme +is +not +implemented in Kannudi. +6 Rules of Convenience +Apart from introducing the novel input method, +Kannudi implements several user-friendly features that +are simply matter of convenience or eliminate or reduce +errors. A few such features are described here. +6.1 Backspace ←BS +During normal course of tying, a user may have typed a +key in error or pressed an adjacent key. When the +mistake is realized, the normal action is to press +backspace key BS. This usually deletes the previous +syllable entirely. If in midst of a multi-phoneme +syllable, all the effort is lost, when the user desires to +undo just last entry or the mistake just made. Hence, +Kannudi recognizes 4 types of deletions: + +6 + +1. Phoneme +Delete: +delete +the +phoneme +immediately left of the cursor, assigned to +←BS. +2. Character Delete: delete the (unicode) character +immediately left of the cursor, assigned to +alt←BS, +3. Syllable +Delete: +delete +the +syllable +immediately left of the cursor, assigned to +shift←BS, +4. Word Delete: delete the word immediately left +of the cursor, assigned to ctrl←BS. +6.2 Shunyification +Sonne or śūn'ya is used in writing to represent an +anunāsika before a consonant as in ಅಂಕ, though it is not +incorrect to write ಅಙಕ. Thus, sonne before a classified +consonant is pronounced as the anunāsika of the same +class. The process of automatic conversion of +anunāsika before a consonant, classified or non- +classified, to a sonne is called shunyification. +For the sake of convenience, and only n/m keys are +considered for Śūn'yīfication in this implementation. +Śūn'yīfication makes the input entry flexible by +allowing both n and m to be automatically Śūn'yīfied. +Śūn'yīfication is straightforward in case of a classified +consonant but can be ambiguous in case of a non- +classified consonant and exceptions occur. +In most cases when sonne precedes a non-classified +consonant (avargīya vyan̄ jana), Śūn'yīfication allows +the entry to be phonetic corresponding to how most +people pronounce. For example, ಸಂಶಯ is pronounced +as ಸಮಶಯ samśaya and ಸಂಸೃತ as sanskr̥ ta or samskr̥ ta +by many, albeit all incorrectly. +Śūn'yīfication does not save any keystrokes; But does +not require one to switch the flow between andante (as +pronounced) and kaṇḍante (as seen). This is especially +convenient for those who are mentally “spelling” the +Kannada words in roman script as they type. And there +are many such casual users. +6.3 Arkification +Though arkāvottu ೯is used mostly in place of ರ್, there +are a few situations where it is not desirable as in ರಾ‍ಯಂಕ, +ಸರ‍ರನ . Kannudi automatically uses the correct form +(arkifies) in such cases allowing the user to type +normally without hindrance. +7 Error Prevention Rules +Certain domain knowledge of the language can be used +to prevent typos and warn the user. A few examples +are described below. +7.1 Non-initial Vowel +Kannada allows a standalone vowel only in beginning +of word. Kannudi prevents such typing. +7.2 Aspirated Ottu +It is not possible to pronounce an aspirated consonant +(mahāprāṇa) when followed by another aspirated +consonant. As such Kannada does not allow a +mahāprāṇa ottu to another mahāprāṇa. However, +certain words can be exceptions due to common usage, +e.g., ವಿಠಠಲ viṭhṭhala. +8 Exception Handling +Convenience and error prevention rules may not be +perfect, and exceptions can be found as mentioned +earlier. Hence it is necessary to ensure that there is a +mechanism to override the normal behavior. Two ways +to override are a) using ZWJ/ZWNJ characters and b) +inputting the phonemes with a space character in +between and then remove it. +9 Conclusions +Here we have introduced an input method called +OHOK! with three possible modes, namey, sva-ottu +(self-conjunct), kaṇḍante (as seen), and andante (as +pronounced/said). It may be noted that kaṇḍante mode +(KO) does not follow the phonemic order. However, +this mode may work well for those who are inclined to +visualize as they type rather than vocalizing the sounds. +OHOK! will work very well on mobile or any device +where input pressure can be sensed where OHOK! can + +7 + +be called Ottu Hāku Ottu Koḍu! (Apply Pressure and +Give Ottu) when implemented on pressure sensitive +devices. it where can be a real time saver. On +mechanical keyboards, it saves keystrokes though any +time saved is dependent on keyboard settings. +We have showed that domain knowledge can be used to +improve user friendliness. Several convenience and +error minimization rules such as Śūn'yīfication and +arkāvottu are described. Four types of deletions, namely +phoneme, character, syllable, and word delete are +identified and assigned to backspace key. Thus, domain +knowledge is shown to be necessary and helpful to +enhance user friendliness as wells as input flow and +speed. +Further, one may consider incorporating these rules into +open type font tables as attached language specific +resources, and eliminate the need for a separate editor +application. +10 References + +[1] V. Dixit, “An Intelligent Screen Editor for Kannada +and Other Indian Languages,” SAIL: A Journal of +Society for the Advancement of Indian Languages, +pp. 12-14, January 1986. +[2] ವಿ. ದೀಕ್ಷಿತ, “ಚಿತರಗುಪ್ತ: ಭಾರತೀಯ ಭಾಷೆಗಳಿಗೆ ಕ್ ಪ್ಯೂಟರ್ +ಮೀಲೆ ಅಚ್ುುಕ್ಟುುವ ಒ ದು ಪ್ದಧತ,” ಅಮರಿಕ್ನ್ನಡ, p. ೫೧, +೨೮ ಏಪ್ರರಲ್ ೧೯೮೫. +[3] ವಿ. ದೀಕ್ಷಿತ, “ಚಿತರಗುಪ್ತ: ಭಾರತೀಯ ಭಾಷೆಗಳಿಗೆ ಕ್ ಪ್ಯೂಟರ್ +ಮೀಲೆ ಅಚ್ುುಕ್ಟುುವ ಒ ದು ಪ್ದಧತ,” ಅಮರಿಕ್ನ್ನಡ, no. ೫, pp. +೪೨-೪೩, ಏಪ್ರರಲ್-ಮೀ ೧೯೮೫. +[4] ಕ್. ರ ಗಾಚಾರ್, “ನ್ನ್ಸಾದ ಕ್ನ್ಸು - ಕ್ನ್ನಡಕೆ್ಕ ದು ಸುಲಭ +ಸಾಧ್ೂ ಕ್ ಪ್ಯೂಟರ್,” ಅಮರಿಕ್ನ್ನಡ, no. 4, pp. ೪೮-೪೯, +ಫೆಬ್ುರವರಿ ೧೯೮೫. +[5] “ನ್ುಡಿ ತ ತಾರ ಶಗಳು,” ಕ್ನ್ನಡ ಗಣಕ್ ಪ್ರಿಷತುತ, +http://www.kagapa.in/kannada/content/ತ ತಾರ ಶಗಳು. +[6] “ಶಿಷುತೆ-ಮತುತ-ಏಕ್ರ್ಪ್ತೆ,” ಕ್ನ್ನಡ ಗಣಕ್ ಪ್ರಿಷತುತ, ಎಪ್ರರಲ್ +೨೦೧೪.http://www.kagapa.in/kannada/content/ಶಿಷುತೆ- +ಮತುತ-ಏಕ್ರ್ಪ್ತೆ. +[7] A. Joshi and A. Rathod, “A Dynamic Text Input +scheme for phonetic scripts like Devanagari,” Media +Labs, Powai, Mumbai. +[8] “Kannada Wiki,” https://kn.wikipedia.com. +[9] “Kannada Kali,” https://kannadakali.com. +[10] V. Dixit, “ಕ್ ನ್ುಡಿ,” Kannada Reasearch Institute of +America, 1 January 2021. +https://kannadakali.com/kannudi/kannudi.html. + + +Terminology +andante +As pronounced/said +anunāsika +Nasal consonant, fifth member of +each of the 5 classes of +consonants, ṅ, ñ, ṇ, n, and m +arkāvottu +rēpha, symbol ೯ for sound r +arkīfication +Automatic conversion of r to +arkāvottu ೯ +dvitva +self-conjunct +kāguṇita +Consonant + Vowel, C+V +kaṇḍante +as seen (as written) +mahāprāṇa +Aspirated consonant +mahāprāṇa +Aspirated consonant +mūla akṣara +vowel a or C+a +null vowel +Represnted with virāma ಂ +ottakshara +Syllable with ottu, Conjunct +Otti hiḍi ottu +koḍu! +Hold pressed give ottu +ottu +1. secondary form of a consonant; +2. accent +ottu Hāku Ottu +Koḍu! +apply pressure give ottu +rēpha +Symbol ೯ for sound r +sanyuktākṣara +conjunct +sonne +śūn'ya, anusvara, symbol ಂ +śūn'ya +sonne, anusvara, symbol ಂ +Śūn'yīfication +Automatic conversion of +anunāsika to śūn'ya +Sva-ottu +self-conjunct +ūhita svara +Default/Presumed vowel +virāma +Null vowel, soundless vowel +symbol ಂ + + diff --git a/OtAyT4oBgHgl3EQf7PqT/content/tmp_files/load_file.txt b/OtAyT4oBgHgl3EQf7PqT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fd2a32dead4aeaa1e97b2a801474c576193f86a3 --- /dev/null +++ b/OtAyT4oBgHgl3EQf7PqT/content/tmp_files/load_file.txt @@ -0,0 +1,280 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf,len=279 +page_content='1 Kannudi - A Reference Editor for Kannada (Based on OPOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' and OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' principles, and Domain Knowledge) Vishweshwar V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Dixit KRIA, Kannada Research Institute of America 714-322-9748 namovish@kannadakali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' namovish@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='com Abstract Kannudi is a reference editor for Kannada based on OPOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' and OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' principles, and domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' It introduces a method of input for Kannada, called OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=', that is, Ottu Hāku Ottu Koḍu!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' (apply pressure and give ottu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' This is especially suited for pressure sensitive input devices, though the current online implementation uses the regular mechanical keyboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' has three possible modes, namely, sva-ottu (self-conjunct), kandante (as you see), and andante (as you say).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' It may be noted that kandante mode does not follow the phonetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' However, this mode may work well for those who are inclined to visualize as they type rather than vocalizing the sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Kannudi also demonstrates how domain knowledge can be effectively used to potentially increase speed, accuracy, and user friendliness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' For example, selection of a default vowel, automatic shunyification, and arkification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Also implemented are four types Deletes that are necessary for phono-syllabic languages like Kannada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Kannudi can be accessed at https://kannadakali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='com/kannudi/kannudi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='html 1 Introduction Many tools are available for digital inputting of Kannada and other Indian language text, such as Input Method Editors (IME), and real time transliteration tools, free and commercial, online as well as offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Most of these are generic in the sense that they are designed to address all Indian languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' That makes sense as the scripts for these languages, as they all descended from the same Brahmi script, share many common features such as being alpha-syllabic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' However, there are subtle differences in the writing styles of these languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' These language specifics can be used to “optimize” and make the editors and IMEs more efficient and user friendly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' An earliest implementation of one such editor attempting to use the ‘domain knowledge’ was described by Dixit [1] [2] [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' It was an ambitious effort as it aimed for a universal framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=" It identified notions such as Śūn'yīfication, Non-initial Vowel, and hinted at a-rule for voice input." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' However, the implementation was limited to Kannada and limited in scope of the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' This was a DOS based editor and no versions were released later on Windows or other platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Another notable DOS editor was developed in Visual Basic by Rangachar [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Described in this article is a reference implementation introducing a new method of input for ottaksharas (conjuncts) while incorporating previous ideas in Chitragupta and Unived [1] [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 2 Input Methods Keyboarding, being the norm, is a required method of input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' A phonetic input method generally assigns a single key to a single phoneme and the keys are typed in the order of pronunciation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' A strict one phoneme - one key (OPOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=') mapping of keys to phonemes may not desirable for the sake of convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Some may assign multiple combinations of 1-3 keys to a single phoneme for convenience to resemble common writing using Latin script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' For example, one may assign B, bh, 2 Bh, or BH for ಭ (mahāprāṇa b) for the sake of convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' A mapping of letters to (ASCII) keys was developed by Kannada Ganaka Parishattu (KGP) and is used in its Nudi editor [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' This has been designated as the official standard by Government of Karnataka [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' A context sensitive dynamic keyboarding has been described by Joshi et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' [7] Handwriting, using stylus on a tablet or mobile screen, is another method of input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' The complexity of graphics processing makes it slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Variations in individual writing styles also introduces errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Correcting these errors as one goes takes time and interrupts the thought process of the user which reduces the speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Voice input is a promising method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' However, Voice recognition is not perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' This is especially problematic in Kannada where regional and other variations in pronunciations abound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Additionally, current implementations are mostly dictionary based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' They suffer from the same difficulties with all Indian languages, namely, variations in pronunciation, dictionary limitations, and indetermination between writing as pronounced and the dictionary entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Current voice input implementations may make it more difficult to type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' One needs to keep looking constantly at the words being entered, select among the choices, or correct manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' A user seems to spend significant time in ‘correcting’ the dictionary words and ultimately, being frustrated, ends up using or taking help of a basic keyboard layout such as Inscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Thus, voice input, though convenient, current implementations fail the user in both accuracy and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 3 Improvement Opportunities Three main considerations in any input method are accuracy, speed, and convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Appropriate tradeoff among the three is also of concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Elimination or minimization of corrections (backspace and deletes) and minimization of required keystrokes become important parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Certainly, there is a need and room to improve the existing methods in these regards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Current implementations aim and try to cater to all Indian languages and therein implementing a set of common minimum features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' They become minimally useful, as most users are not interested in 15+ languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Hence, an implementation must be extensible so that domain knowledge specific to each language and script can be incorporated Mobile platforms present interesting opportunities for novel methods using soft keyboards, dynamic context, and new input mechanisms such as swipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 4 Letter Frequencies Whereas earlier studies have found 36% mūla akṣaras (vowel a or C+a) and 14% conjuncts, using a sample of articles from Kannada Wikipedia [8] and Kannada Kali [9],we found the letter frequencies as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' moola akshara 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='4% Gunitakshara 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content="3% anusvāra / sonne / śūn'ya 5." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='6% Ottu and end virāma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='2% Sajāti/Dvitva (self-conjunct) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='7% Dvitva post-vowel 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='1% Vijāti (non-self-conjunct) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='8% End-virāma 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='7% Table 1: Syllable Frequencies 5 Kannudi Kannudi [10] is a reference editor introducing several innovative and experimental features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Currently, an online implementation invoked via a web browser, is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Some features in this implementation require a keyboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Kannudi input method follows the phonetic order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=', phonemes are entered in the order of their pronunciation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Major principles in Kannudi are OPOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=', OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=', user friendliness, prevention/minimization of errors and illegal combinations and letter formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='1 OPOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Principle First major principle in Kannudi implementation of One Phoneme One Key (OPOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=') Here, keys are assigned to phonemes, not graphemes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' and further, a one-one correspondence exists between a key and a phoneme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Key assignments adhere to the standard specified by Government of Karnataka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' ೧ 1 ೨ 2 ೩ 3 ೪ 4 ೫ 5 ೬ 6 ೭ 7 ೮ 8 ೯ 9 ೦ 0 ಟ q ಡ w ಎ e ರ r ತ t ಯ y ಉ u ಇ i ಒ o ಪ p ಅ a ಸ s ದ d ್ f ಗ g ಹ h ಜ j ಕ k ಲ l Shift ⇧ ಙ z ಷ x ಚ c ವ v ಬ b ನ n ಮ m Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Keyboard Layout – non-Shift State ಠ Q ಢ W ಏ E ಋ R ಥ T ಐ Y ಊ U ಈ I ಓ O ಫ P ಆ A ಶ S ಧ D F ಘ G ್ H ಝ J ಖ K ಳ L Shift ⇧ ಞ Z X ಛ C ಔ V ಭ B ಣ N ಂ M Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Keyboard Layout – Shift State 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='2 Default Vowel – null ಂ or a ಅ A pure phonetic method of input would be to type every vowel and consonant in the order pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' This assumes no vowel inherently attached to a consonant in the alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Thus, with OPOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' in force, ಕ್ needs only one keystroke but each kāguṇita ka kā ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' kau ಕ, ಕಾ, ಕಿ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='ಕೌ requires 2 keystrokes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' As can be seen from the Table 1, majority of the syllables are mūla akṣaras, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=', independent vowels and consonants with vowel a ಅ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Hence, in the alphabet, the graphemes for consonants have been designed with an assumed or default vowel (ūhita svara) a ಅ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Ottu is the secondary form of a consonant that appears in a conjunct (sanyuktākṣara).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' An otttu is produced when there is no vowel between the two consonants, indicated by null vowel or virama ್ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' If the default vowel is virama ್ , then one simply types consonant keys in succession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' However, if the default vowel a ಅ, then the null vowel must be typed in with an additional keystroke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' For example, to type ಕತ three keys need to be pressed as shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' It adds an “extra” key, thus significantly negating the savings provided by the default vowel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Even then, having a ಅ as the default vowel saves 24% of keystrokes (42-18=24%) compared to having no default vowel (or assuming null ್ as the default vowel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' However, one may find this somewhat unnatural and not a pure phonetic method, and experience a loss in the speed of typing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Default Vowel a ಅ null ್ Key pressed k f t k t a Result ಕ ಕ್ ಕತ ಕ್ ಕ್ತ ಕತ Table 2: Typing an ottakshara conjunct As such, Kannudi provides a choice of default vowels a ಅ and null ್ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' A user can choose one as a preference or switch between the two as suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='3 OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Principle OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' uses pressure as input to produce secondary forms of consonants (ottu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' It is to apply pressure or press and hold a key to produce an ottu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' In case of mechanical keyboards, which are not pressure sensitive, OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=', which can be thought of as Otti Hidi Ottu Kodu!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' ಒತ್ತತ ಹಿಡಿ ಒತತತ ಕ ೊಡತ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=', is simply time based, that is, dependent on how long the key is held pressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Though keyboard timings, touch, and pressure sensitivities can be optimized to speed OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' we understand that it may be beyond the normal capabilities of a user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 4 In case of pressure sensitive (mobile) devices, OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' is called Ottu Hāku Ottu Koḍu!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' ಒತತತ ಹಾಕತ ಒತತತ ಕ ೊಡತ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' (apply pressure and give ottu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' This offers greater potential for savings in time as well as key strokes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' can have three modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Here OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' of a key, namely applying pressure (or holding pressed) is denoted by superscript + sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Sva-ottu Mode (SO): This is Self-Ottu, also known as Dvitva where a consonant gets its own secondary form (ottu) attached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' For example, k+ will produce ಕ್ಕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' And kn+w, equivalent to typing knfnw, produces ಕನನಡ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Kaṇḍante Ottu Mode (KO): This is visual mode where the input follows the written order – holding a consonant will add its ottu to preceding consonant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' For example, the key sequence st+r+I (= sftfrI) produces ಸ್ತ್ರೀ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Andante Ottu Mode (AO): This is “as you say” or phonetic mode where the input follows the order of pronunciation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Pressing and holding (ಒತ್ತತ ಹಿಡಿ) of a consonant key will prepare it for an ottu by adding the null vowel (virama).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' This agrees with the order of pronunciation as the ottu (accent) is on this consonant and the next consonant is turned into an ottu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Here k+ is equivalent to kf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' To produce ಸ್ತ್ರೀ enter s+t+rI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' In essence this is an alternative to f key in normal mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Default Vowel = a ಅ Syllable Type Mode Normal OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' SO Dvitva OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' KO Kandante OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' AO Andante Self-conjunct (dvitva) Keys Typed k f k k+ k k+ k+ k Display ಕ ಕ್ ಕಕ ಕಕ ಕ ಕಕ ಕ್ ಕಕ Non-self- conjunct Keys Typed g f r g f r g r+ g+ r Display ಗ ಗ್ ಗರ ಗ ಗ್ ಗರ ಗ ಗರ ಗ್ ಗರ End-Virama Keys Typed n f n f n f n+ Display ನ ನ್ ನ ನ್ ನ ನ್ ನ್ V[MH]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' + self- conjunct Keys Typed ak f k ak+ ak k+ ak+ k Display ಅಕ ಅಕ್ ಅಕಕ ಅಕಕ ಅಕ ಅಕಕ ಅಕ್ ಅಕಕ Keys saved per 1000 syllables 0 174 196 193 Table 3: Key savings with a-default 5 Default Vowel = null ್ Syllable Type Mode Normal OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Dvitva OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' KO Kandante OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' AO Andante Self-conjunct (dvitva) Keys Typed k k k+ k k+ k+ k Display ಕ್ ಕ್ಕ ಕ್ಕ ಕ್ ಕ್ಕ ಕ್ ಕ್ಕ Non-self- conjunct Keys Typed g r g r g r+ g+ r Display ಗ್ ಗ್ರ ಗ್ ಗ್ರ ಗ್ ಗ್ರ ಗ್ ಗ್ರ Virama Keys Typed n n n+ Display ನ್ ನ್ ನ್ V[MH]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' + self-conjunct Keys Typed ak k ak+ ak+ ak+ k Display ಅಕ್ ಅಕ್ಕ ಅಕ್ಕ ಅಕ್ಕ ಅಕ್ ಅಕ್ಕ keys saved per 1000 syllables 0 87 11 0 Table 4: Key Savings with null-default 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='4 Key Savings Table 3 shows the key savings, with a-default, for various conjunct syllable types for the three modes of OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='. For example, a dvitva requiring 3 keys normally, can be produced with only 1 key in dvitva mode (SO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Similarly, Table 4 lists the key savings with null- default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' These tables also show the keys saved per 1000 syllables in a typical document, based on the frequency data in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Thus, under a-default, key savings of 174 can be achieved in OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Dvitva mode (SO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' It may be noted that, most key savings occur in a- default mode while the differences among the three OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' modes remain insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Another scheme: when a key is pressed and held next key becomes an ottu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='g press-holding a k then typing r will produce ಕರ in a-default-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' This scheme has physical limitations due to positions of keys being fixed on a keyboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' It becomes necessary to cross hands or fingers or quickly decide which hand to use for a key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' This is contrary to the trained typist who expects to blindly use the same finger at the same physical location for a given key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' It can be physically impossible or totally confusing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Hence this scheme is not implemented in Kannudi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 6 Rules of Convenience Apart from introducing the novel input method, Kannudi implements several user-friendly features that are simply matter of convenience or eliminate or reduce errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' A few such features are described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='1 Backspace ←BS During normal course of tying, a user may have typed a key in error or pressed an adjacent key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' When the mistake is realized, the normal action is to press backspace key \uf0dfBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' This usually deletes the previous syllable entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' If in midst of a multi-phoneme syllable, all the effort is lost, when the user desires to undo just last entry or the mistake just made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Hence, Kannudi recognizes 4 types of deletions: 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Phoneme Delete: delete the phoneme immediately left of the cursor, assigned to ←BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Character Delete: delete the (unicode) character immediately left of the cursor, assigned to alt←BS, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Syllable Delete: delete the syllable immediately left of the cursor, assigned to shift←BS, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Word Delete: delete the word immediately left of the cursor, assigned to ctrl←BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content="2 Shunyification Sonne or śūn'ya is used in writing to represent an anunāsika before a consonant as in ಅಂಕ, though it is not incorrect to write ಅಙಕ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Thus, sonne before a classified consonant is pronounced as the anunāsika of the same class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' The process of automatic conversion of anunāsika before a consonant, classified or non- classified, to a sonne is called shunyification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=" For the sake of convenience, and only n/m keys are considered for Śūn'yīfication in this implementation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=" Śūn'yīfication makes the input entry flexible by allowing both n and m to be automatically Śūn'yīfied." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=" Śūn'yīfication is straightforward in case of a classified consonant but can be ambiguous in case of a non- classified consonant and exceptions occur." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=" In most cases when sonne precedes a non-classified consonant (avargīya vyan̄ jana), Śūn'yīfication allows the entry to be phonetic corresponding to how most people pronounce." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' For example, ಸಂಶಯ is pronounced as ಸಮಶಯ samśaya and ಸಂಸೃತ as sanskr̥ ta or samskr̥ ta by many, albeit all incorrectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=" Śūn'yīfication does not save any keystrokes;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' But does not require one to switch the flow between andante (as pronounced) and kaṇḍante (as seen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' This is especially convenient for those who are mentally “spelling” the Kannada words in roman script as they type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' And there are many such casual users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='3 Arkification Though arkāvottu ೯is used mostly in place of ರ್, there are a few situations where it is not desirable as in ರಾ\u200dಯಂಕ, ಸರ\u200dರನ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Kannudi automatically uses the correct form (arkifies) in such cases allowing the user to type normally without hindrance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 7 Error Prevention Rules Certain domain knowledge of the language can be used to prevent typos and warn the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' A few examples are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='1 Non-initial Vowel Kannada allows a standalone vowel only in beginning of word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Kannudi prevents such typing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='2 Aspirated Ottu It is not possible to pronounce an aspirated consonant (mahāprāṇa) when followed by another aspirated consonant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' As such Kannada does not allow a mahāprāṇa ottu to another mahāprāṇa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' However, certain words can be exceptions due to common usage, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=', ವಿಠಠಲ viṭhṭhala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 8 Exception Handling Convenience and error prevention rules may not be perfect, and exceptions can be found as mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Hence it is necessary to ensure that there is a mechanism to override the normal behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Two ways to override are a) using ZWJ/ZWNJ characters and b) inputting the phonemes with a space character in between and then remove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 9 Conclusions Here we have introduced an input method called OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' with three possible modes, namey, sva-ottu (self-conjunct), kaṇḍante (as seen), and andante (as pronounced/said).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' It may be noted that kaṇḍante mode (KO) does not follow the phonemic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' However, this mode may work well for those who are inclined to visualize as they type rather than vocalizing the sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' will work very well on mobile or any device where input pressure can be sensed where OHOK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' can 7 be called Ottu Hāku Ottu Koḍu!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' (Apply Pressure and Give Ottu) when implemented on pressure sensitive devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' it where can be a real time saver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' On mechanical keyboards, it saves keystrokes though any time saved is dependent on keyboard settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' We have showed that domain knowledge can be used to improve user friendliness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=" Several convenience and error minimization rules such as Śūn'yīfication and arkāvottu are described." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Four types of deletions, namely phoneme, character, syllable, and word delete are identified and assigned to backspace key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Thus, domain knowledge is shown to be necessary and helpful to enhance user friendliness as wells as input flow and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Further, one may consider incorporating these rules into open type font tables as attached language specific resources, and eliminate the need for a separate editor application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 10 References [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Dixit, “An Intelligent Screen Editor for Kannada and Other Indian Languages,” SAIL: A Journal of Society for the Advancement of Indian Languages, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 12-14, January 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' [2] ವಿ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' ದೀಕ್ಷಿತ, “ಚಿತರಗುಪ್ತ: ಭಾರತೀಯ ಭಾಷೆಗಳಿಗೆ ಕ್ ಪ್ಯೂಟರ್ ಮೀಲೆ ಅಚ್ುುಕ್ಟುುವ ಒ ದು ಪ್ದಧತ,” ಅಮರಿಕ್ನ್ನಡ, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' ೫೧, ೨೮ ಏಪ್ರರಲ್ ೧೯೮೫.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' [3] ವಿ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' ದೀಕ್ಷಿತ, “ಚಿತರಗುಪ್ತ: ಭಾರತೀಯ ಭಾಷೆಗಳಿಗೆ ಕ್ ಪ್ಯೂಟರ್ ಮೀಲೆ ಅಚ್ುುಕ್ಟುುವ ಒ ದು ಪ್ದಧತ,” ಅಮರಿಕ್ನ್ನಡ, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' ೫, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' ೪೨-೪೩, ಏಪ್ರರಲ್-ಮೀ ೧೯೮೫.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' [4] ಕ್.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' ರ ಗಾಚಾರ್, “ನ್ನ್ಸಾದ ಕ್ನ್ಸು - ಕ್ನ್ನಡಕೆ್ಕ ದು ಸುಲಭ ಸಾಧ್ೂ ಕ್ ಪ್ಯೂಟರ್,” ಅಮರಿಕ್ನ್ನಡ, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' ೪೮-೪೯, ಫೆಬ್ುರವರಿ ೧೯೮೫.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' [5] “ನ್ುಡಿ ತ ತಾರ ಶಗಳು,” ಕ್ನ್ನಡ ಗಣಕ್ ಪ್ರಿಷತುತ, http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='kagapa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='in/kannada/content/ತ ತಾರ ಶಗಳು.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' [6] “ಶಿಷುತೆ-ಮತುತ-ಏಕ್ರ್ಪ್ತೆ,” ಕ್ನ್ನಡ ಗಣಕ್ ಪ್ರಿಷತುತ, ಎಪ್ರರಲ್ ೨೦೧೪.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='kagapa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='in/kannada/content/ಶಿಷುತೆ- ಮತುತ-ಏಕ್ರ್ಪ್ತೆ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Joshi and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Rathod, “A Dynamic Text Input scheme for phonetic scripts like Devanagari,” Media Labs, Powai, Mumbai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' [8] “Kannada Wiki,” https://kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' [9] “Kannada Kali,” https://kannadakali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' [10] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Dixit, “ಕ್ ನ್ುಡಿ,” Kannada Reasearch Institute of America, 1 January 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' https://kannadakali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='com/kannudi/kannudi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Terminology andante As pronounced/said anunāsika Nasal consonant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' fifth member of each of the 5 classes of consonants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' ṅ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' ñ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' ṇ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' and m arkāvottu rēpha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' symbol ೯ for sound r arkīfication Automatic conversion of r to arkāvottu ೯ dvitva self-conjunct kāguṇita Consonant + Vowel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' C+V kaṇḍante as seen (as written) mahāprāṇa Aspirated consonant mahāprāṇa Aspirated consonant mūla akṣara vowel a or C+a null vowel Represnted with virāma ಂ ottakshara Syllable with ottu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Conjunct Otti hiḍi ottu koḍu!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' Hold pressed give ottu ottu 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' secondary form of a consonant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=' accent ottu Hāku Ottu Koḍu!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} +page_content=" apply pressure give ottu rēpha Symbol ೯ for sound r sanyuktākṣara conjunct sonne śūn'ya, anusvara, symbol ಂ śūn'ya sonne, anusvara, symbol ಂ Śūn'yīfication Automatic conversion of anunāsika to śūn'ya Sva-ottu self-conjunct ūhita svara Default/Presumed vowel virāma Null vowel, soundless vowel symbol ಂ" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQf7PqT/content/2301.00836v1.pdf'} diff --git a/PdAyT4oBgHgl3EQftfm7/content/2301.00597v1.pdf b/PdAyT4oBgHgl3EQftfm7/content/2301.00597v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c89161b4f431bab8a54001c07e834138a56383fb --- /dev/null +++ b/PdAyT4oBgHgl3EQftfm7/content/2301.00597v1.pdf @@ -0,0 +1,3 @@ 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883408 diff --git a/QdFPT4oBgHgl3EQfozUy/vector_store/index.pkl b/QdFPT4oBgHgl3EQfozUy/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..25432e0ac3c6b5b4d6b10db042ae2feb8b69d2d9 --- /dev/null +++ b/QdFPT4oBgHgl3EQfozUy/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c1fb36fcc9006ccfb980e57d0bfd0973392424b4a361c35d5dff84f0250288e6 +size 186005 diff --git a/RNE1T4oBgHgl3EQfHgMk/content/tmp_files/2301.02926v1.pdf.txt b/RNE1T4oBgHgl3EQfHgMk/content/tmp_files/2301.02926v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..42679ba08960390ddb7b7e7ccac73297b8e52164 --- /dev/null +++ b/RNE1T4oBgHgl3EQfHgMk/content/tmp_files/2301.02926v1.pdf.txt @@ -0,0 +1,640 @@ +arXiv:2301.02926v1 [cs.LG] 7 Jan 2023 +Markov Chain Concentration with an Application +in Reinforcement Learning +Debangshu Banerjee +February 2021 +Abstract +Given X1, ·, XN random variables whose joint distribution is given as +µ we will use the Martingale Method to show any Lipshitz Function f over +these random variables is subgaussian. The Variance parameter however +can have a simple expression under certain conditions. For example under +the assumption that the random variables follow a Markov Chain and +that the function is Lipschitz under a Weighted Hamming Metric. We +shall conclude with certain well known techniques from concentration of +suprema of random processes with applications in Reinforcement Learning +1 +Introduction +In the class we have been introduced to the concentration of measure for func- +tions of independent random variables. A natural extension of this would be +to consider the concentration of measure phenomenon for dependent random +variables. In this report we shall investigate this. We shall see how Markov +Chain concentration form an example in this setting. We shall highlight the +following result which has already been highlighted in a form for independent +random variables within the class lectures +Lipschitz Functions defined on Markov Chains are Sub-Gaussian under certain +Ergodic Properties of the Markov Chain +The way this report is organized is as follows: +• Theoretical Platform and Definitions +• Martingale Method for Concentration of Measure +• Wasserstein Matrix +• General Result and Extensions +We shall also present a hypothesis of how concentration of Markov Chains +may be utilized in the field of Reinforcement Learning. These results are still in +their infancy but we shall nevertheless include them here with hopes of future +work. +1 + +2 +Theoretical Platform and Definitions +We shall assume the following : Let T be a countable index set, (Ωt, Bt) be a +measurable space where Ωt is a Polish Space for each t ∈ T and dt be a metric +function for each Ωt. +We shall define the product space Ω = ⊗tΩt, the product sigma algebra +B = ⊗tBt and define a borel probability measure µ on Ω (Note that that is not +a product measure). We define the metric d on the product space as d = � +t dt +Local Oscillation of f Let f : Ω → R be a measurable function. We will +define the Local oscillation of f at i ∈ T as ∆i(f) = sup +x,y +x−i=y−i +f(x)−f(y) +di(xi,yi) +That is we keep all except the ith coordinate fixed and vary the i coordinate in +forming the ratio. +Note: This definition is siimilar to the bounded difference assumption that +was introduced in the class. +Note: Under this definition f(x) − f(y) ≤ � +i∈T ∆i(f)di(xi, yi). +Note: Under the assumtption that f is Lipshitz under the weighted Hamming +Distance |f(x) − f(y)| ≤ � +i∈T ci1xi̸=yi, we have ∆i(f) ≤ ci. +Markov Kernel K : Ω × B → [0, 1] is a Markov Kernel if K is a Borel +Probability measure for each x ∈ Ω. +Note: If f is any Bounded Measurable function we have Kf(x) = +� +f(y)K(x, dy) +a measurable function for almost every x in Ω +Note: If µ is any borel probability measure we can define µK(A) = +� +K(x, A)dµ(x) +a borel probability measure ∀A ∈ B +The following result has been done in class and is presented here without +proof: +Azuma-Hoeffding: Let {Fk}k≤n be any filtration and Mk be a martingale +difference sequence such that Ak ≤ Mk ≤ Bk a.s., then �N +k=1 Mk is subgaussian +with variance paramter 1/4 �N +k=1 ||Bk − Ak||2 +∞ +3 +Martingale Method for Concentration of Mea- +sure +Here we see how to use the Martingale Method introduced in class to get the +concentration of measure phenomenon. The proof will will take the following +course. +1. Construct the Martingales +E[f(X1, · · · , Xn|X1, · · · , Xi] − E[f(X1, · · · , Xn|X1, · · · , Xi−1] +2 + +2. Bound each term above as f(x1, · · · , xi, Xi+1, · · · , XN)−f(x1, · · · , Xi, · · · , XN) +under the assumption of existence of an Wasserstein Matrix +3. Finally use Azuma-Hoeffding to get bounds in term of a matrix norm of +this Wasserstein Matrix +4. Motivate how Wassertein Matrices can be represented as Upper Triangular +Matrices and get simple interpretations under stronger assumptions. +Thus let us begin +3.1 +Construct the Martingale Differences Mk +Given any borel probability µ measure on Ω. (Recall Ω is the product space) let +us define {Xi}i∈T random variables defined on each Ωi with joint distribution +µ. Let us define the natural filtration {Fk}i∈T . For each i ∈ T we definte the +Markov Kernel +Ki(x, dy) = δx[i−1](dy[i−1]) ⊗ µ[i,n](dy[i,n]|x[i−1]) +. That is Ki is nothing but the conditional measure of µ given µ[i−1]. Note that +[n] = {1, · · · , n}. +Under this definition of the Markov Kernel we have for any bounded mea- +surable function f : Ω → R, +Kif(x) = Eµ[f(X)|X[i−1] = x[i−1]] +Now let us denote the Martingale Difference under the natural filtration +Mi = Ki+1f(x) − Ki(f) += Eµ[f(X)|X[i] = x[i]] − Eµ[f(X)|X[i−1] = x[i−1]] +(1) +Therefore Note that f(X1, · · · , XN) − E[f(X1, · · · , XN)] = �N +i=1 Mi +3.2 +Bounding the Martingale differences +Using the tower property of conditional expectations +Mi = Eµ[f(X)|X[i] = x[i]] − Eµ[f(X)|X[i−1] = x[i−1]] += Eµ[f(X)|X[i] = x[i]] − Eµ[Eµ[f(X)|X[i−1] = x[i−1], Xi]|X[i−1] = x[i−1]] += +� +Ω[i,n] +� � +Ω(i,n] +f(x[i−1]xiy(i,n])µ(i,n](dy(i,n]|x[i])− +� +Ω(i,n] +f(x[i−1]yiy(i,n])µ(i,n](dy(i,n]|x[i], yi) +� +µ[i,n](dy[i,n]|x[i−1]) +(2) +3 + += +� +Ω[i,n] +� +Ki+1f(x[i−1]xiy(i,n])−Ki+1f(x[i−1]yiy(i,n]) +� +µ[i,n](dy[i,n]|x[i−1]) +(3) +This implies that each Mi can be bounded by +Ai = +� +X[i,n] +inf +xi∈Xi +� +Ki+1f(x[i−1]xiy(i,n])−Ki+1f(x[i−1]yiy(i,n]) +� +µ[i,n](dy[i,n]|x[i−1]) +Bi = +� +X[i,n] +sup +xi∈Xi +� +Ki+1f(x[i−1]xiy(i,n])−Ki+1f(x[i−1]yiy(i,n]) +� +µ[i,n](dy[i,n]|x[i−1]) +Now using the bounded differnces definition of ∆i(Ki+1f), we have +||Bi − Ai||∞ < ||di||∆i(Ki+1f) +where ||di|| = supxi,zi di(xi, zi) +So now we are in the vicinity of using Azuma-Hoeffding. The only remaining +step remains is how to bound ∆i(Ki+1f) using some known function of ∆i(f). +4 +Wasserstein Matrices +From the theory of general contractive Markov Kernels K with Dobrushin Co- +efficient θ defined as: +sup +x,y ||K(x, .) − K(y, .)||T V < θ +, under the Weighted Hamming Distance ∀f ∈ Lip(X, d) +∆i(Kf) < θ +αi +� +j∈T +αj∆j(f) +We refer to [1] for a proof of the above. +As displayed above it is suggested that all ∆j(f) ∀j ∈ T influence ∆i(Kf). +With such a relation as a motivation a Wasserstein Matrix is defined: +Wasserstein MatrixA Markov Kernel K is said to have a Wasserstrien +Matrix V = (Vij)i,j∈T Vij > 0 if ∀f ∈ Lip(X, d) +∆i(Kf) < +� +j∈T +Vij∆j(f)∀i ∈ T +or in vector form +∆(Kf) < V∆(f) +4 + +5 +General Result and Extensions +5.1 +SubGaussianity of Markov Kernels under assumption +of existence of Wassertein Matrix +Let us make the following assumption +For each Markov Kernel Ki, i ∈ T defined as before assume there exists +Wasserstein Matrices Vi = (V i +lm)l,m∈T s.t for each i ∈ T +∆(Kif) < Vi∆(f) +for each f ∈ Lip(X, d) +What this assumption is essentially saying is a rather strong statement. It states +that we assume that for each markov kernel Ki as defined before, there exists a +Wasserstein Matrix Vi. We shall soon see how we can back up our assumption +by explicit construction of Wasserstein Matrices for general probability mea- +sures µ, and later give simple expressions of Wasserstein Matrices under further +assumptions about the structure of µ and the distance metric d. +For the time being let us see how this assumption immediately gives us a Sub- +Gaussianity result +As before, from Azuma-Hoeffidfing, f(X1, ....XN) − Ef(X1, ...XN) is sub- +gaussian with variance paramter �N +k=1 ||Bk − Ak||2 +∞ where +N +� +k=1 +||Bk − Ak||2 +∞ <= +N +� +k=1 +||dk||2∆k(Kk+1f)2 += +N +� +k=1 +( +� +j∈T +||dk||V k+1 +kj +∆j(f))2 += +N +� +k=1 +( +� +j∈T +Γkj∆j(f))2 += +N +� +k=1 +Γ∆(f)2 +k = ||Γ∆(f)||2 +l2(T ) +where we define Γij = ||di||V i+1 +ij +The remaining of the document is sectioned as follows: +• Construction of Wassersterin Matrices Using Couplings +• Wassersterin Matrices under the discrete Metric and Goldstein Coupling +• Contractive Markov Chains +• Uniformly Ergodic Markov Chains +5 + +5.2 +Construction of Wassersterin Matrices Using Couplings +This section illustrates the most general form of a Wasserstein Matrix. +Let P[i] +xy be any coupling of conditional probabilities µ(i,n](.|x[i]) and µ(i,n](.|y[i]) +where x and y only differ in the i coordinate. Then recall that we are trying to +relate ∆i(Ki+1f), we see +Ki+1f(x) − Ki+1f(y) = +� +X(i,n] +� +X(i,n] +(f(x[i], u(i,n]) − f(y[i], v(i,n])µ(i,n](du(i,n]|x[i])µ(i,n](dv(i,n]|y[i]) += +� +X(i,n]×X(i,n] +P[i] +xy(du(i,n], dv(i,n])(f(x[i], u(i,n]) − f(y[i], v(i,n]) +(4) +Further note that by our construction of the Markov Kernels Ki, we have +∆i(Kjf) = 0 for i > j Thus we can express the above as +≤ ∆i(f)di(xi, yi)+ +� +j>i +∆j(f) +� +X(i,n]×X(i,n] +P[i] +xy(du(i,n], dv(i,n])dj(uj, vj) +(5) +or, +Ki+1f(x) − Ki+1f(y) +di(xi, yi) +≤ ∆i(f) + +� +j>i +� +P[i] +xydj +di(xi, yi)∆j(f) +and therefore, +∆i(Ki+1f) ≤ ∆i(f) + +� +j>i +sup +x,y +x−i=y−i +� +P[i] +xydj +di(xi, yi)∆j(f) +Thus from our definition of Wasserstien Matrices +V i+1 +ij += + + + + + + + +0 +if i > j +1 +if i = j +sup +x,y +x−i=y−i +� P[i] +xydj +di(xi,yi) +if i < j +5.3 +Wassersterin Matrices under the discrete Metric and +Goldstein Coupling +Let us introduce a further assumption, that we are working with a discrete met- +ric. +Under the discrete metric, +Γij = +sup +x,y +x−i=y−i +P[i] +xy[Y (0) +j +̸= Y (1) +j +] +6 + +where (Y (0), Y (1)) = ((Y (0) +i+1, · · · , Y (0) +n +), (Y (1) +i+1, · · · , Y (1) +n ) is a random variable +taking value in X(i,n] × X(i,n] with Y (0) distributed as µ(i,n](.|x[i]) and Y (1) +distributed as µ(i,n](.|y[i]) +P[i] +xy[Y (0) +j +̸= Y (1) +j +] ≤ P[i] +xy[(Y (0) +j +, · · · , Y (0) +n +) ̸= (Y (1) +j +, · · · , Y (1) +n +)] +Goldstein Coupling[2] +There exists a coupling P[i] +xy known as the Goldstein Maximal Coupling s.t. +P[i] +xy[(Y (0) +j +, · · · , Y (0) +n +) ̸= (Y (1) +j +, · · · , Y (1) +n +)] = ||µ[j,n](.|x[i]) − µ[j,n](.|y[i])||T V +Thus using the Goldstein Coupling +Γij = + + + + + + + +0 +if i > j +1 +if i = j +sup +x,y +x−i=y−i ||µ[j,n](.|x[i]) − µ[j,n](.|y[i])||T V +if i < j +5.4 +Contractive Markov Chains +Let us revisit the example of Contractive Markov Chain +Here we have a directed Markov Model, where µ = µ0K1....Kn where Ki is +a Markov Kernel from Xi to Xi+1 under the Doeblin Contraction Coefficient +sup +x,y ||Ki(x, .) − Ki(y, .)||T V < θi +Under this assumption it can be shown using a recursion argument that, +||µ[j,n](.|x[i−1], xi) − µ[j,n](.|x[i], yi)||T V ≤ θj−i +where θ = max θi. +We refer to [3] or [4] for a full proof. +Thus from the previous sections, we have +Γ = + + + + + +1 +θ +θ2 +. . . +θn−1 +0 +1 +θ +. . . +θn−2 +... +... +... +... +... +0 +0 +0 +. . . +1 + + + + + +Now from our general result, and noting that if f is Lipshitz under the weighted +Hamming Distance |f(x)−f(y)| ≤ � +i∈T ci1xi̸=yi, we have ∆i(f) ≤ ci. we have +subgaussianity with ||Γ||2||c||2 where c = (ci)i≤n +5.5 +Uniformly Ergodic Markov Chains +Be recalling a general result that uniform ergodic Markov Chains have finite +mixing times, we have the following variant of the general result. +7 + +Let us first define what the mixing time for a Markov Chain is: +τ(ǫ) = min +t [{ +max +1 J/2. +it is quadratic in fermions [2,18]. For κ > 0 a second order phase transition in the Ising universality +class separates a ferromagnetically ordered phase from a paramagnetic one. For κ < J/2 and small +values of h the locus of the critical line can be determined by second order perturbation theory, +which yields [15] +J − 2κc = hc − 1 +2J +κch2 +c +J − κc +. +(3) +In terms of the spins the transition is characterized by the order parameter 〈σx +j 〉 taking a non-zero +value in the ferromagnetic phase. In terms of the fermions this is a non-local (string) operator and +the transition is topological [23]. Our analysis of quench dynamics close to quantum critical points +in one dimension therefore pertains to both topological transitions and conventional transitions +3 + +SciPost Physics +Submission +with local order parameters. Moreover, our mean-field analysis developed below is exact along the +line κ = 0 and correctly accounts for the symmetry and critical exponents of the Ising transition +for κ > 0. Hence it is expected to give a quantitatively accurate description of the ANNNI model +in the region h ≈ J and κ ≈ 0. +In what follows we consider quantum quenches from initial thermal states of the TFIM with +transverse field hi and inverse temperature β, i.e. our initial density matrix is +ρ(t = 0) = +exp +� +− βH(hi,0) +� +Trexp +� +− βH(hi,0) +� . +(4) +Including thermal states at finite temperatures rather than only ground states is useful as it allows +us to tune the energy density of the stationary state reached at late times in a simple manner. We +then consider the time evolution induced by the ANNNI Hamiltonian H(hf ,κ), i.e. +ρ(t > 0) = e−iH(hf ,κ)tρ(t = 0)eiH(hf ,κ)t . +(5) +We will restrict ourselves to the case hi = hf ≡ h and quenches with κ < J/2. To simplify notations +we also set J = 1. As the ANNNI model is non-integrable when both h and κ are non-zero we +expect the model to thermalize [4,24], i.e. in the thermodynamic limit the system should locally +relax to a thermal stationary state described by an effective temperature that is set by the energy +density of the initial state +e0 = lim +L→∞ +1 +L Tr +� +ρ(t = 0)H(hf ,κ) +� +. +(6) +In our setup the correlation length typically starts off small as a result of a large pre-quench gap, +while at late times the system settles into a thermal state at a low effective temperature in the +vicinity of a quantum critical point. Hence the correlation length in the stationary state is typically +much larger than in the initial state. Intuitively therefore the physics should be that of a system +whose correlation length grows following the quench. +3 +Mean-field theory for the Stationary State +Since the ANNNI model is believed to thermalize and has no local conservation laws other than +the total energy, we expect local observables O to reach their Gibbs ensemble values at late times +after a quantum quench +〈O〉(t) +t=∞ +−→ Z−1Tr[e−βf H(hf ,κ)O] . +(7) +Here Z is the partition function and βf the inverse effective temperature, set by the initial energy +density (6) generated by the quench protocol. For sufficiently small values of κ this thermal +state should be amenable to a description in terms of a simple self-consistent mean-field theory of +spinless fermions +Z−1Tr[e−βf HO] ≈ Z−1 +MFTTr[e−βMFTHMFTO] , +(8) +where +HMFT = +� +i +2 +� +a=0 +� +J(a) +Eff (c† +i ci+a + hc) + (∆(a) +Eff c† +i c† +i+a + hc) +� ++ E0 . +(9) +4 + +SciPost Physics +Submission +This mean-field theory is the result of requiring that Wick’s theorem holds, or equivalently that +higher cumulants vanish. The effective couplings J(a) +Eff and ∆(a) +Eff and the constant E0 are generated +by decoupling the quartic interaction terms self-consistently via +ABCD �→ 〈AB〉MFTCD + AB〈CD〉MFT − 〈AB〉MFT〈CD〉MFT + all other Wick contractions , +(10) +where +〈O〉MFT ≡ Z−1 +MFTTr[e−βMFTHMFTO] . +(11) +Defining the (self-consistent) expectation values +ta ≡ 〈c† +j cj+a〉MFT , +a = 0,1,2 , +∆b ≡ 〈c† +j c† +j+b〉MFT , +b = 1,2 , +(12) +we have +J(0) +eff = h − 2κ(t2 + Re∆2) , +J(1) +eff = −(J − 4κ(t1 + Re∆1)) , +∆(1) +eff = −(J − 4κ(t1 + ∆∗ +1)) , +J(2) +eff = κ(1 − 2t0) , +∆(2) +eff = κ(1 − 2t0) , +E0 = −hL − 4Lκ(|∆1|2 + t2 +1 − t0t2 + 2Re∆1t1 − Re∆2t0) +. +(13) +In order to fully specify our self-consistent mean-field theory we require the self-consistent values +of the five mean-fields as well as the value of the inverse effective temperature βMFT, which is +fixed by the condition that the energy density in the stationary state is the same as in the initial +state (6), i.e. +e0 = lim +L→∞ +〈HMFT〉MFT +L +. +(14) +The various self-consistency equations are most easily solved in momentum space. As stated +above it is sufficient to work in the Neveu-Schwarz sector for even system sizes L, so that +ck ≡ +1 +� +L +� +m +eikmcm , +k ∈ +� +2πn + 1/2 +L +, n = − L +2,..., L +2 − 1 +� +. +(15) +The mean-field Hamiltonian then becomes +HMFT = +� +k>0 +Ak(c† +kck − c† +−kc−k) + iBk(c† +kc† +−k) − iB∗ +k(c−kck) + const , +Ak = 2 +2 +� +a=0 +J(a) +eff cos ak , +Bk = 2 +2 +� +a=1 +∆(a) +eff sin ak . +(16) +We remark that in equilibrium not just the ta but also the ∆b are in fact real despite the absence +of a unitary symmetry enforcing this, see Appendix A. This in turn makes it possible to diagonalize +the Hamiltonian by a one-parameter Bogoliubov transformation +bκ(k) =cos θκ(k) +2 +c(k) − i sin θκ(k) +2 +c†(−k) , +eiθκ(k) = Ak − iBk +� +A2 +k + B2 +k +, +(17) +5 + +SciPost Physics +Submission +which gives 1 +HMFT = +� +k>0 +ϵκ(k)b† +κ(k)bκ(k) + const , +ϵκ(k) = +� +A2 +k + |Bk|2. +(18) +The self-consistency conditions on the mean-fields are given by calculating the expectation +values using (11) +ta = 1 +L +� +k +e−iak〈c† +kck〉MFT = 1 +L +� +k>0 +cos ak +� +1 − cosθκ(k)tanh βMFTϵκ(k) +2 +� +, +(19) +∆a = 1 +L +� +k +e−iak〈c† +kc† +−k〉MFT = 1 +L +� +k>0 +sin ak sinθκ(k)tanh βMFTϵκ(k) +2 +, +(20) +while the equation fixing the effective temperature (6) takes the form +4κ +� +(t1 + ∆1)2 − (t0 − 1/2)(t2 + ∆2) +� +κ=0 + h − 1 +L +� +k>0 +ϵκ=0(k)tanh βiϵκ=0(k) +2 += E0 + J(0) +Eff − 1 +L +� +k>0 +ϵκ(k)tanh βMFTϵ(k) +2 +. +(21) +The initial energy density given by the left hand side of (21) is a constant for fixed values +of κ,h, however the right-hand side depends upon the values of the mean-fields and thus this +equation must be solved self-consistently along with the other conditions on the mean-fields. +Eqs (19)-(21) need to be solved numerically, where the Bogoliubov angles are defined by Eq +(17) and Eq (13). The solutions can be directly compared to numerical results obtained in Ref. [1] +via a numerical linked cluster expansion [25,26]. In Fig. 2 we plot the mean-field results for the +(a) +MFT +NLCE +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.6 +0.7 +0.8 +0.9 +1.0 +κ +C1 +x +(b) +MFT +NLCE +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +-8 +-6 +-4 +-2 +0 +κ +dC1 +x/dκ +Figure 2: (a) C x +1 = 2(t1 + ∆2) in the thermal state reached at late times after a quench +from the TFIM ground state at h = 0.2 as a function of κ. The solid blue line is the +result obtained from our self-consistent mean-field theory and the dashed black line +shows numerical linked cluster expansion (NLCE) results extracted from [1]. (b) Same +comparison as (a) but for χ1 = ∂κC x +1 (κ). The vertical lines indicate κc. +longitudinal nearest-neighbour correlator +C x +1 ≡ 〈σx +i σx +i+1〉 = 2(t1 + Re∆1) , +(22) +1Here we write |Bk|2 which gives the correct dispersion for complex Bk, as it will be out-of-equilibrium, although +the form of the required canonical transformation in (17) will be more complicated. +6 + +SciPost Physics +Submission +in the (thermal) steady state following a quench from the ground state of the TFIM with h = 0.2 +along with the susceptibility dC x +1 /dκ defined using an ensemble of quenches. We see that the +agreement of our mean-field analysis with the numerical results of Ref. [1] is excellent up to +fairly large values of κ. We observe similarly good agreement with the transverse magnetization +mz ≡ 〈σz +j〉 and the next-nearest neighbour longitudinal correlator C x +2 ≡ 〈σx +i σx +i+2〉. In Fig. 3 we +compare the self-consistent inverse temperature βMFT to numerical results of Ref. [1]. We observe +excellent agreement essentially over the full range of κ considered. +MFT +NLCE +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.05 +0.10 +0.20 +0.50 +κ +T +[h] +Figure 3: Comparison of T = β−1 +MFT to effective temperatures reported in [1]. The dashed +black curve shows the NLCE results reported in Fig. 8 of [1], while the blue data points +are the values found by our self-consistent mean-field theory. The vertical line indicates +κc. +Given the good agreement with state-of-the-art numerical results we conclude that our self- +consistent fermionic mean-field theory provides a good description of the steady state reached at +late times after the quenches considered. +3.1 +Scaling regime at finite energy densities +The key objective of Ref. [1] was to establish that quantum quenches can be used to locate the +positions of quantum phase transitions in some parameter space. An important question is to +what extent the observed signatures are indeed associated with the scaling behaviour induced +by the proximate quantum critical point. To answer this question by purely numerical methods +would require the analysis of the long-distance behaviour of correlation functions or entanglement +entropies of large sub-systems, in order to ascertain whether they display scaling behaviour char- +acteristic of the proximate quantum critical point. Our mean-field theory gives us a much simpler +way of answering this question: as the field theory describing the quantum critical point is a gap- +less relativistic Majorana fermion the scaling regime extends at most to energies per particle at +which the mean-field dispersion is still to a good approximation linear. These considerations set +an energy cut-off for the field theory. In Fig. 4 we plot the mean-field dispersion relation (18) and +compare it to the respective effective temperatures. We see from Fig. 4(c,d) that when h is close +to 1 and κ small, the scale over which the dispersion is linear is much larger than the effective +temperature. This implies that for these quenches the steady state is in fact in the scaling regime of +the Ising transition and properties of the underlying quantum critical point are readily accessible. +By contrast in Fig. 4(a,b) we show the mean-field dispersion relation (18) in the steady state +7 + +SciPost Physics +Submission +-π +0 +π +0 +1 +2 +3 +4 +k +ϵ(k) +(a) h=0.2 +(b) h=0.5 +(c) h=0.8 +(d) h=0.9 +Figure 4: Effective dispersion relations in the steady state following a quench with (a) +h = 0.2, κ = 0.407 ≈ κc, (b) h = 0.5, κ = 0.269 ≈ κc, (c) h = 0.8, κ = 0.114 ≈ κc, +(d) h = 0.9, κ = 0.058 ≈ κc . The black horizontal line is the effective temperature +T = β−1 +MFT. The dashed black line is a fit to ϵfit(k) = +� +ϵκ(0)2 + v2 +fitk2 and the gray +shaded region indicates the regime of energy densities where spectral non-linearities +become significant and corrections to scaling limit behaviour can no longer be expected +to be negligible. +for quenches with small h and large κ can be fitted with a relativistic dispersion only for a small +energy window. Here the scale over which the Majorana dispersion is linear is very small and of +the same order of magnitude as the effective temperature. This means that for these quenches +the steady state is outside the scaling regime of the Ising transition, and so we can’t actually glean +any useful information about the underlying quantum critical point using quench dynamics. +We expect the fact that the cut-off decreases for smaller values of h to be an accurate prediction +of the mean-field theory presented here in light of the good agreement with the numerics seen +in Fig. 2. The point that the energy density needs to be sufficiently below the cut-off scale of the +quantum critical point one is trying to probe is of course both obvious and very general. +4 +Self-consistent time-dependent mean-field theory (SCTDMFT) +Following Refs [27–33] we now turn to the dynamics after our quantum quenches in the frame- +work of a self-consistent time-dependent Gaussian approximation. This amounts to considering +time evolution with a time-dependent mean-field Hamiltonian +HMFT(t) = +� +i +2 +� +a=0 +� +J(a) +Eff (t)(c† +i ci+a + hc) + (∆(a) +Eff (t)c† +i c† +i+a + hc) +� ++ E0(t) , +(23) +where the time-dependent couplings are given by the time-dependent analogs of (13), i.e. +ta(t) = Tr +� +ρMFT(t)c† +j cj+a +� +, +a = 0,1,2 , +∆b(t) = Tr +� +ρMFT(t)c† +j c† +j+b +� +, +b = 1,2 , +ρMFT(t) = +� +T e−i +� t +0 HMFT(t′)dt′� +ρ(t = 0) +� +T e−i +� t +0 HMFT(t′)dt′�† +. +(24) +Here T denotes time ordering; the initial density matrix ρ(t = 0) (4) is by construction Gaussian +and concomitantly so is ρMFT(t). This is the essence of the SCTDMFT, which by construction is +8 + +SciPost Physics +Submission +expected to work best at short times. This is because it is based on the assumption that all higher +cumulants vanish, which is strictly true at time t = 0. At short times the higher cumulants will +become non-zero, but their growth is expected to be slow for small κ. At late times SCTDMFT is +not expected to work well in general [34,35] and in some models is known to describe relaxation +towards a “prethermalization plateau” [36–38] rather than thermalization. However, as we will +see, it works reasonably well even at late times for some of the quenches considered here. +As a consequence of the translation invariance of the problem the time-evolved Gaussian den- +sity matrix ρMFT(t) is fully characterised by the two momentum space two-point averages +˜tk(t) = Tr +� +ρMFT(t) c† +kck +� +, +˜∆k(t) = Tr +� +ρMFT(t) c† +kc† +−k +� +. +(25) +The self-consistent equations of motion for these k space two-point functions can be obtained +using the Heisenberg equations of motion associated to the (now time-dependent) analog of the +momentum space Hamiltonian (16). The result is +d ˜∆k(t) +dt +=2iAk(t) ˜∆k(t) + B∗ +k +� +1 − 2˜tk(t) +� +d˜tk(t) +dt +=2Re +� +Bk(t) ˜∆k(t) +� +, +(26) +where +Ak = 2 +2 +� +a=0 +J(a) +eff (t)cos ak , +Bk = 2 +2 +� +b=1 +∆(b) +eff (t)sin ak . +(27) +We now integrate the equations (26) using a second-order midpoint scheme with a timestep of +10−3, which we choose to ensure that the mean-fields are converged with respect to the timestep. +At each timestep we must update the real space mean-fields ta and ∆b using ˜tk and ˜∆k +ta = 1 +L +� +k +˜tk(t)e−ika , +∆b = 1 +L +� +k +˜∆k(t)e−ikb . +(28) +Physical quantities such as spin-spin correlation functions can then be calculated in terms of (sums +of products of) the fermionic two-point functions. +4.1 +Short and intermediate-time behaviour of local correlation functions +In Fig. 5 we compare the results of the above SCTDMFT approximation to iTEBD results taken +from [1], which are believed to be essentially numerically exact. For small values of κ compared +to the critical value κc we find excellent agreement over the entire time range accessible to iTEBD. +For larger values of κ the agreement is still very good at short times, but gets worse at late times. +While Ref. [1] focused on spin correlations, the time evolution of the fermionic two-point +functions is of interest as well, in particular in relation to the question of detecting topological +transitions by quench dynamics. In Fig. 6 we present results obtained by SCTDMFT for t1(t) +and Re(∆1(t)) following quenches from the ground state of H(h,κ = 0) with h = 0.2,0.8 to +κ = 0.05,0.20. We observe the following: +• For quenches with small transverse fields h there are persistent oscillations around a con- +stant value, which is in good agreement with the corresponding expectation value after +thermalization. +9 + +SciPost Physics +Submission +κ=0.30 +κ=0.35 +κ=0.40 +0 +5 +10 +15 +20 +25 +0.90 +0.92 +0.94 +0.96 +0.98 +1.00 +C1 +x +t +Figure 5: Comparison of SCTDMFT results for C x +1 (t) to iTEBD results taken from [1] +for a quench from the ground state at h = 0.2,κ = 0 to κ > 0. Here the solid lines are +SCTDMFT results for L = 2000 and the dashed lines in the respective color are iTEBD. +The agreement is seen to be very good except for near the critical point (κc ≈ 0.407). +(a) +0.240 +0.245 +t1 +h = 0.2, κ = 0.20 +0 +10 +20 +30 +40 +50 +t +0.16 +0.17 +t1 +h = 0.8, κ = 0.05 +(b) +0.245 +0.250 +0.255 +Re ∆1 +h = 0.2, κ = 0.20 +0 +10 +20 +30 +40 +50 +t +0.224 +0.226 +0.228 +Re ∆1 +h = 0.8, κ = 0.05 +Figure 6: Nearest neighbour fermion two-point functions t1(t), Re∆1(t) after quenches +from the ground state of H(h,κ = 0) with h = 0.2 and h = 0.8 to H(h,κ). Horizontal +lines indicate the stationary values found in Section 3. +• For quenches at large fields h there are no long-lived oscillations. Instead the expectation +values relax to stationary values that differ from the ones predicted by thermalization by an +amount that scales at O(κ2). This is expected by virtue of the perturbative nature of the +mean-field approximation. +An explanation of the oscillatory behaviour is provided below in section 4.3. +As suggested in [1], a signature of the proximate quantum phase transition can be obtained by +processing data for the expectation value of a local observable for an ensemble of quenches at a +fixed time t after the quench. In Fig. 7 we show results for C x +1 (t) and dC x +1 (t)/dκ for an ensemble +of quenches starting in the ground state of H(h,κ = 0) and quenching to H(h,κ) for h = 0.2,0.8 +and a wide range of κ values. +In Fig. 7(a-b) we find very good agreement between our SCTDMFT results and the iTEBD +simulations of Ref. [1] for h = 0.2 and in Fig. 7(c-d) we show the results for h = 0.8. The +generalized susceptibility dC x +1 /dκ in Fig. 7(b,d) shows a strong dip even at the relatively early +time t = 20 around the critical value κc. Intuitively one expects that the reason for this strong +10 + +SciPost Physics +Submission +(b) h=0.2, t=20 +MFT +iTEBD +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +-5 +-4 +-3 +-2 +-1 +0 +κ +dC1 +x/dκ +(a) h=0.2, t=20 +MFT +iTEBD +0.6 +0.7 +0.8 +0.9 +1. +C1 +x +(d) h=0.8 +t=15 +t=20 +t=25 +0.01 +0.05 +0.1 +0.15 +0.2 +κ +(c) h=0.8 +t=15 +t=20 +t=25 +Figure 7: Performing quenches from H(h,0) to H(h,κ) we build a picture of observables +as a function of final κ. (a-b) Comparison with iTEBD data taken from [1] for h = 0.2 +(κc ≈ 0.407, indicated by thick gray line). (c-d) Equivalent calculation at h = 0.8 +(κc ≈ 0.114). All quenches done starting from the ground state for system size L = 2000. +response to the varying post-quench parameters is that the correlation length at time t = 20 +is already large and the system “feels” the proximity of the QPT; this implies a large correlation +length and consequently a strong linear response of the system, reflected in the dips in generalized +susceptibilities. We return to this point in the next section where, in Fig. 9, we extract correlation +lengths for the non-equilibrium state of the system following the quench for h = 0.8 and find that +the correlation length has grown from ξ ≈ 1.9 at t = 0 to ξ ≈ 12 at t = 20. Conversely, in cases +where the correlation length is short we do not expect the susceptibility to be large. This is indeed +the case for small values of κ in Fig. 7. In Fig. 8 we show the time evolution of the generalized +susceptibilities. Fig. 8 shows , for two values of h, quench data for various κ, including near the +critical value κc. For κ far from κc we observe a quick relaxation to a plateau, whilst for κ close to +the QPT we observe a longer relaxation time. Fig. 8(b) features growing oscillations due a ‘beat’ +phenomenon when numerically differentiating between the different quench data with slightly +different persistent oscillation frequencies. +4.2 +Growth of the correlation length in time +As we have noted above, the correlation length grows in time for many of the quenches we con- +sider. To show this explicitly we focus on the connected order-parameter two-point function +C x +c,ℓ(t) = Tr +� +ρMFT(t) σx +nσx +n+ℓ +� +� +�� +� +C x +ℓ (t) +− +� +Tr +� +ρMFT(t) σx +n +��2 +, +(29) +as it is easier to extract a correlation length for than σz +j. Since the order parameter expectation +value is itself difficult to calculate even in the TFIM [39, 40] we follow Ref. [41] in using the +11 + +SciPost Physics +Submission +(a) +0 +100 +200 +300 +t +−1.50 +−1.25 +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +dCx +1/dκ +κ =0.05 +κ =0.08 +κ =0.11 +(b) +0 +10 +20 +30 +t +−5 +−4 +−3 +−2 +−1 +0 +1 +dCx +1/dκ +κ =0.3 +κ =0.35 +κ =0.41 +Figure 8: Short time dynamics of the generalized susceptibility for quenches from an +initial thermal state with β = 2.0 and (a) h = 0.8 (κc ≈ 0.114) and (b) h = 0.2 +(κc ≈ 0.407) on a system with L = 2000. +Lieb-Robinson bound [42] to express the connected correlator as +C x +c,ℓ(t) = C x +ℓ (t) − C x +R (t) , +R ≫ vmaxt, +(30) +where vmax is the Lieb-Robinson velocity. In our self-consistent mean-field approximation we can +use Wick’s theorem to express C x +ℓ (t) as a block-Toeplitz Pfaffian [43] +C x +ℓ (t) =Pf +� +� +� +� +� +G0(t) +G1(t) +... +Gℓ−1(t) +−GT +1 (t) +... +... +... +... +... +... +... +−GT +ℓ−1(t) +... +... +G0(t) +� +� +� +� +� , +(31) +where +Gn(t) =2 +� +i Im∆n(t) +Re(t1−n(t) + ∆1+n(t)) − 1 +2δ0,n+1 +−Re(t1−n(t) + ∆1−n(t)) + 1 +2δ0,1−n +i Im∆n(t) +� +. +(32) +We note that if we replace the time-dependent Gaussian density matrix by a thermal equilibrium +state Eq (31) reduces to a determinant because ∆n ∈ R. +In Fig. 9 we show the connected order-parameter two-point function for a quench from the +ground state of the TFIM with h = 0.8 and turning on next nearest neighbour interactions of +strength κ = 0.11. In the initial state the connected correlator displays exponential decay with a +correlation length ξ(0) ≈ 1.9. Extracting correlation lengths at t > 0 is complicated by the fact +that the connected correlator for outside the “light-cone” remains unchanged and we are therefore +restricted to separations ℓ < 2vmaxt, where vmax is the maximal propagation velocity [4,44,45]. On +the other hand, in order to extract a correlation length ξ(t) we require that ℓ ≫ ξ(t). This causes +us to be unable to convincingly fit correlation lengths for short times (other than t = 0 which is an +equilibrium state by design), although we obtain relatively good fits to the exponential behaviour +at times t ≥ 20 which show the correlation length has grown to about ξ(25) ≈ 14.3. +4.3 +Oscillations in the low energy-density regime +A striking feature seen in Figs. 5, 6, 8 are the high-frequency oscillations in local observables for +quenches at reasonably small h which do not appear to decay in time in the mean-field theory. +12 + +SciPost Physics +Submission +ξ=1.92 +ξ=10.8 +ξ=12.4 +ξ=14.3 +t=0 +t=15 +t=20 +t=25 +0 +20 +40 +60 +80 +100 +10-6 +10-5 +10-4 +0.001 +0.010 +0.100 +1 +ℓ +|Cℓ +x-C150 +x | +Figure 9: Connected order-parameter two-point function C x +c,ℓ(t) for a quench from the +ground state of the TFIM at h = 0.8 to the ANNNI with h = 0.8,κ = 0.11 (κc ≈ 0.114). +Vertical lines indicate the lightcone distance at t = 15,20,25 using the maximal group +velocity of the effective dispersion in the steady state. Gray lines indicate fits to functions +of the form C x +c,fit = aℓ−ν exp(−ℓ/ξ) where ξ is the fitted correlation length. +These do not occur in quenches in the TFIM and hence seem to be a result of fermion interactions. +We stress that these oscillations were previously observed in the iTEBD simulations of Ref. [1] and +are not an artifact of the mean-field approximation. Importantly they are observed in quenches +that result in small energy densities compared to the fermion gap, which puts us in a regime where +we are dealing with the non-equilibrium dynamics of a very dilute gas of fermions. This suggests +that these oscillations could be related to the formation of long-lived bound states of (pairs of) +fermions, cf. Refs [46–50]. A simple limiting case in which this bound state formation can be seen +is h = 0. Here excitations are (highly degenerate) domain-wall states, whilst the antiferromagnetic +next-nearest neighbour term partially lifts this degeneracy by introducing an energy penalty of +4κ when the domain-walls are on exactly neighbouring bonds. That is, at h = 0 the next-nearest +neighbour interaction produces a spin-flip (anti-)bound state. In order to investigate the possibility +of these bound states persisting to the non-zero values of h we consider we have determined the +spectrum of low-lying excitations of the ANNNI model by exact diagonalization using the QuSpin +[51] package on L = 24 sites. These results provide useful information for physical properties at +finite energy densities that are small compared to the excitation gap over the ground state. As in +the ferromagnetic phase of the TFIM the lowest excitations can then be thought of as a continuum +of pairs of ferromagnetic domain-walls. This is indeed observed in the exact diagonalization results +in Fig. 10. In addition we observe a bosonic bound state of two domain-walls that occurs at +energies above the two domain-wall continuum. With regards to the oscillations observed in local +observables after some of our quenches we note the following: +• The bound state energy at k = 0 agrees with the oscillation frequency observed after the +quantum quenches. +• For reasonably large values of h the bound state ceases to exist around k = 0. It can be +seen from a Lehmann representation that only excited states with k = 0 contribute to the +dynamics when performing quenches from translationally invariant states as we do here. As +13 + +SciPost Physics +Submission +(a) +−π +−π/2 +0 +π/2 +π +Momentum +0 +1 +2 +3 +4 +Energy +(b) +−π +−π/2 +0 +π/2 +π +Momentum +0 +1 +2 +3 +4 +Energy +Figure 10: Spectrum of the ANNNI Hamiltonian for (a) h = 0.1, κ = 0.15 and (b) +h = 0.2, κ = 0.2 from exact diagonalisation using QuSpin [51] on L = 24 sites. As +physical states have even fermion parity, the lowest excited states are the two domain- +wall continuum and a sharp bosonic mode corresponding to the anti-bound state. For +h = 0.2,κ = 0.2 the four-particle continuum is low enough in energy to be visible on +this scale. +such this is consistent with the fact that when we perform quenches with larger h we do not +see persistent oscillations. +An important caveat is that in the quench set-up we are dealing with there is a small, but finite, +energy density above the ground state and thus in the thermodynamic limit the system is in fact +at an energy infinitely above what is pictured in Fig. 10. There the bound states always “sit” on +top of multi domain-wall excitations and are not expected to be stable. However, as the density +of domain-walls is very small the life-time of the bound state can be very large compared to the +time scale we observe in our quenches. We believe that this is indeed the case. +A rough estimate of the decay time of the bound states can be obtained by thinking in the +quasiparticle picture described above. If there were truly a single bound state then energy and +momentum conservation would prevent it from decaying, however the decay is allowed due a +background density of domain walls that the bound state may scatter from. A semi-classical ap- +proach to compute the scattering time is to introduce the mean-free-path of the domain-walls +λmfp = +Eg +ϵ , +(33) +where ϵ is the energy density relative to the ground state after the quench and Eg the quasiparticle +gap. If the mean-free-path is larger than the system size λmfp > L, then the state has in expectation +fewer than one quasi-particle in the entire system and the system does not require a many-body +description and the bound states will have nothing to scatter from. Even for thermodynamically +large systems however if we consider times less than +2vmaxt ≲ λmfp , +(34) +where vmax is the Lieb-Robinson velocity of the domain-wall excitations, we may consider the +bound state quasiparticles as having little interaction with the domain-wall background. We now +estimate all the relevant quantities in the case of interest. The post-quench energy density e0 +defined in (14) may be calculated using Wick’s theorem. The energy density ϵ appearing in Eq +14 + +SciPost Physics +Submission +0.1 +0.2 +0.3 +0.4 +κ +0 +2000 +4000 +6000 +8000 +10000 +λmfp +0.35 +0.40 +0.45 +0 +50 +100 +Figure 11: Mean free path of the quasiparticles generated by quantum quenches from the +TFIM ground state at transverse field h = 0.2 to the ANNNI model with 0.1 < κ < 0.45 +(κc ≈= 0.407). +(33) is however not the e0 of (14) but rather one must subtract the ground state energy density +of the ANNNI, which is not known analytically. We estimate the latter by exact diagonalization +for L = 18 sites, for which it is essentially converged. The resulting mean-free-path for quenches +from the ground state of the TFIM with h = 0.2 to the ANNNI model with 0.1 < κ < 0.45 is shown +in Fig. 11. We see that for these quenches the mean free path is extremely large unless κ is very +close to the QPT. The time range accessible to us in our SCTDMFT analysis is limited by finite- +size effects, which strongly influence observables after the traversal time L/(2vmax) [4, 52, 53]. +To access very late times without encountering finite-size effects therefore requires larger system +sizes and more memory. In order to test whether or not the oscillations eventually decay in mean- +field theory we instead change our initial density matrix in a way that reduces the mean free path, +e.g. for a quench with h = 0.1 and κ = 0.15 from an initial temperature β = 2.0, we estimate that +the mean free path should be roughly 50 sites and the scattering time about ts ∼ 56, see Table (1). +Nonetheless there is no visible damping in the mean-field theory up to very late times (t = 103), +see Fig. 12. We conclude that in SCTDMFT the oscillations are undamped while we expect in an +e0(β = 2.0) +eGS(h = 0.1,κ = 0.15) +ϵ +2Eg +λmfp +vmax +ts +-0.82739 +-0.85295 +0.02556 +2.410 +47.14 +0.4187 +56.29 +Table 1: Postquench energy density e0 obtained from Eq (14), ground state energy den- +sity eGS and two particle gap estimated with ED on L = 20 sites. Lieb-Robinson velocity +is estimated as the maximal group velocity for the dispersion εκ(k) given in Eq (18) +using the values of the mean-fields at t = 100. +exact theory they would decay. +15 + +SciPost Physics +Submission +0 +20 +40 +0.2380 +0.2385 +0.2390 +0.2395 +0.2400 +0.2405 +960 +980 +1000 +t +t1 +Figure 12: Time evolution of the mean field t1 following a quench from β = 2.0, +h = 0.1,κ = 0.15. +5 +Non-equal time correlation functions +A natural question is whether the existence of a bound state can be detected more directly in +the quench setup. One proposal in the literature is to use certain Fourier transforms of equal-time +correlation functions [54,55], but these do not provide useful insights in our case. In thermal equi- +librium it is well established that dynamical response functions give detailed information about +the particle content of the theory. An obvious question then is to what extent their non-equilibrium +analogs can be used to do the same. In order to address this question we now determine certain +non-equal time correlation functions in our SCTDMFT. We do not attempt to address the problem +of calculating non-equal time two-point functions of the order parameter, as this is difficult even +for the transverse field Ising chain itself [40,56]. In MFT the Heisenberg equations of motion for +the fermion operators ck are linear +d +dt ck(t) = i[HMFT(t), ck(t)] = −iAk(t)ck(t) + Bkc† +−k(t) , +(35) +and can be solved by a time-dependent Bogoliubov transformation +ck(t) =αk(t)ck(0) + βk(t)c† +−k(0) , +(36) +where the time-dependent coefficients are solutions to +dαk(t) +dt += − iAk(t)αk(t) + Bk(t)β∗ +−k(t) , +dβk(t) +dt += −iAk(t)βk(t) + Bk(t)α∗ +−k(t) . +(37) +16 + +SciPost Physics +Submission +As we are dealing with a Gaussian theory all non-equal time correlation functions are then ex- +pressible in terms of the two non-equal time Green’s functions given by +Gk(t, t′) = 〈c† +k(t)ck(t′)〉 =α∗ +k(t)αk(t′)fk + α∗ +k(t)βk(t′)gk ++ β∗ +k(t)αk(t′)g∗ +k + β∗ +k(t)βk(t′)(1 − f−k) = G−k(t, t′) , +(38) +˜Gk(t, t′) = 〈c† +k(t)c† +−k(t′)〉 =α∗ +k(t)α∗ +−k(t′)gk + α∗ +k(t)β∗ +−k(t′)f−k ++ β∗ +k(t)α∗ +−k(t′)(1 − fk) + β∗ +k(t)β∗ +−k(t′)g∗ = − ˜G−k(t, t′) , +(39) +where expectation values are always taken with respect to ρ(t = 0), i.e. 〈O〉 = Tr[ρ(t = 0)O]. +The final equalities hold due to the parity symmetry and fk, gk encode the initial conditions +fk = Gk(0,0), +gk = ˜Gk(0,0) . +(40) +As an example of the use of these formulas we consider the non-equilibrium analog of the density +response function +χρρ(r, t, t′) = 1 +L2 +� +k1,...k4 +ei(k1−k2)r〈[c† +k1(t)ck2(t), c† +k3(t′)ck4(t′)]〉 . +(41) +After Fourier transforming in the spatial co-ordinate this takes the following form in SCTDMFT +˜χ(q, t, t′) = 1 +L +� +k +� +˜Gk(t, t′) ˜G∗ +k−q(t′, t) − ˜Gk(t′, t) ˜G∗ +k−q(t, t′) ++ Gk(t, t′) +� +α∗ +k−q(t′)αk−q(t) + β∗ +k−q(t′)βk−q(t) +� +− +� +α∗ +k(t)αk(t′) + β∗ +k(t)βk(t′) +� +Gk−q(t′, t) +� +. +(42) +We note that χ(q, t, t′) is in principle measurable via linear-response measurements, see Ap- +pendix B. Employing a Lehmann representation suggests that spectral properties of the post- +quench Hamiltonian should be inferrable by taking appropriate “Fourier transforms” in time. In +practice we consider +χt f (q,ω) = +� t f +0 +dt′ ˜χ(q, t f , t′) eiωt′. +(43) +The imaginary part of this generalized dynamical susceptibility is shown in Fig. 13 for a quench +from κ = 0 to κ = 0.15 and initial temperature β = 1.0. We can clearly identify the continuum +of two domain-wall excitations but there is no evidence for a bound state above it. In order to +capture the latter one has to go beyond the SCTDMFT. +6 +Conclusion +We have formulated both equilibrium (at finite energy density) and time-dependent mean-field de- +scriptions for quantum quenches in the ANNNI model starting from a Gaussian state. We first used +this to compute properties of the expected stationary state following a quantum quench, assuming +that the system looks thermal again at late times and then used the time-dependent formulation +to probe the approach to stationarity. Comparisons in both the stationary and time-dependent +17 + +SciPost Physics +Submission +0 +π/4 +π/2 +3π/4 +π +0.0 +1.0 +2.0 +3.0 +4.0 +5.0 +6.0 +q +ω +Im[χ(q,ω)] +Figure 13: Out-of-equilibrium density-density susceptibility calculated for the mean-field +theory with L = 200,h = 0.1,κ = 0.15,β = 1.0 +cases with the numerical results of Ref. [1] show that this simple description is surprisingly accu- +rate even for large next-nearest neighbour interactions close to the critical value. Importantly it +fully reproduces the signatures of the equilibrium phase transition previously found numerically. +Our approach makes it clear that the observed signatures are associated with the growth of the +correlation length following a quantum quench and sheds light on the applicability of this mech- +anism for detecting quantum phase transitions in general. Our theory is based on a fermionic +description with a topological transition and so it is clear that topological as well as conventional +transitions may be detected in this manner. Moreover, we give an explanation for a potentially +puzzling feature of the real time dynamics, namely long-lived oscillations, by showing that the +oscillation frequency is the mass of a bound state in the interacting theory. +Finally, we showed that the time-dependent mean-field approach used here is capable of cal- +culating non-equal time correlation functions, however it is unable to capture the bound state +produced by the quartic interaction in the theory. +Acknowledgements +This work was supported by the EPSRC under grant EP/S020527/1. We are grateful to A. Das for +drawing our attention to Ref. [1] and helpful discussions. +A +Reality of certain mean-fields +When evaluating our self-consistent mean-fields we observe that some of them are real. In this +appendix we explain why this is the case, beginning with a clarification of the site parity σα +j �→ σα +−j. +This does not act on the fermions as cj �→ c−j due to the presence of the Jordan-Wigner string. The +18 + +SciPost Physics +Submission +simplest way to deduce the effect of site parity in the fermion basis is to look at the action of site +parity on fermion bilinears, which can be simply related to spin operators without semi-infinite +strings. In particular, we consider the following spin bilinears of definite parity +A =σx +i σx +i+1 , +B =σy +i σy +i+1 , +C± =σx +i σx +i+1 ± σy +i σy +i+1 . +(44) +We then note that the fermionic bilinears can be decomposed in terms of these via +c† +i cj =1 +4(A+ B − iC−) , +c† +j ci =1 +4(A+ B + iC−) , +c† +i c† +j =1 +4(A− B − iC+) , +cicj =1 +4(−A+ B − iC+) . +(45) +We thus see that the action of site parity on the bilinears is to exchange c† +i cj with c† +j ci and therefore +ti j = 〈c† +i cj〉 = t ji ∈ R as stated in the main text. +Additionally, the ANNNI Hamiltonian satisfies H = H∗ = H T in both the spin and fermion +bases. In particular, in the fermion basis cicj is also real. By the spectral theorem for real symmetric +matrices we then know that the eigenvectors of H are real in the same basis and so +〈cicj〉β = +1 +Z(β) +� +n +〈En|cicj|En〉e−βEn ∈ R +(46) +is manifestly real in equilibrium. However, after the quench the corresponding time-evolved quan- +tity becomes +〈cicj〉t = +1 +Z(β) +� +n,n′,m′ +e−βE0 +n〈En|Em′〉〈Em′|cicj|En′〉〈En′|En〉e−it(Em′−En′) , +(47) +where E0 +n are the pre-quench energies and Em′ the post-quench energies. Even if the post-quench +Hamiltonian is also real and thus the post-quench energy eigenstates |En′〉 real, the phase factors +will cause it to be generically complex. However, at very late times we would expect that the +system would come back to equilibrium via these factors dephasing and so the correlation function +should become real again at late times. Since tn are all real due to the site parity Z2 this implies +that all effective couplings are real in equilibrium, and out of equilibrium the only complex one +will be ∆(1) +Eff(t). +B +Linear response +In this appendix we summarize how to derive Kubo linear response relations after a quantum +quench that occurs at time t = 0, see e.g. Ref. [57]. The Hamiltonian is of the form +H(t) = θ(−t)Hi + θ(t)H f + f (t)V , +(48) +19 + +SciPost Physics +Submission +where θ(t) is the Heaviside step function. If f = 0 this corresponds to a quench at t = 0. The +linear response regime is when f (t) ≪ 1 and for this to be genuinely non-equilibrium we require +f (t) to have support in the time period before the system thermalizes after the quench. +We work in an interaction picture such that H = H0 + f (t)V, where H0 is generally not free. +The interaction picture states |ψ(t)〉I are defined by +|ψ(t)〉I = eiH0tU(t, t0)|ψ(t0)〉 , +(49) +where U(t, t0) is the full time-evolution operator associated with H(t), |ψ(t0)〉 is the Schrödinger +picture state at t0 and H0 is considered time independent by requiring, according to (48), that +t0 ≥ 0. Consistently, in the interaction picture the general operator O evolves in time as +OI(t) = eiH0tOe−iH0t . +(50) +The time-evolution according to H(t) of the expectation value of O in the state defined at t0 by +ρ(t0) = |ψ(t0)〉〈ψ(t0)| can be expressed in the interaction picture as +Tr(ρ(t)O) = Tr(ρI(t)OI(t)) ≈ Tr(ρ(t0)OI(t)) − i +� t +t0 +f (t′)χ(t, t′)dt′ , +(51) +where the susceptibility χ(t, t′) is given by +χ(t, t′) ≡ Tr +� +ρ(t0)[OI(t), VI(t′)] +� +. +(52) +In the last step of (51) we have expressed ρI(t) = |ψ(t)〉I I〈ψ(t)| by the first two terms in its +power series in the small function f (t). Eq (51) is the usual linear response formula except that +the time-translation invariance of the susceptibility is broken by the quench and hence χ(t, t′) +does not depend only on the time-difference t − t′. +References +[1] A. Haldar, K. Mallayya, M. Heyl, F. Pollmann, M. Rigol and A. 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Silva, +Quantum quenches, linear response +and superfluidity out of equilibrium, +EPL (Europhysics Letters) 107(3), 30002 (2014), +doi:10.1209/0295-5075/107/30002. +24 + diff --git a/UdE2T4oBgHgl3EQftgg8/content/tmp_files/load_file.txt b/UdE2T4oBgHgl3EQftgg8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9884968a67084c03a11a3f1178a7e01393913936 --- /dev/null +++ b/UdE2T4oBgHgl3EQftgg8/content/tmp_files/load_file.txt @@ -0,0 +1,1023 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf,len=1022 +page_content='SciPost Physics Submission A simple theory for quantum quenches in the ANNNI model Jacob H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Robertson1⋆, Riccardo Senese1 and Fabian H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Essler1 1 The Rudolf Peierls Centre for Theoretical Physics, Oxford University, Oxford OX1 3NP, UK ⋆ jacob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='robertson@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='uk Abstract In a recent numerical study [1] it was shown that signatures of proximate quantum critical points can be observed at early and intermediate times after certain quantum quenches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Said work focused mainly on the case of the axial next-nearest neighbour Ising (ANNNI) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Here we construct a simple time-dependent mean-field theory that allows us to ob- tain a quantitatively accurate description of these quenches at short times and a surprisingly good approximation to the thermalization dynamics at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Our approach provides a simple framework for understanding the reported numerical results as well as fundamental limitations on detecting quantum critical points through quench dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We moreover explain the origin of the peculiar oscillatory behaviour seen in various observables as arising from the formation of a long-lived bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Contents 1 Introduction 2 2 Definition of the model and quench protocol 2 3 Mean-field theory for the Stationary State 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='1 Scaling regime at finite energy densities 7 4 Self-consistent time-dependent mean-field theory (SCTDMFT) 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='1 Short and intermediate-time behaviour of local correlation functions 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2 Growth of the correlation length in time 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='3 Oscillations in the low energy-density regime 12 5 Non-equal time correlation functions 16 6 Conclusion 17 A Reality of certain mean-fields 18 B Linear response 19 References 20 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='04070v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='stat-mech] 10 Jan 2023 SciPost Physics Submission 1 Introduction Quantum phase transitions (QPT) provide a key framework for our understanding of equilibrium phases of correlated quantum matter [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' More recently physical properties in the vicinity of quantum critical points in out-of-equilibrium settings have been investigated theoretically [3–5] and in ultra-cold atom experiments [6–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' An interesting question that has been raised is whether it is possible to detect the location of QPTs, and associated physical properties, through the dy- namics at short and intermediate times after a quantum quench [1,10–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' [1] Haldar et al proposed a set of generalized susceptibilities that quantify the sensitivity of the time evolution and stationary values of local observables to changes in the quench protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Based on numerical studies in the axial next-nearest neighbour Ising model (ANNNI) the authors concluded that such susceptibilities can indeed provide signatures of a proximate QPT not only in the stationary regime but already at short/intermediate times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' An important question is how general this approach is, and what its limitations are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In order to address these issues we show that the findings of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' [1] for the ANNNI model can be understood in terms of a simple (time-dependent) mean-field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' This approach provides a clear insight into the window of applicability of any approach using gen- eralized susceptibilities to search for the location of critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' En route we clarify the origin of interesting oscillatory behaviours of local observables observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The outline of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In Section 2 we introduce the ANNNI model and de- scribe the quench protocol we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In Section 3 we then construct a mean-field description of the stationary state under the assumption that the system thermalizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In Section 4 we con- struct a time-dependent self-consistent mean-field description of the time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Within this approximation the density matrix is Gaussian at all times and Wick’s theorem may be employed to calculate any correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' This method is expected to be quantitatively accurate for short times as long as the initial state is itself Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In Section 5 we show that non-equal time correlation functions are easily accessible with this method and use it to compute the trans- verse component of the generalized dynamical structure factor following a quench in the ANNNI, demonstrating that this object contains information about the spectrum of the post-quench Hamil- tonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 2 Definition of the model and quench protocol The ANNNI model is a well studied non-integrable model with competing interactions, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The model consists of a transverse-field Ising model with an additional next-nearest neighbour Ising exchange, which we take to have the opposite sign to the nearest-neighbour Ising interaction H(h,κ) = −J L � i σx i σx i+1 − h � i σz i + κ L � i σx i σx i+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (1) Here σα j are the usual Pauli matrices on sites j of a ring of circumference L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The Hamiltonian (1) can be mapped to a model of spinless lattice fermions by means of a Jordan-Wigner transformation 2 SciPost Physics Submission [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' As we adopt periodic boundary conditions for the spins the fermions must obey either anti- periodic (Neveu-Schwarz) or periodic (Ramond) boundary conditions depending on whether the fermion number is even or odd, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Appendix A of [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' For our purposes it is sufficient to work in the Neveu-Schwarz sector for even system sizes L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The Hamiltonian then reads H(h,κ) = − J � j � c† j cj+1 + c† j c† j+1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' � + κ � j (c† j cj+2 + c† j c† j+2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=') + 2h � j c† j cj + 2κ � j � cj c† j+1cj+1c† j+2 − c† j c† j+1cj+1c† j+2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (2) The next-nearest neighbour spin-spin interaction is seen to give rise to a quartic interaction amongst the fermions, making the model non-integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The Hamiltonian (1) has a global Z2 ⊗ Z2 sym- metry corresponding to rotations around the z-axis by π – which is broken spontaneously in the ferromagnetic phase – and site parity σα n �→ σα −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The latter remains unbroken in the situations we consider and enforces ti j ≡ 〈c† i cj 〉 = t ji ∈ R (see Appendix A), while the former translates into fermion number parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The ground state phase diagram of the ANNNI model for κ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='5 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 1 [16,20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' At κ = 0 the model (1) reduces to the transverse field Ising model (TFIM) and is exactly solvable as Ferromagnetic Paramagnetic ▲ ▲ ▲ ▲ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='407 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='9 1 κ/J h/J Figure 1: Ground state phase diagram of the ANNNI model for 0 < κ/J < 1/2 - the solid curve is the boundary obtained by second order perturbation theory (3), red triangles indicate the critical points found by our self-consistent mean-field theory at select fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' There exist other phases at κ > J/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' it is quadratic in fermions [2,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' For κ > 0 a second order phase transition in the Ising universality class separates a ferromagnetically ordered phase from a paramagnetic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' For κ < J/2 and small values of h the locus of the critical line can be determined by second order perturbation theory, which yields [15] J − 2κc = hc − 1 2J κch2 c J − κc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (3) In terms of the spins the transition is characterized by the order parameter 〈σx j 〉 taking a non-zero value in the ferromagnetic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In terms of the fermions this is a non-local (string) operator and the transition is topological [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Our analysis of quench dynamics close to quantum critical points in one dimension therefore pertains to both topological transitions and conventional transitions 3 SciPost Physics Submission with local order parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Moreover, our mean-field analysis developed below is exact along the line κ = 0 and correctly accounts for the symmetry and critical exponents of the Ising transition for κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Hence it is expected to give a quantitatively accurate description of the ANNNI model in the region h ≈ J and κ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In what follows we consider quantum quenches from initial thermal states of the TFIM with transverse field hi and inverse temperature β, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' our initial density matrix is ρ(t = 0) = exp � − βH(hi,0) � Trexp � − βH(hi,0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (4) Including thermal states at finite temperatures rather than only ground states is useful as it allows us to tune the energy density of the stationary state reached at late times in a simple manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We then consider the time evolution induced by the ANNNI Hamiltonian H(hf ,κ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' ρ(t > 0) = e−iH(hf ,κ)tρ(t = 0)eiH(hf ,κ)t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (5) We will restrict ourselves to the case hi = hf ≡ h and quenches with κ < J/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' To simplify notations we also set J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' As the ANNNI model is non-integrable when both h and κ are non-zero we expect the model to thermalize [4,24], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' in the thermodynamic limit the system should locally relax to a thermal stationary state described by an effective temperature that is set by the energy density of the initial state e0 = lim L→∞ 1 L Tr � ρ(t = 0)H(hf ,κ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (6) In our setup the correlation length typically starts off small as a result of a large pre-quench gap, while at late times the system settles into a thermal state at a low effective temperature in the vicinity of a quantum critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Hence the correlation length in the stationary state is typically much larger than in the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Intuitively therefore the physics should be that of a system whose correlation length grows following the quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 3 Mean-field theory for the Stationary State Since the ANNNI model is believed to thermalize and has no local conservation laws other than the total energy, we expect local observables O to reach their Gibbs ensemble values at late times after a quantum quench 〈O〉(t) t=∞ −→ Z−1Tr[e−βf H(hf ,κ)O] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (7) Here Z is the partition function and βf the inverse effective temperature, set by the initial energy density (6) generated by the quench protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' For sufficiently small values of κ this thermal state should be amenable to a description in terms of a simple self-consistent mean-field theory of spinless fermions Z−1Tr[e−βf HO] ≈ Z−1 MFTTr[e−βMFTHMFTO] , (8) where HMFT = � i 2 � a=0 � J(a) Eff (c† i ci+a + hc) + (∆(a) Eff c† i c† i+a + hc) � + E0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (9) 4 SciPost Physics Submission This mean-field theory is the result of requiring that Wick’s theorem holds, or equivalently that higher cumulants vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The effective couplings J(a) Eff and ∆(a) Eff and the constant E0 are generated by decoupling the quartic interaction terms self-consistently via ABCD �→ 〈AB〉MFTCD + AB〈CD〉MFT − 〈AB〉MFT〈CD〉MFT + all other Wick contractions , (10) where 〈O〉MFT ≡ Z−1 MFTTr[e−βMFTHMFTO] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (11) Defining the (self-consistent) expectation values ta ≡ 〈c† j cj+a〉MFT , a = 0,1,2 , ∆b ≡ 〈c† j c† j+b〉MFT , b = 1,2 , (12) we have J(0) eff = h − 2κ(t2 + Re∆2) , J(1) eff = −(J − 4κ(t1 + Re∆1)) , ∆(1) eff = −(J − 4κ(t1 + ∆∗ 1)) , J(2) eff = κ(1 − 2t0) , ∆(2) eff = κ(1 − 2t0) , E0 = −hL − 4Lκ(|∆1|2 + t2 1 − t0t2 + 2Re∆1t1 − Re∆2t0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (13) In order to fully specify our self-consistent mean-field theory we require the self-consistent values of the five mean-fields as well as the value of the inverse effective temperature βMFT, which is fixed by the condition that the energy density in the stationary state is the same as in the initial state (6), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' e0 = lim L→∞ 〈HMFT〉MFT L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (14) The various self-consistency equations are most easily solved in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' As stated above it is sufficient to work in the Neveu-Schwarz sector for even system sizes L, so that ck ≡ 1 � L � m eikmcm , k ∈ � 2πn + 1/2 L , n = − L 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=', L 2 − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (15) The mean-field Hamiltonian then becomes HMFT = � k>0 Ak(c† kck − c† −kc−k) + iBk(c† kc† −k) − iB∗ k(c−kck) + const , Ak = 2 2 � a=0 J(a) eff cos ak , Bk = 2 2 � a=1 ∆(a) eff sin ak .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (16) We remark that in equilibrium not just the ta but also the ∆b are in fact real despite the absence of a unitary symmetry enforcing this, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' This in turn makes it possible to diagonalize the Hamiltonian by a one-parameter Bogoliubov transformation bκ(k) =cos θκ(k) 2 c(k) − i sin θκ(k) 2 c†(−k) , eiθκ(k) = Ak − iBk � A2 k + B2 k , (17) 5 SciPost Physics Submission which gives 1 HMFT = � k>0 ϵκ(k)b† κ(k)bκ(k) + const , ϵκ(k) = � A2 k + |Bk|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (18) The self-consistency conditions on the mean-fields are given by calculating the expectation values using (11) ta = 1 L � k e−iak〈c† kck〉MFT = 1 L � k>0 cos ak � 1 − cosθκ(k)tanh βMFTϵκ(k) 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (19) ∆a = 1 L � k e−iak〈c† kc† −k〉MFT = 1 L � k>0 sin ak sinθκ(k)tanh βMFTϵκ(k) 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (20) while the equation fixing the effective temperature (6) takes the form 4κ � (t1 + ∆1)2 − (t0 − 1/2)(t2 + ∆2) � κ=0 + h − 1 L � k>0 ϵκ=0(k)tanh βiϵκ=0(k) 2 = E0 + J(0) Eff − 1 L � k>0 ϵκ(k)tanh βMFTϵ(k) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (21) The initial energy density given by the left hand side of (21) is a constant for fixed values of κ,h, however the right-hand side depends upon the values of the mean-fields and thus this equation must be solved self-consistently along with the other conditions on the mean-fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Eqs (19)-(21) need to be solved numerically, where the Bogoliubov angles are defined by Eq (17) and Eq (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The solutions can be directly compared to numerical results obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' [1] via a numerical linked cluster expansion [25,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 2 we plot the mean-field results for the (a) MFT NLCE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='0 κ C1 x (b) MFT NLCE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='45 8 6 4 2 0 κ dC1 x/dκ Figure 2: (a) C x 1 = 2(t1 + ∆2) in the thermal state reached at late times after a quench from the TFIM ground state at h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2 as a function of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The solid blue line is the result obtained from our self-consistent mean-field theory and the dashed black line shows numerical linked cluster expansion (NLCE) results extracted from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (b) Same comparison as (a) but for χ1 = ∂κC x 1 (κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The vertical lines indicate κc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' longitudinal nearest-neighbour correlator C x 1 ≡ 〈σx i σx i+1〉 = 2(t1 + Re∆1) , (22) 1Here we write |Bk|2 which gives the correct dispersion for complex Bk, as it will be out-of-equilibrium, although the form of the required canonical transformation in (17) will be more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 6 SciPost Physics Submission in the (thermal) steady state following a quench from the ground state of the TFIM with h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2 along with the susceptibility dC x 1 /dκ defined using an ensemble of quenches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We see that the agreement of our mean-field analysis with the numerical results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' [1] is excellent up to fairly large values of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We observe similarly good agreement with the transverse magnetization mz ≡ 〈σz j〉 and the next-nearest neighbour longitudinal correlator C x 2 ≡ 〈σx i σx i+2〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 3 we compare the self-consistent inverse temperature βMFT to numerical results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We observe excellent agreement essentially over the full range of κ considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' MFT NLCE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='50 κ T [h] Figure 3: Comparison of T = β−1 MFT to effective temperatures reported in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The dashed black curve shows the NLCE results reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 8 of [1], while the blue data points are the values found by our self-consistent mean-field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The vertical line indicates κc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Given the good agreement with state-of-the-art numerical results we conclude that our self- consistent fermionic mean-field theory provides a good description of the steady state reached at late times after the quenches considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='1 Scaling regime at finite energy densities The key objective of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' [1] was to establish that quantum quenches can be used to locate the positions of quantum phase transitions in some parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' An important question is to what extent the observed signatures are indeed associated with the scaling behaviour induced by the proximate quantum critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' To answer this question by purely numerical methods would require the analysis of the long-distance behaviour of correlation functions or entanglement entropies of large sub-systems, in order to ascertain whether they display scaling behaviour char- acteristic of the proximate quantum critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Our mean-field theory gives us a much simpler way of answering this question: as the field theory describing the quantum critical point is a gap- less relativistic Majorana fermion the scaling regime extends at most to energies per particle at which the mean-field dispersion is still to a good approximation linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' These considerations set an energy cut-off for the field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 4 we plot the mean-field dispersion relation (18) and compare it to the respective effective temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 4(c,d) that when h is close to 1 and κ small, the scale over which the dispersion is linear is much larger than the effective temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' This implies that for these quenches the steady state is in fact in the scaling regime of the Ising transition and properties of the underlying quantum critical point are readily accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' By contrast in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 4(a,b) we show the mean-field dispersion relation (18) in the steady state 7 SciPost Physics Submission π 0 π 0 1 2 3 4 k ϵ(k) (a) h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2 (b) h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='5 (c) h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8 (d) h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='9 Figure 4: Effective dispersion relations in the steady state following a quench with (a) h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2, κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='407 ≈ κc, (b) h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='5, κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='269 ≈ κc, (c) h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8, κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='114 ≈ κc, (d) h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='9, κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='058 ≈ κc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The black horizontal line is the effective temperature T = β−1 MFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The dashed black line is a fit to ϵfit(k) = � ϵκ(0)2 + v2 fitk2 and the gray shaded region indicates the regime of energy densities where spectral non-linearities become significant and corrections to scaling limit behaviour can no longer be expected to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' for quenches with small h and large κ can be fitted with a relativistic dispersion only for a small energy window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Here the scale over which the Majorana dispersion is linear is very small and of the same order of magnitude as the effective temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' This means that for these quenches the steady state is outside the scaling regime of the Ising transition, and so we can’t actually glean any useful information about the underlying quantum critical point using quench dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We expect the fact that the cut-off decreases for smaller values of h to be an accurate prediction of the mean-field theory presented here in light of the good agreement with the numerics seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The point that the energy density needs to be sufficiently below the cut-off scale of the quantum critical point one is trying to probe is of course both obvious and very general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 4 Self-consistent time-dependent mean-field theory (SCTDMFT) Following Refs [27–33] we now turn to the dynamics after our quantum quenches in the frame- work of a self-consistent time-dependent Gaussian approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' This amounts to considering time evolution with a time-dependent mean-field Hamiltonian HMFT(t) = � i 2 � a=0 � J(a) Eff (t)(c† i ci+a + hc) + (∆(a) Eff (t)c† i c† i+a + hc) � + E0(t) , (23) where the time-dependent couplings are given by the time-dependent analogs of (13), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' ta(t) = Tr � ρMFT(t)c† j cj+a � , a = 0,1,2 , ∆b(t) = Tr � ρMFT(t)c† j c† j+b � , b = 1,2 , ρMFT(t) = � T e−i � t 0 HMFT(t′)dt′� ρ(t = 0) � T e−i � t 0 HMFT(t′)dt′�† .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (24) Here T denotes time ordering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' the initial density matrix ρ(t = 0) (4) is by construction Gaussian and concomitantly so is ρMFT(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' This is the essence of the SCTDMFT, which by construction is 8 SciPost Physics Submission expected to work best at short times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' This is because it is based on the assumption that all higher cumulants vanish, which is strictly true at time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' At short times the higher cumulants will become non-zero, but their growth is expected to be slow for small κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' At late times SCTDMFT is not expected to work well in general [34,35] and in some models is known to describe relaxation towards a “prethermalization plateau” [36–38] rather than thermalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' However, as we will see, it works reasonably well even at late times for some of the quenches considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' As a consequence of the translation invariance of the problem the time-evolved Gaussian den- sity matrix ρMFT(t) is fully characterised by the two momentum space two-point averages ˜tk(t) = Tr � ρMFT(t) c† kck � , ˜∆k(t) = Tr � ρMFT(t) c† kc† −k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (25) The self-consistent equations of motion for these k space two-point functions can be obtained using the Heisenberg equations of motion associated to the (now time-dependent) analog of the momentum space Hamiltonian (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The result is d ˜∆k(t) dt =2iAk(t) ˜∆k(t) + B∗ k � 1 − 2˜tk(t) � d˜tk(t) dt =2Re � Bk(t) ˜∆k(t) � , (26) where Ak = 2 2 � a=0 J(a) eff (t)cos ak , Bk = 2 2 � b=1 ∆(b) eff (t)sin ak .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (27) We now integrate the equations (26) using a second-order midpoint scheme with a timestep of 10−3, which we choose to ensure that the mean-fields are converged with respect to the timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' At each timestep we must update the real space mean-fields ta and ∆b using ˜tk and ˜∆k ta = 1 L � k ˜tk(t)e−ika , ∆b = 1 L � k ˜∆k(t)e−ikb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (28) Physical quantities such as spin-spin correlation functions can then be calculated in terms of (sums of products of) the fermionic two-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='1 Short and intermediate-time behaviour of local correlation functions In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 5 we compare the results of the above SCTDMFT approximation to iTEBD results taken from [1], which are believed to be essentially numerically exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' For small values of κ compared to the critical value κc we find excellent agreement over the entire time range accessible to iTEBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' For larger values of κ the agreement is still very good at short times, but gets worse at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' While Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' [1] focused on spin correlations, the time evolution of the fermionic two-point functions is of interest as well, in particular in relation to the question of detecting topological transitions by quench dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 6 we present results obtained by SCTDMFT for t1(t) and Re(∆1(t)) following quenches from the ground state of H(h,κ = 0) with h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8 to κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We observe the following: For quenches with small transverse fields h there are persistent oscillations around a con- stant value, which is in good agreement with the corresponding expectation value after thermalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 9 SciPost Physics Submission κ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='30 κ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='35 κ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='40 0 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='00 C1 x t Figure 5: Comparison of SCTDMFT results for C x 1 (t) to iTEBD results taken from [1] for a quench from the ground state at h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2,κ = 0 to κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Here the solid lines are SCTDMFT results for L = 2000 and the dashed lines in the respective color are iTEBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The agreement is seen to be very good except for near the critical point (κc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='407).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='245 t1 h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2, κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='20 0 10 20 30 40 50 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='17 t1 h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8, κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='05 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='245 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='255 Re ∆1 h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2, κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='20 0 10 20 30 40 50 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='226 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='228 Re ∆1 h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8, κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='05 Figure 6: Nearest neighbour fermion two-point functions t1(t), Re∆1(t) after quenches from the ground state of H(h,κ = 0) with h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2 and h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8 to H(h,κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Horizontal lines indicate the stationary values found in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' For quenches at large fields h there are no long-lived oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Instead the expectation values relax to stationary values that differ from the ones predicted by thermalization by an amount that scales at O(κ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' This is expected by virtue of the perturbative nature of the mean-field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' An explanation of the oscillatory behaviour is provided below in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' As suggested in [1], a signature of the proximate quantum phase transition can be obtained by processing data for the expectation value of a local observable for an ensemble of quenches at a fixed time t after the quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 7 we show results for C x 1 (t) and dC x 1 (t)/dκ for an ensemble of quenches starting in the ground state of H(h,κ = 0) and quenching to H(h,κ) for h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8 and a wide range of κ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 7(a-b) we find very good agreement between our SCTDMFT results and the iTEBD simulations of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' [1] for h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2 and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 7(c-d) we show the results for h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The generalized susceptibility dC x 1 /dκ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 7(b,d) shows a strong dip even at the relatively early time t = 20 around the critical value κc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Intuitively one expects that the reason for this strong 10 SciPost Physics Submission (b) h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2, t=20 MFT iTEBD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='4 5 4 3 2 1 0 κ dC1 x/dκ (a) h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2, t=20 MFT iTEBD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' C1 x (d) h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8 t=15 t=20 t=25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2 κ (c) h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8 t=15 t=20 t=25 Figure 7: Performing quenches from H(h,0) to H(h,κ) we build a picture of observables as a function of final κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (a-b) Comparison with iTEBD data taken from [1] for h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2 (κc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='407, indicated by thick gray line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (c-d) Equivalent calculation at h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8 (κc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='114).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' All quenches done starting from the ground state for system size L = 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' response to the varying post-quench parameters is that the correlation length at time t = 20 is already large and the system “feels” the proximity of the QPT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' this implies a large correlation length and consequently a strong linear response of the system, reflected in the dips in generalized susceptibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We return to this point in the next section where, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 9, we extract correlation lengths for the non-equilibrium state of the system following the quench for h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8 and find that the correlation length has grown from ξ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='9 at t = 0 to ξ ≈ 12 at t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Conversely, in cases where the correlation length is short we do not expect the susceptibility to be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' This is indeed the case for small values of κ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 8 we show the time evolution of the generalized susceptibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 8 shows , for two values of h, quench data for various κ, including near the critical value κc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' For κ far from κc we observe a quick relaxation to a plateau, whilst for κ close to the QPT we observe a longer relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 8(b) features growing oscillations due a ‘beat’ phenomenon when numerically differentiating between the different quench data with slightly different persistent oscillation frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2 Growth of the correlation length in time As we have noted above, the correlation length grows in time for many of the quenches we con- sider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' To show this explicitly we focus on the connected order-parameter two-point function C x c,ℓ(t) = Tr � ρMFT(t) σx nσx n+ℓ � � �� � C x ℓ (t) − � Tr � ρMFT(t) σx n ��2 , (29) as it is easier to extract a correlation length for than σz j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Since the order parameter expectation value is itself difficult to calculate even in the TFIM [39, 40] we follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' [41] in using the 11 SciPost Physics Submission (a) 0 100 200 300 t −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='50 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='25 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='00 dCx 1/dκ κ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='05 κ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='08 κ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='11 (b) 0 10 20 30 t −5 −4 −3 −2 −1 0 1 dCx 1/dκ κ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='3 κ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='35 κ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='41 Figure 8: Short time dynamics of the generalized susceptibility for quenches from an initial thermal state with β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='0 and (a) h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8 (κc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='114) and (b) h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2 (κc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='407) on a system with L = 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Lieb-Robinson bound [42] to express the connected correlator as C x c,ℓ(t) = C x ℓ (t) − C x R (t) , R ≫ vmaxt, (30) where vmax is the Lieb-Robinson velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In our self-consistent mean-field approximation we can use Wick’s theorem to express C x ℓ (t) as a block-Toeplitz Pfaffian [43] C x ℓ (t) =Pf � � � � � G0(t) G1(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Gℓ−1(t) −GT 1 (t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' −GT ℓ−1(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' G0(t) � � � � � , (31) where Gn(t) =2 � i Im∆n(t) Re(t1−n(t) + ∆1+n(t)) − 1 2δ0,n+1 −Re(t1−n(t) + ∆1−n(t)) + 1 2δ0,1−n i Im∆n(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (32) We note that if we replace the time-dependent Gaussian density matrix by a thermal equilibrium state Eq (31) reduces to a determinant because ∆n ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 9 we show the connected order-parameter two-point function for a quench from the ground state of the TFIM with h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8 and turning on next nearest neighbour interactions of strength κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In the initial state the connected correlator displays exponential decay with a correlation length ξ(0) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Extracting correlation lengths at t > 0 is complicated by the fact that the connected correlator for outside the “light-cone” remains unchanged and we are therefore restricted to separations ℓ < 2vmaxt, where vmax is the maximal propagation velocity [4,44,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' On the other hand, in order to extract a correlation length ξ(t) we require that ℓ ≫ ξ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' This causes us to be unable to convincingly fit correlation lengths for short times (other than t = 0 which is an equilibrium state by design), although we obtain relatively good fits to the exponential behaviour at times t ≥ 20 which show the correlation length has grown to about ξ(25) ≈ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='3 Oscillations in the low energy-density regime A striking feature seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 5, 6, 8 are the high-frequency oscillations in local observables for quenches at reasonably small h which do not appear to decay in time in the mean-field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 12 SciPost Physics Submission ξ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='92 ξ=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8 ξ=12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='4 ξ=14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='3 t=0 t=15 t=20 t=25 0 20 40 60 80 100 10-6 10-5 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='100 1 ℓ |Cℓ x-C150 x | Figure 9: Connected order-parameter two-point function C x c,ℓ(t) for a quench from the ground state of the TFIM at h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8 to the ANNNI with h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='8,κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='11 (κc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='114).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Vertical lines indicate the lightcone distance at t = 15,20,25 using the maximal group velocity of the effective dispersion in the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Gray lines indicate fits to functions of the form C x c,fit = aℓ−ν exp(−ℓ/ξ) where ξ is the fitted correlation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' These do not occur in quenches in the TFIM and hence seem to be a result of fermion interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We stress that these oscillations were previously observed in the iTEBD simulations of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' [1] and are not an artifact of the mean-field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Importantly they are observed in quenches that result in small energy densities compared to the fermion gap, which puts us in a regime where we are dealing with the non-equilibrium dynamics of a very dilute gas of fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' This suggests that these oscillations could be related to the formation of long-lived bound states of (pairs of) fermions, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Refs [46–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' A simple limiting case in which this bound state formation can be seen is h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Here excitations are (highly degenerate) domain-wall states, whilst the antiferromagnetic next-nearest neighbour term partially lifts this degeneracy by introducing an energy penalty of 4κ when the domain-walls are on exactly neighbouring bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' That is, at h = 0 the next-nearest neighbour interaction produces a spin-flip (anti-)bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In order to investigate the possibility of these bound states persisting to the non-zero values of h we consider we have determined the spectrum of low-lying excitations of the ANNNI model by exact diagonalization using the QuSpin [51] package on L = 24 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' These results provide useful information for physical properties at finite energy densities that are small compared to the excitation gap over the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' As in the ferromagnetic phase of the TFIM the lowest excitations can then be thought of as a continuum of pairs of ferromagnetic domain-walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' This is indeed observed in the exact diagonalization results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In addition we observe a bosonic bound state of two domain-walls that occurs at energies above the two domain-wall continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' With regards to the oscillations observed in local observables after some of our quenches we note the following: The bound state energy at k = 0 agrees with the oscillation frequency observed after the quantum quenches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' For reasonably large values of h the bound state ceases to exist around k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' It can be seen from a Lehmann representation that only excited states with k = 0 contribute to the dynamics when performing quenches from translationally invariant states as we do here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' As 13 SciPost Physics Submission (a) −π −π/2 0 π/2 π Momentum 0 1 2 3 4 Energy (b) −π −π/2 0 π/2 π Momentum 0 1 2 3 4 Energy Figure 10: Spectrum of the ANNNI Hamiltonian for (a) h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='1, κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='15 and (b) h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2, κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2 from exact diagonalisation using QuSpin [51] on L = 24 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' As physical states have even fermion parity, the lowest excited states are the two domain- wall continuum and a sharp bosonic mode corresponding to the anti-bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' For h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2,κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2 the four-particle continuum is low enough in energy to be visible on this scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' such this is consistent with the fact that when we perform quenches with larger h we do not see persistent oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' An important caveat is that in the quench set-up we are dealing with there is a small, but finite, energy density above the ground state and thus in the thermodynamic limit the system is in fact at an energy infinitely above what is pictured in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' There the bound states always “sit” on top of multi domain-wall excitations and are not expected to be stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' However, as the density of domain-walls is very small the life-time of the bound state can be very large compared to the time scale we observe in our quenches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We believe that this is indeed the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' A rough estimate of the decay time of the bound states can be obtained by thinking in the quasiparticle picture described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' If there were truly a single bound state then energy and momentum conservation would prevent it from decaying, however the decay is allowed due a background density of domain walls that the bound state may scatter from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' A semi-classical ap- proach to compute the scattering time is to introduce the mean-free-path of the domain-walls λmfp = Eg ϵ , (33) where ϵ is the energy density relative to the ground state after the quench and Eg the quasiparticle gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' If the mean-free-path is larger than the system size λmfp > L, then the state has in expectation fewer than one quasi-particle in the entire system and the system does not require a many-body description and the bound states will have nothing to scatter from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Even for thermodynamically large systems however if we consider times less than 2vmaxt ≲ λmfp , (34) where vmax is the Lieb-Robinson velocity of the domain-wall excitations, we may consider the bound state quasiparticles as having little interaction with the domain-wall background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We now estimate all the relevant quantities in the case of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The post-quench energy density e0 defined in (14) may be calculated using Wick’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The energy density ϵ appearing in Eq 14 SciPost Physics Submission 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='4 κ 0 2000 4000 6000 8000 10000 λmfp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='45 0 50 100 Figure 11: Mean free path of the quasiparticles generated by quantum quenches from the TFIM ground state at transverse field h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2 to the ANNNI model with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='1 < κ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='45 (κc ≈= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='407).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (33) is however not the e0 of (14) but rather one must subtract the ground state energy density of the ANNNI, which is not known analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We estimate the latter by exact diagonalization for L = 18 sites, for which it is essentially converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The resulting mean-free-path for quenches from the ground state of the TFIM with h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2 to the ANNNI model with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='1 < κ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='45 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We see that for these quenches the mean free path is extremely large unless κ is very close to the QPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The time range accessible to us in our SCTDMFT analysis is limited by finite- size effects, which strongly influence observables after the traversal time L/(2vmax) [4, 52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' To access very late times without encountering finite-size effects therefore requires larger system sizes and more memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In order to test whether or not the oscillations eventually decay in mean- field theory we instead change our initial density matrix in a way that reduces the mean free path, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' for a quench with h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='1 and κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='15 from an initial temperature β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='0, we estimate that the mean free path should be roughly 50 sites and the scattering time about ts ∼ 56, see Table (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Nonetheless there is no visible damping in the mean-field theory up to very late times (t = 103), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We conclude that in SCTDMFT the oscillations are undamped while we expect in an e0(β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='0) eGS(h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='1,κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='15) ϵ 2Eg λmfp vmax ts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='82739 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='85295 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='02556 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='410 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='4187 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='29 Table 1: Postquench energy density e0 obtained from Eq (14), ground state energy den- sity eGS and two particle gap estimated with ED on L = 20 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Lieb-Robinson velocity is estimated as the maximal group velocity for the dispersion εκ(k) given in Eq (18) using the values of the mean-fields at t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' exact theory they would decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 15 SciPost Physics Submission 0 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2380 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2385 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2390 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2395 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='2405 960 980 1000 t t1 Figure 12: Time evolution of the mean field t1 following a quench from β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='0, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='1,κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 5 Non-equal time correlation functions A natural question is whether the existence of a bound state can be detected more directly in the quench setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' One proposal in the literature is to use certain Fourier transforms of equal-time correlation functions [54,55], but these do not provide useful insights in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In thermal equi- librium it is well established that dynamical response functions give detailed information about the particle content of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' An obvious question then is to what extent their non-equilibrium analogs can be used to do the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In order to address this question we now determine certain non-equal time correlation functions in our SCTDMFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We do not attempt to address the problem of calculating non-equal time two-point functions of the order parameter, as this is difficult even for the transverse field Ising chain itself [40,56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In MFT the Heisenberg equations of motion for the fermion operators ck are linear d dt ck(t) = i[HMFT(t), ck(t)] = −iAk(t)ck(t) + Bkc† −k(t) , (35) and can be solved by a time-dependent Bogoliubov transformation ck(t) =αk(t)ck(0) + βk(t)c† −k(0) , (36) where the time-dependent coefficients are solutions to dαk(t) dt = − iAk(t)αk(t) + Bk(t)β∗ −k(t) , dβk(t) dt = −iAk(t)βk(t) + Bk(t)α∗ −k(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (37) 16 SciPost Physics Submission As we are dealing with a Gaussian theory all non-equal time correlation functions are then ex- pressible in terms of the two non-equal time Green’s functions given by Gk(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' t′) = 〈c† k(t)ck(t′)〉 =α∗ k(t)αk(t′)fk + α∗ k(t)βk(t′)gk + β∗ k(t)αk(t′)g∗ k + β∗ k(t)βk(t′)(1 − f−k) = G−k(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' t′) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (38) ˜Gk(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' t′) = 〈c† k(t)c† −k(t′)〉 =α∗ k(t)α∗ −k(t′)gk + α∗ k(t)β∗ −k(t′)f−k + β∗ k(t)α∗ −k(t′)(1 − fk) + β∗ k(t)β∗ −k(t′)g∗ = − ˜G−k(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' t′) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (39) where expectation values are always taken with respect to ρ(t = 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 〈O〉 = Tr[ρ(t = 0)O].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The final equalities hold due to the parity symmetry and fk, gk encode the initial conditions fk = Gk(0,0), gk = ˜Gk(0,0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (40) As an example of the use of these formulas we consider the non-equilibrium analog of the density response function χρρ(r, t, t′) = 1 L2 � k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='k4 ei(k1−k2)r〈[c† k1(t)ck2(t), c† k3(t′)ck4(t′)]〉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (41) After Fourier transforming in the spatial co-ordinate this takes the following form in SCTDMFT ˜χ(q, t, t′) = 1 L � k � ˜Gk(t, t′) ˜G∗ k−q(t′, t) − ˜Gk(t′, t) ˜G∗ k−q(t, t′) + Gk(t, t′) � α∗ k−q(t′)αk−q(t) + β∗ k−q(t′)βk−q(t) � − � α∗ k(t)αk(t′) + β∗ k(t)βk(t′) � Gk−q(t′, t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (42) We note that χ(q, t, t′) is in principle measurable via linear-response measurements, see Ap- pendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Employing a Lehmann representation suggests that spectral properties of the post- quench Hamiltonian should be inferrable by taking appropriate “Fourier transforms” in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In practice we consider χt f (q,ω) = � t f 0 dt′ ˜χ(q, t f , t′) eiωt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (43) The imaginary part of this generalized dynamical susceptibility is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 13 for a quench from κ = 0 to κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='15 and initial temperature β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We can clearly identify the continuum of two domain-wall excitations but there is no evidence for a bound state above it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In order to capture the latter one has to go beyond the SCTDMFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' 6 Conclusion We have formulated both equilibrium (at finite energy density) and time-dependent mean-field de- scriptions for quantum quenches in the ANNNI model starting from a Gaussian state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We first used this to compute properties of the expected stationary state following a quantum quench, assuming that the system looks thermal again at late times and then used the time-dependent formulation to probe the approach to stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Comparisons in both the stationary and time-dependent 17 SciPost Physics Submission 0 π/4 π/2 3π/4 π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='0 q ω Im[χ(q,ω)] Figure 13: Out-of-equilibrium density-density susceptibility calculated for the mean-field theory with L = 200,h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='1,κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='15,β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='0 cases with the numerical results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' [1] show that this simple description is surprisingly accu- rate even for large next-nearest neighbour interactions close to the critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Importantly it fully reproduces the signatures of the equilibrium phase transition previously found numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Our approach makes it clear that the observed signatures are associated with the growth of the correlation length following a quantum quench and sheds light on the applicability of this mech- anism for detecting quantum phase transitions in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Our theory is based on a fermionic description with a topological transition and so it is clear that topological as well as conventional transitions may be detected in this manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Moreover, we give an explanation for a potentially puzzling feature of the real time dynamics, namely long-lived oscillations, by showing that the oscillation frequency is the mass of a bound state in the interacting theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Finally, we showed that the time-dependent mean-field approach used here is capable of cal- culating non-equal time correlation functions, however it is unable to capture the bound state produced by the quartic interaction in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Acknowledgements This work was supported by the EPSRC under grant EP/S020527/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We are grateful to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Das for drawing our attention to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' [1] and helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' A Reality of certain mean-fields When evaluating our self-consistent mean-fields we observe that some of them are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In this appendix we explain why this is the case, beginning with a clarification of the site parity σα j �→ σα −j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' This does not act on the fermions as cj �→ c−j due to the presence of the Jordan-Wigner string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The 18 SciPost Physics Submission simplest way to deduce the effect of site parity in the fermion basis is to look at the action of site parity on fermion bilinears, which can be simply related to spin operators without semi-infinite strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In particular, we consider the following spin bilinears of definite parity A =σx i σx i+1 , B =σy i σy i+1 , C± =σx i σx i+1 ± σy i σy i+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (44) We then note that the fermionic bilinears can be decomposed in terms of these via c† i cj =1 4(A+ B − iC−) , c† j ci =1 4(A+ B + iC−) , c† i c† j =1 4(A− B − iC+) , cicj =1 4(−A+ B − iC+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (45) We thus see that the action of site parity on the bilinears is to exchange c† i cj with c† j ci and therefore ti j = 〈c† i cj〉 = t ji ∈ R as stated in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Additionally, the ANNNI Hamiltonian satisfies H = H∗ = H T in both the spin and fermion bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' In particular, in the fermion basis cicj is also real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' By the spectral theorem for real symmetric matrices we then know that the eigenvectors of H are real in the same basis and so 〈cicj〉β = 1 Z(β) � n 〈En|cicj|En〉e−βEn ∈ R (46) is manifestly real in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' However, after the quench the corresponding time-evolved quan- tity becomes 〈cicj〉t = 1 Z(β) � n,n′,m′ e−βE0 n〈En|Em′〉〈Em′|cicj|En′〉〈En′|En〉e−it(Em′−En′) , (47) where E0 n are the pre-quench energies and Em′ the post-quench energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Even if the post-quench Hamiltonian is also real and thus the post-quench energy eigenstates |En′〉 real, the phase factors will cause it to be generically complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' However, at very late times we would expect that the system would come back to equilibrium via these factors dephasing and so the correlation function should become real again at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Since tn are all real due to the site parity Z2 this implies that all effective couplings are real in equilibrium, and out of equilibrium the only complex one will be ∆(1) Eff(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' B Linear response In this appendix we summarize how to derive Kubo linear response relations after a quantum quench that occurs at time t = 0, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The Hamiltonian is of the form H(t) = θ(−t)Hi + θ(t)H f + f (t)V , (48) 19 SciPost Physics Submission where θ(t) is the Heaviside step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' If f = 0 this corresponds to a quench at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The linear response regime is when f (t) ≪ 1 and for this to be genuinely non-equilibrium we require f (t) to have support in the time period before the system thermalizes after the quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' We work in an interaction picture such that H = H0 + f (t)V, where H0 is generally not free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' The interaction picture states |ψ(t)〉I are defined by |ψ(t)〉I = eiH0tU(t, t0)|ψ(t0)〉 , (49) where U(t, t0) is the full time-evolution operator associated with H(t), |ψ(t0)〉 is the Schrödinger picture state at t0 and H0 is considered time independent by requiring, according to (48), that t0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Consistently, in the interaction picture the general operator O evolves in time as OI(t) = eiH0tOe−iH0t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (50) The time-evolution according to H(t) of the expectation value of O in the state defined at t0 by ρ(t0) = |ψ(t0)〉〈ψ(t0)| can be expressed in the interaction picture as Tr(ρ(t)O) = Tr(ρI(t)OI(t)) ≈ Tr(ρ(t0)OI(t)) − i � t t0 f (t′)χ(t, t′)dt′ , (51) where the susceptibility χ(t, t′) is given by χ(t, t′) ≡ Tr � ρ(t0)[OI(t), VI(t′)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' (52) In the last step of (51) we have expressed ρI(t) = |ψ(t)〉I I〈ψ(t)| by the first two terms in its power series in the small function f (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQftgg8/content/2301.04070v1.pdf'} +page_content=' Eq (51) is the usual linear 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b/WNAyT4oBgHgl3EQfV_f-/content/tmp_files/2301.00157v1.pdf.txt @@ -0,0 +1,1626 @@ +Ponder: Point Cloud Pre-training via Neural Rendering +Di Huang1,2 +Sida Peng3 +Tong He2,† +Xiaowei Zhou3 +Wanli Ouyang2 +The University of Sydney1 +Shanghai AI Laboratory2 +Zhejiang University3 +Abstract +We propose a novel approach to self-supervised learning +of point cloud representations by differentiable neural ren- +dering. Motivated by the fact that informative point cloud +features should be able to encode rich geometry and ap- +pearance cues and render realistic images, we train a point- +cloud encoder within a devised point-based neural renderer +by comparing the rendered images with real images on mas- +sive RGB-D data. The learned point-cloud encoder can be +easily integrated into various downstream tasks, including +not only high-level tasks like 3D detection and segmenta- +tion, but low-level tasks like 3D reconstruction and image +synthesis. Extensive experiments on various tasks demon- +strate the superiority of our approach compared to exist- +ing pre-training methods. +The code will be released at +https://dihuangdh.github.io/ponder. +1. Introduction +We have witnessed the widespread success of supervised +learning in developing vision tasks, such as image classi- +fication [11, 17] and object detection [16, 42]. In contrast +to the 2D image domain, current 3D point cloud bench- +marks only maintain limited annotations, in terms of quan- +tity and diversity, due to the extremely high cost of labo- +rious labeling. +Self-supervised learning (SSL) for point +cloud [7,18,20,22,25,31,36,40,47,52,55,58,60,61], con- +sequently, becomes one of the main driving forces and has +attracted increasing attention in the 3D research community. +Previous SSL methods for learning effective 3D rep- +resentation can be roughly categorized into two groups: +contrast-based [7, 18, 20, 22, 40, 52, 61] and completion- +based [25, 31, 36, 47, 55, 58, 60]. Contrast-based methods +are designed to maintain invariant representation under dif- +ferent transformations. To achieve this, informative sam- +ples are required. +In the 2D image domain, the above +challenge is addressed by (1) introducing efficient posi- +tive/negative sampling methods, (2) using a large batch size +and storing representative samples, and (3) applying vari- +†denote corresponding author. +3D Object Detection +3D Semantic Segmentation +3D Scene Reconstruction +Image synthesis +RGB-D +Neural Scene +Representation +Render +Compare +Figure 1. This work proposes a novel point cloud pre-training +method via neural rendering, named Ponder. Ponder is directly +trained with RGB-D image supervision, and can be used for vari- +ous applications, e.g. 3D object detection, 3D semantic segmenta- +tion, 3d scene reconstruction, and image synthesis. +ous data augmentation policies. Inspired by these works, +many works [7, 18, 20, 22, 40, 52, 61] are proposed to learn +geometry-invariant features on 3D point cloud. +Completion-based methods are another line of research +for 3D SSL, which utilizes a pre-training task of recon- +structing the masked point cloud based on partial observa- +tions. By maintaining a high masking ratio, such a simple +task encourages the model to learn a holistic understand- +ing of the input beyond low-level statistics. Although the +masked autoencoders have been successfully applied for +SSL in images [14] and videos [12,46], it remains challeng- +ing and still in exploration due to the inherent irregularity +and sparsity of the point cloud data. +Different from the two groups of methods above, we pro- +pose point cloud pre-training via neural rendering (Pon- +der). Our motivation is that neural rendering, one of the +most amazing progress and domain-specific design in 3D +vision, can be leveraged to enforce the point cloud features +being able to encode rich geometry and appearance cues. +As illustrated in Figure 1, we address the task of learning +representative 3D features via point cloud rendering. To the +best of our knowledge, this is the first exploration of neural +rendering for pre-training 3D point cloud models. Specif- +ically, given one or a sequence of RGB-D images, we lift +them to 3D space and obtain a set of colored points. Points +arXiv:2301.00157v1 [cs.CV] 31 Dec 2022 + +QRGB-D Supervision +Augmented Point Cloud +Point Cloud +Point Cloud Supervision +Encoder +Decoder +Encoder +Neural Rendering +3D Feature Volume +Contrast +Augmented Point Cloud +Augmented Point Cloud +Contrast-based +Completion-based +Ponder +Figure 2. Different types of point cloud pre-training. +are then forwarded to a 3D encoder to learn the geome- +try and appearance of the scene via a neural representation. +Provided specific parameters of the camera and the neural +representation from the encoder, neural rendering is lever- +aged to render the RGB and depth images in a differentiable +way. The network is trained to minimize the difference be- +tween rendered and observed 2D images. In doing so, our +approach enjoys multiple advantages: +• Our method is able to learn effective point cloud rep- +resentation, which encodes rich geometry and appear- +ance clues by leveraging neural rendering. +• Our method can be flexibly integrated into various +tasks. For the first time, we validate the effectiveness +of the proposed pre-training method to low-level tasks +like surface reconstruction and image synthesis tasks. +• The proposed method can leverage rich RGB-D im- +ages for pre-training. The easier accessibility of the +RGB-D data enables the possibility of 3D pre-training +on a large amount of data. +We conduct comprehensive experiments on a host of tasks. +The consistent improvements demonstrate the effectiveness +of our proposed Ponder. Our approach can serve as a strong +alternative to contrast-based methods and completion-based +methods in 3D point cloud pre-training. +2. Related Work +Neural rendering. +Neural Rendering is a type of render- +ing technology that uses neural networks to differentiablely +render images from 3D scene representation. NeRF [30] is +one of the representative neural rendering methods, which +represents the scene as the neural radiance field and renders +the images via volume rendering. Based on NeRF, there are +a series of works [4,33,34,41,49,50,56,57,59] trying to im- +prove the NeRF representation, including accelerate NeRF +training, boost the quality of geometry, and so on. Another +type of neural rendering leverages neural point clouds as +the scene representation. [2, 39] take points locations and +corresponding descriptors as input, rasterize the points with +z-buffer, and use a rendering network to get the final image. +Later work of PointNeRF [53] renders realistic images from +neural point cloud representation using a NeRF-like render- +ing process. Our work is inspired by the recent progress of +neural rendering. +Self-supervised learning in point clouds. +Current meth- +ods can be roughly categorized into two categories: +contrast-based and completion-based. +Inspired by the +works [6,15] from the 2D image domain, PointContrast [52] +is one of the pioneering works for 3D contrastive learn- +ing. Similarly, it encourages the network to learn invariant +3D representation under different transformations. Some +works [7,18,20,22,40,61] follow the pipeline by either de- +vising new sampling strategies to select informative pos- +itive/negative training pairs, or explore various types of +data augmentations. Another line of work is completion- +based [25,31,36,55,58,60] methods, which get inspiration +from Masked Autoencoders [14]. PointMAE [36] proposes +restoring the masked points via a set-to-set Chamfer Dis- +tance. VoxelMAE [31] instead recovers the underlying ge- +ometry by distinguishing if the voxel contains points. An- +other work MaskPoint [25] pre-train point cloud encoder +by performing binary classification to check if a sampled +point is occupied. Later, IAE [55] proposes to pre-train +point cloud encoder by recovering continuous 3D geometry +in an implicit manner. Different from the above pipelines, +we propose a novel framework for point cloud pre-training +via neural rendering. +Multi-modal point cloud pre-training. +Some recent +works explore the pre-training pipeline with multi-modality +data of 2D images and 3D point clouds. Pri3D [19] use 3D +point cloud and multi-view images to pre-train the 2D im- +age networks. CrossPoint [1] aligns the 2D image features +and 3D point cloud features through a contrastive learning +pipeline. [23] proposes a unified framework for exploring +the invariances with different input data formats, including +2D images and 3D point clouds. +Different from previous methods, most of which attempt +to align 2D images and 3D point clouds in the feature space, +our method proposes to connect 2D and 3D in the RGB-D +image domain via differentiable rendering. +3. Methods +An overview of our Ponder is presented in Figure 3. Pro- +vided the camera pose, 3D point clouds are obtained by pro- +jecting the RGB-D images back to 3D space (Section 3.1). +Then, we extract point-wise feature using a point cloud en- + +Multi-view +RGB-D Images +Point Cloud +Point Cloud +Encoder +SDF +RGB +Query Point +Feature +(PointNet, PointNet++ or DGCNN) +Supervision +3D Feature Volume +Rendered +RGB-D Images +View 1 +View 2 +AB6ni +cbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqseiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+g +fHjU0nGqGDZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0EPaTfrniVt05yCrxclKBHI1+as3iFkaoTRMUK27npsYP6PKcC +ZwWuqlGhPKxnSIXUsljVD72fzUKTmzyoCEsbIlDZmrvycyGmk9iQLbGVEz0sveTPzP6YmvPYzLpPUoGSLRWEqiInJ7G8y4AqZERNLKFPc3krYiCrKjE2 +nZEPwl9eJa2LqlereveXlfpNHkcRTuAUzsGDK6jDHTSgCQyG8Ayv8OYI58V5dz4WrQUnzmGP3A+fwBRuI3Tfp +AB8nicbVDLSsNAFL2pr1pfVZdugkVwVRIRdVl047KCfUAbymQ6aYdOZsLMjVBCP8 +ONC0Xc+jXu/BsnbRbaemDgcM69zLknTAQ36HnfTmltfWNzq7xd2dnd2z+oHh61jUo1ZS2qhNLdkBgmuGQt5ChYN9GMxKFgnXByl/udJ6YNV/IRpwkLYjKSPOKUoJV6/ZjgmBKRtWeDas2re3O4q8QvSA0KNAfVr/5Q0TRmEqkgxvR8L8EgIxo5FWxW6aeGJYROyIj1LJUkZibI5pFn7plVhm6ktH0S3bn6eyMjsTHT +OLSTeUSz7OXif14vxegmyLhMUmSLj6KUuGicvP73SHXjKYWkKo5jarS8dE4q2pYotwV8+eZW0L+r+Vd1/uKw1bos6ynACp3AOPlxDA+6hCS2goOAZXuHNQefFeXc+FqMlp9g5hj9wPn8Aki+Rcg=V +AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqseiF48V7Qe0oWy2k3bpZhN2N0IJ/Q +lePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0EPZ1v1xq+4cZJV4OalAjka/NUbxCyNUBomqNZdz02Mn1FlOBM4LfVSjQlYzrErqWSRqj9bH7qlJxZUDCWNmShszV3xMZjbSeRIHtjKgZ +6WVvJv7ndVMTXvsZl0lqULFojAVxMRk9jcZcIXMiIklClubyVsRBVlxqZTsiF4y+vktZF1atVvfvLSv0mj6MIJ3AK5+DBFdThDhrQBAZDeIZXeHOE8+K8Ox+L1oKTzxzDHzifP1ZEjdY=fs +AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqseiF48V7Qe0oWy2k3bpZhN2N0IJ/Q +lePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0EPZv1xq+4cZJV4OalAjka/NUbxCyNUBomqNZdz02Mn1FlOBM4LfVSjQlYzrErqWSRqj9bH7qlJxZUDCWNmShszV3xMZjbSeRIHtjKgZ +6WVvJv7ndVMTXvsZl0lqULFojAVxMRk9jcZcIXMiIklClubyVsRBVlxqZTsiF4y+vktZF1atVvfvLSv0mj6MIJ3AK5+DBFdThDhrQBAZDeIZXeHOE8+K8Ox+L1oKTzxzDHzifPz4EjcY=fc +AB8ni +cbVDLSsNAFL2pr1pfVZduBovgqiQi6rLoxmUF+4A2lMl0g6dTMLMjVBCP8ONC0Xc+jXu/BsnbRbaemDgcM69zLknSKQw6LrfTmltfWNzq7xd2dnd2z+o +Hh61TZxqxlslrHuBtRwKRvoUDJu4nmNAok7wSTu9zvPHFtRKwecZpwP6IjJULBKFqp148ojhmVWXc2qNbcujsHWSVeQWpQoDmofvWHMUsjrpBJakzPcx +P0M6pRMlnlX5qeELZhI54z1JFI278bB5Rs6sMiRhrO1TSObq742MRsZMo8BO5hHNspeL/3m9FMbPxMqSZErtvgoTCXBmOT3k6HQnKGcWkKZFjYrYWO +qKUPbUsW4C2fvEraF3Xvqu49XNYat0UdZTiBUzgHD6hAfQhBYwiOEZXuHNQefFeXc+FqMlp9g5hj9wPn8AlTmRdA=X +Figure 3. The pipeline of our point cloud pre-training via neural rendering (Ponder). Given multi-view RGB-D images, we first +construct the point cloud by back-projection, then use a point cloud encoder fp to extract per-point features E. E are organized to a 3D +feature volume (visualized as an image in this figure) by average pooling. Finally, the 3D feature volume is rendered to multi-view RGB-D +images via a differentiable neural rendering, which are compared with the original input multi-view RGB-D images as the supervision. +Point cloud encoder fp and color decoder fc are used for transfer learning. +coder (Section 3.2) and organize it to a 3D feature volume +(Section 3.3), which is used to reconstruct the neural scene +representation and render images in a differentiable manner +(Section 3.4). +3.1. Constructing point cloud from RGB-D images +The proposed method makes use of sequential RGB- +D images {(Ii, Di)}N +i=1, the camera intrinsic parameters +{Ki}N +i=1, and extrinsic poses {ξi}N +i=1 ∈ SE(3). N is the +input view number. SE(3) refers to the Special Euclidean +Group representing 3D rotations and translations. The cam- +era parameters can be easily obtained from SfM or SLAM. +We construct the point cloud X by back-projecting +RGB-D images to point clouds in a unified coordinate: +X = +N +� +i +π−1(Ii, Di, ξi, Ki), +(1) +where π−1 back-projects the RGB-D image to 3D world +space using camera poses. Note that different from pre- +vious methods which only consider the point location, our +method attributes each point with both point location and +RGB color. The details of π−1 are provided in the supple- +mentary material. +3.2. Point cloud encoder for feature extraction +Given the point cloud X constructed from RGB-D im- +ages, a point cloud encoder fp is used to extract per-point +feature embedding E: +E = fp(X). +(2) +The encoder fp pre-trained with the method mentioned in +the Section 3.4 serves as a good initialization for various +downstream tasks. +3.3. Building feature volume +Once the feature extraction is done, we map the point +embeddings E to a 3D sparse feature volume. To fill in the +empty space, we perform average pooling, followed by a +3D CNN, to aggregate features from the nearby points. The +dense 3D volume is denoted as V. +3.4. Pre-training with Neural Rendering +This section introduces how to reconstruct the implicit +scene representation and render images differentiablely. We +first give a brief introduction to neural scene representation, +then illustrate how to integrate it into our point cloud pre- +training pipeline. Last, we show the differentiable render- +ing formulation to render color and depth images from the +neural scene representation. +Brief introduction of neural scene representation. +Neural scene representation aims to represent the scene ge- +ometry and appearance through a neural network. In this +paper, we use the Signed Distance Function (SDF), which +measures the distance between a query point and the sur- +face boundary, to represent the scene geometry implicitly. +SDF is capable of representing high-quality geometry de- +tails. For any query point of the scene, the neural network +takes points features as input and outputs the corresponding +SDF value and RGB value. In this way, the neural network +captures both the geometry and appearance information of +a specific scene. Following NeuS [49], the scene can be +reconstructed as: +s(p) = ˜fs(p), +c(p, d) = ˜fc(p, d), +(3) +where ˜fs is the SDF decoder and ˜fc is the RGB color de- +coder. ˜fs takes point location p as input, and predicts the +SDF value s. ˜fc takes point location p and viewing direc- +tion d as input, and outputs the RGB color value c. Both ˜fs +and ˜fc are implemented by simple MLP networks. +Neural scene representation from point cloud input in +Ponder. +To predict a neural scene representation from the +input point cloud, we change the scene formulation to take + +3D feature volume V as an additional input. Specifically, +given a 3D query point p and viewing direction d, the fea- +ture embedding V(p) can be extracted from the processed +feature volume V by trilinear interpolation. The scene is +then represented as: +s(p) = fs(p, V(p)), +c(p, d) = fc(p, d, V(p)), +(4) +where V is predicted by the point cloud encoder fp and en- +codes information of each scene. fs and fc are SDF and +RGB decoders shared for all scenes. Different from Equa- +tion (3), which is used for storing single-scene information +in the { ˜fs, ˜fc}, the formulation in Equation (4) includes an +extra input V(p) to facilitate representing the information +of multiple scenes. +Differentiable rendering. +Given the dense 3D volume V +and viewing point, we use differentiable volume render- +ing to render the projected color images and depth images. +For each rendering ray with camera origin o and viewing +direction d, we sample a set of ray points {p(z)|p(z) = +o + zd, z ∈ [zn, zf]} along the ray, where z denotes the +length of the ray. Note that o and d can be calculated from +paired camera parameters {(Ki, ξi)}. zn and zf denote +the near and far bounds of the ray. Different from previ- +ous methods [30,49], we automatically determine {zn, zf} +by the ray intersection with the 3D feature volume box, us- +ing axis-aligned bounding boxes (AABB) algorithm. Then, +the ray color and depth value can be aggregated as: +ˆC = +� zf +zn +ω(z)c(p(z), d)dz, +(5) +ˆD = +� zf +zn +ω(z)zdz, +(6) +where the ˆC is the ray color and the ˆD is the ray depth. +We follow NeuS [49] to build an unbiased and occlusion- +awareness weight function w(z): +w(z) = T(z) · ρ(z). +(7) +T(z) measures the accumulated transmittance from zn to z +and ρ(z) is the occupied density function which are defined +as: +T(z) = exp(− +� zf +zn +ρ(z)dz), +(8) +ρ(z) = max +�− dΦh +dz (s(p(z))) +Φh(s(p(z))) , 0 +� +. +(9) +Φh(x) is the Sigmoid function Φh(x) = (1 + e−hx)−1 +where h−1 is treated as a trainable parameter, h−1 ap- +proaches to zero as the network training converges. In prac- +tice, we use a numerically approximated version by quadra- +ture. +We make the decode networks {fs, fc} relatively +smaller than [30,49] to accelerate the training process. +Projected Points +Rendered Color +Reference Color +Rendered Depth +Reference Depth +Figure 4. Rendered images by Ponder on the ScanNet validation +set. The projected point clouds are visualized in the first column. +Even though input point clouds are very sparse, our model is still +capable of rendering color and depth images similar to the refer- +ence images. +Rendered examples. +The rendered color images and +depth images are shown in Figure 4. As shown in the fig- +ure, even though the input point cloud is pretty sparse, our +method is still capable of rendering color and depth images +similar to the reference image. +3.5. Pre-training loss +We leverage the input {Ii, Di} to supervise neural scene +representation reconstruction. The total loss function con- +tains five parts, +L = λcLc + λdLd + λeLe + λsLs + λfLf, +(10) +which are loss functions responsible for color supervision +Lc, depth supervision Ld, Eikonal regularization Le, near- +surface SDF supervision Ls, and free space SDF supervi- +sion Lf. These loss functions are illustrated in the follow- +ing section. +Color and depth loss. +Lc and Ld are the color loss and +depth loss, which measure consistency between the ren- +dered pixels and the ground-truth pixels. Assume that we +sample Nr rays for each image and Np points for each ray, +then the Lc and Ld can be written as: +Lc = 1 +Nr +Nr +� +i +|| ˆC − C||2 +2 +(11) +Ld = 1 +Nr +Nr +� +i +|| ˆD − D||2 +2, +(12) +where C and D are the ground-truth color and depth re- +spectively for each ray, ˆC and ˆD are their corresponding +rendered ones in Eq. (5) and Eq. (6). + ++**+Loss for SDF regularization. +Le is the widely used +Eikonal loss [13] for SDF regularization: +Le = +1 +NrNp +Nr,Np +� +i,j +(|∇s(pi,j)| − 1)2, +(13) +where ∇s(pi,j) denotes the gradient of SDF s at location +pi,j. Since SDF is a distance measure, Le encourages this +distance to have a unit norm gradient at the query point. +Near-surface and free space loss for SDF. +To stabilize +the training and improve the reconstruction performance, +similar to iSDF [35] and GO-Surf [48], we add additional +approximate SDF supervision to help the SDF estimation. +Specifically, for near-surface points, the difference between +rendered depth and ground-truth depth can be viewed as the +pseudo-SDF ground-truth supervision; for points far from +the surface, a free space loss is used to regularize the irreg- +ular SDF value additionally. To calculate the approximate +SDF supervision, we first define an indicator b(z) for each +sampled ray point with ray length z and corresponding GT +depth D: +b(z) = D − z. +(14) +b(z) can be viewed as the approximate SDF value, which is +credible only when b(z) is small. Let t be a human-defined +threshold, which is set as 0.05 in this paper. For sampled ray +points that satisfy b(z) ≤ t, we leverage the near-surface +SDF loss to constrain the SDF prediction s(zi,j): +Ls = +1 +NrNp +Nr,Np +� +i,j +|s(zi,j) − b(zi,j)|. +(15) +For the remaining sampled ray points, we use a free space +loss: +Lf = +1 +NrNp +Nr,Np +� +i,j +max(0, e−α·s(zi,j)−1, s(zi,j)−b(zi,j)), +(16) +where α is set as 5 following the same with [35, 48]. Note +that due to the noisy depth images, we only apply Ls and +Lf on the rays that have valid depth values. +In our experiments, we follow a similar loss of weight +with GO-Surf [48], which sets λc as 10.0, λd as 1.0, λs as +10.0, and λf as 1.0. We observe that the Eikonal term in +our method can easily lead to over-smooth reconstructions, +thus we use a small weight of 0.01 for the Eikonal loss. +4. Experiments +4.1. Pre-training +Datasets. +We use ScanNet [10] RGB-D images as our +pre-training data. ScanNet is a widely used real-world in- +door dataset, which contains more than 1500 indoor scenes. +Each scene is carefully scanned by an RGB-D camera, lead- +ing to about 2.5 million RGB-D frames in total. We follow +the same train/val split with VoteNet [38]. +Data preparation. +During pre-training, a mini-batch of +batch size 8 includes point clouds from 8 scenes. The point +cloud of a scene, serving as the input of the point cloud en- +coder in our approach, is back-projected from the 5 RGB-D +frames of the video for the scene with an interval of 20. The +5 frames are also used as the supervision of the network. +Data augmentation. +We augment the point cloud by ran- +dom sampling, normalization, and random masking. First, +we randomly down-sample the point cloud to 20,000 points. +Then, the point cloud is normalized into a 3D unit cube. Fi- +nally, we apply the same masking strategy as used in Mask +Point [25]. Specifically, we use FPS to split the point cloud +into 2,048 groups, each group containing 64 points, then +mask the point groups with a mask ratio of 90%. +Implementation details. +We train the proposed pipeline +for 100 epochs using an AdamW optimizer [29] with a +weight decay of 0.05. The learning rate is initialized as 1e- +4 with Exponential scheduling. For the rendering process, +we randomly choose 128 rays for each image and sample +128 points for each ray. More implementation details can +be found in the supplementary materials. +4.2. Transfer Learning +In contrast to previous methods, our approach is able to +encode rich geometry and appearance cues into the point +cloud representations via neural rendering. These strengths +make it flexible to be applied to various tasks, including not +only 3D semantic segmentation and 3D detection tasks but +also low-level surface reconstruction and image synthesis. +4.2.1 +High-level 3D Tasks +3D object detection. +For transfer learning on 3D ob- +ject detection task, we use VoteNet [38] as the baseline. +VoteNet leverage a voting mechanism to generate object +centers, which are used for 3D bounding box proposals. +Two datasets are applied to verify the effectiveness of our +method: ScanNet [10] and SUN RGB-D [44]. +Differ- +ent from ScanNet, which contains fully reconstructed 3D +scenes, SUN RGB-D is a single-view RGB-D dataset with +3D bounding box annotations. It has 10,335 RGB-D images +for 37 object categories. For pre-training, we use Point- +Net++ as the point cloud encoder fp, which is identical to +the backbone used in VoteNet. We pre-train the point cloud +encoder on the ScanNet dataset and transfer the weight as +the VoteNet initialization. Following [38], we use average +precision with 3D detection IoU threshold 0.25 and thresh- +old 0.5 as the evaluation metrics. + +Method +Detection +Pre-training +Pre-training +Pre-training +ScanNet +SUN RGB-D +Model +Type +Data +Epochs +AP50 ↑ +AP25 ↑ +AP50 ↑ +AP25 ↑ +3DETR [32] +3DETR +- +- +- +37.5 +62.7 +30.3 +58.0 +Point-BERT [58] +3DETR +Completion +3D Model +300 +38.3 +61.0 +- +- +MaskPoint [25] +3DETR +Completion +Depth +300 +40.6 +63.4 +- +- +VoteNet [38] +VoteNet +- +- +- +33.5 +58.6 +32.9 +57.7 +STRL [20] +VoteNet +Contrast +Depth +100 +38.4 +59.5 +35.0 +58.2 +RandomRooms [40] +VoteNet +Contrast +Synthesis +300 +36.2 +61.3 +35.4 +59.2 +PointContrast [52] +VoteNet +Contrast +3D Model +- +38.0 +59.2 +34.8 +57.5 +PC-FractalDB [54] +VoteNet +Contrast +Synthesis +- +38.3 +61.9 +33.9 +59.4 +DepthContrast [61] +VoteNet +Contrast +Depth +1000 +39.1 +62.1 +35.4 +60.4 +IAE [55] +VoteNet +Completion +3D Model +1000 +39.8 +61.5 +36.0 +60.4 +Ponder +VoteNet +Rendering +Depth +100 +40.9 +64.2 +36.1 +60.3 +Ponder +VoteNet +Rendering +Color & Depth +100 +41.0 +63.6 +36.6 +61.0 +Table 1. 3D object detection AP25 and AP50 on ScanNet and SUN RGB-D. VoteNet [38] and 3DETR [32] are two baseline 3D object de- +tection models. The DepthContrast [61] and Point-BERT [58] results are adopted from IAE [55] and MaskPoint [25]. Ponder outperforms +both VoteNet-based and 3DETR-based point cloud pre-training methods with fewer training epochs. +The 3D detection results are shown in Table 1. +Our +method improves the baseline of VoteNet without pre- +training by a large margin, boosting AP50 by 7.5% and +3.7% for ScanNet and SUN RGB-D, respectively. IAE [55] +is a pre-training method that represents the inherent 3D ge- +ometry in a continuous manner. Our learned point cloud +representation achieves higher accuracy because it is able +to recover both the geometry and appearance of the scene. +The AP50 and AP25 of our method are higher than that of +IAE by 1.2% and 2.1% on ScanNet, respectively. Mask- +Point [25] is another method aiming to learn a continuous +surface by classifying if the query point is occupied. How- +ever, its performance can be constrained due to the noisy +labeling of the query point occupancy value. As presented +in Table 1, even with an inferior backbone (PointNet++ vs +3DETR), our method is able to achieve better accuracy with +fewer pre-training epochs. +3D semantic segmentation. +3D semantic segmentation +is another fundamental scene understanding task. Follow- +ing [43,47,55], we choose DGCNN [51] as our baseline for +a fair comparison. DGCNN applies a dynamic graph CNN +as the backbone. For pre-training, we use DGCNN as the +point cloud encoder fp, and pre-train the model on ScanNet. +We validate the effectiveness of our method by transfer- +ring the weights to Stanford Large-Scale3D Indoor Spaces +(S3DIS) [3] dataset, which is an indoor 3D understanding +dataset containing 6 large-scale indoor scenes with point se- +mantic annotations. Following the same setting of [51], we +use the overall accuracy (OA) mean IoU(mIoU) on points +as the evaluation metric, and report the average evaluation +results across six folds. +Table 2 shows the quantitative results. Compared with +the DGCNN baseline, the proposed method boost the seg- +mentation performance by a large margin, boosting OA +and mIoU for 2.1% and 5%, respectively. +Jigsaw and +OcCo use ShapeNet as the pre-train dataset. +Although +they get improvements compared with the baseline, the +limited scale of training data constrains the transferring +ability. IAE achieves significant improvements by lever- +aging the large-scale dataset and an implicit reconstruc- +tion manner. Compared with IAE, the proposed approach +achieves a higher semantic segmentation performance with +the DGCNN backbone (+0.3% for OA and +0.4% for +mIoU). Besides, IAE requires a large amount of 3D mesh +for supervision. Our approach, in contrast, only requires +RGB-D images as the supervision, which is much cheaper +and easy to fetch. +4.2.2 +Low-level 3D Tasks +Low-level 3D tasks like scene reconstruction and image +synthesis are getting increasing attention due to their wide +applications. +However, most of them are trained from +scratch. How to pre-train a model with a good initialization +is desperately needed. We are the first pre-training work to +demonstrate a strong transferring ability to such low-level +3D tasks. +3D scene reconstruction. +3D scene reconstruction task +aims to recover the scene geometry, e.g. +mesh, from +the point cloud input. We choose ConvONet [37] as the +baseline model, whose architecture are widely adopted +in [9,26,56]. Following the same setting as ConvONet, we +conduct experiments on the Synthetic Indoor Scene Dataset +(SISD) [37], which is a synthetic dataset and contains 5000 +scenes with multiple ShapeNet [5] objects. We pre-train the +PointNet encoder, which is the same as the original Con- + +Method +OA↑ +mIoU↑ +DGCNN [51] +84.1 +56.1 +Jigsaw [43] +84.4 +56.6 +OcCo [47] +85.1 +58.5 +IAE [55] +85.9 +60.7 +Ponder +86.2 +61.1 +Table 2. +3D semantic segmentation OA +and mIoU on S3DIS dataset with DGCNN +model. Ponder outperforms previous state-of- +the-art models. +Method +Encoder +IoU↑ +Normal Consistency↑ +F-Score↑ +ConvONet [37] +PointNet +84.9 +0.915 +0.964 +Ponder +PointNet +85.7 +0.917 +0.965 +ConvONet +PointNet++ +77.8 +0.887 +0.906 +Ponder +PointNet++ +80.2 +0.893 +0.920 +Table 3. 3D scene reconstruction IoU, NC, and F-Score on SISD dataset with +PointNet and PointNet++ model. For both PointNet and PointNet++, +Ponder is able to boost the reconstruction performance. +Figure 5. Comparison of image synthesis from point clouds. +Compared with training from scratch, our Ponder model is able to +converge faster and achieve better image synthesis results. +vONet implementation, and test the reconstruction quality +on the SISD dataset. Additionally, to compare with another +self-supervised learning method IAE [55], we add extra ex- +periments using VoteNet-style PointNet++ as the encoder +of ConvONet. Following [37], we use Volumetric IoU, Nor- +mal Consistency, and F-Score [45] with the threshold value +of 1% as the evaluation metrics. +The results are shown in Table 3. +Compared to the +baseline ConvONet model with PointNet, the proposed ap- +proach is able to improve the reconstruction quality (+0.8% +for IoU). By replacing the encoder of ConvONet from +PointNet to PointNet++, ours is able to achieve more accu- +racy improvement (+2.4% for IoU and +0.014 for F-Score). +Our method also gets better reconstruction results than IAE. +Check our supplementary materials for more details. +Image synthesis from point clouds. +We also validate the +effectiveness of our method on another low-level task of im- +age synthesis from point clouds. We use Point-NeRF [53] +as the baseline. Point-NeRF uses neural 3D point clouds +with associated neural features to render images. It can be +used both for a generalizable setting for various scenes and a +single-scene fitting setting. In our experiments, we mainly +focus on the generalizable setting of Point-NeRF. We re- +Supervision +ScanNet +SUN RGB-D +Depth +40.9 +36.1 +Color +40.5 +35.8 +Color+RGB +41.0 +36.6 +Table 4. Ablation study for supervision type. 3D detection AP50 +on ScanNet and SUN RGB-D. Combining color supervision and +depth supervision can lead to better detection performance than +using a single type of supervision. +place the 2D image features of Point-NeRF with point fea- +tures extracted by a DGCNN network. Following the same +setting with PointNeRF, we use DTU [21] as the evaluation +dataset. DTU dataset is a multiple-view stereo dataset con- +taining 80 scenes with paired images and camera poses. We +transfer both the DGCNN encoder and color decoder as the +weight initialization of Point-NeRF. We use PSNR as the +metric for synthesized image quality evaluation. +The results are shown in Figure 5. By leveraging the pre- +trained weights of our method, the image synthesis model is +able to converge faster with fewer training steps and achieve +better final image quality than training from scratch. +4.3. Ablation study +In this section, we do two ablation experiments. First, +we show the effectiveness of using different 2D supervi- +sion. Then, we test how the view number affects the final +performance. For both experiments, we use the 3D object +detection task as the transfer learning task. +Influence on Rendering Targets. +The rendering part of +our method contains two items: RGB color image and depth +image. We study the influence of each item with the trans- +ferring task of 3D detection. The results are presented in Ta- +ble 4. Combining depth and color images for reconstruction +shows the best detection results. In addition, using depth re- +construction presents better performance than color recon- +struction for 3D detection. + +Image Synthesis +20 +18 +PSNR +16 +14 +12 +scratch +ours +10 +100 +200 +500 +2000 +5000 +Train StepsInput Point Cloud +Projected Point Cloud +Reconstruction +Image Synthesis +Reference Image +Depth Synthesis +Reference Depth +Figure 6. Direct applications of Ponder on the ScanNet validation set. The proposed Ponder model can be directly used for various +applications, such as 3D reconstruction and image synthesis. The input point clouds are drawn as spheres for better clarity. +Number of input RGB-D view. +Our method utilizes N +RGB-D images, where N is the input view number. We +study the influence of N and conduct experiments on 3D +detection, as shown in Table 5. Using multi-view super- +vision helps to reduce single-view ambiguity. Similar ob- +servations are also found in the multi-view reconstruction +task [27]. Compared with the single view, multiple views +achieve higher accuracy, boosting AP50 by 0.9% and 1.2% +for ScanNet and Sun RGB-D datasets, respectively. +4.4. Other applications +In previous sections, we show that the proposed pipeline +can be used for transfer learning. In this section, we show +that the pre-trained model from our pipeline Ponder itself +can also be directly used for surface reconstruction and im- +age synthesis from sparse point clouds. +3D reconstruction from sparse point clouds. +The +learned model has the capability to recover the scene sur- +face from sparse point clouds. Specifically, after learning +the neural scene representation, we query the SDF value in +the 3D space and leverage the Marching Cubes [28] to ex- +tract the surface. We show the reconstruction results in Fig- +ure 6. The results show that even though the input is sparse +point clouds from complex scenes, our method is able to +recover high-fidelity meshes. +Image synthesis from sparse point clouds. +Another in- +teresting experiment to explore is that our pipeline is able +to render realistic images from sparse point cloud input. As +shown in Figure 6, our method is able to recover similar +color images with the ground truth. Also, the recovered +depth may even look better compared with the ground-truth +depth image which has irregular values. +View number +ScanNet +SUN RGB-D +1 view +40.1 +35.4 +3 views +40.8 +36.0 +5 views +41.0 +36.6 +Table 5. Ablation study for view number. 3D detection AP50 +on ScanNet and SUN RGB-D. Using multi-view supervision for +point cloud pre-training can achieve better performance. +5. Conclusion +In this paper, we show that differentiable neural render- +ing is a powerful tool for point cloud representation learn- +ing. The proposed pre-training pipeline, Ponder, is able to +encode rich geometry and appearance cues into the point +cloud representation via neural rendering. For the first time, +our model can be transferred to not only high-level 3D per- +ception tasks but also 3D low-level tasks, like 3D recon- +struction and image synthesis from point clouds. Also, the +learned Ponder model can be directly used for 3D recon- +struction and image synthesis from sparse point clouds. +Several directions could be explored in future works. +First, there are various types of neural rendering, which +could also be leveraged for point cloud representation learn- +ing. Second, other 3D domain-specific designs could be +integrated into point cloud pre-training pipelines. Third, +exploring the proposed pre-training pipeline Ponder on a +larger dataset and more downstream tasks is also a potential +research direction. + +++ +++#+083 +9References +[1] Mohamed Afham, Isuru Dissanayake, Dinithi Dissanayake, +Amaya Dharmasiri, Kanchana Thilakarathna, and Ranga Ro- +drigo. Crosspoint: Self-supervised cross-modal contrastive +learning for 3d point cloud understanding. 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Building a 3D hierarchical feature volume has been +wildly used for recovering detailed 3D geometry, e.g. [8,9]. +After processing the 3D feature volume with a 3D CNN, +we use trilinear interpolation to get the feature of the query +point p, denoted as V(p). We use the drop-in replacement +of grid sampler from [48] to accelerate the training. +Ray sampling strategy. +Similar to [30, 49], we sample +twice for each rendering ray. First, we uniformly sample +coarse points between the near bound zn and far bound zf. +Then, we use importance sampling with the coarse proba- +bility estimation to sample fine points. Folowing [49], the +coarse probability is calculated based on Φh(s). By this +sampling strategy, our method can automatically determine +sample locations and can collect more points near the sur- +face, which makes the training process more efficient. +Back projection +Here we give details of the back projec- +tion function π−1 to get point clouds from depth images. +Let K be camera intrinsic parameters, ξ = [R|t] be camera +extrinsic parameters, where R is the rotation matrix and t is +the translation matrix. Xuv is the projected point location +and Xw is the point location in the 3D world coordinate. +Then, according to the pinhole camera model: +sXuv = K(RXw + t), +(17) +where s is the depth value. After expanding the Xuv and +Xw: +s +� +� +u +v +1 +� +� = K(R +� +� +X +Y +Z +� +� + t). +(18) +Then, the 3D point location can be calculated as follows: +� +� +X +Y +Z +� +� = R−1(K−1s +� +� +u +v +1 +� +� − t) +(19) +The above Equation 19 is the back-projection equation π−1 +used in this paper. +Training Time. +The Ponder model is trained with 8 +NVIDIA A100 GPUs for 96 hours. +A.2. Transfer Learning Details +3D scene reconstruction. +ConvONet [37] reconstructs +scene geometry from the point cloud input. It follows a +two-step manner, which first encodes the point cloud into +a 3D feature volume or multiple feature planes, then de- +codes the occupancy probability for each query point. To +test the transfer learning ability of our point cloud encoder, +we directly replace the point cloud encoder of ConvONet, +without any other modification. We choose the highest per- +forming configuration of ConvONet as the baseline setting, +which uses a 3D feature volume with a resolution of 64. +For the training of ConvONet, we follow the same training +setting as the released code1. +Image synthesis from point clouds. +Point-NeRF [53] +renders images from neural point cloud representation. It +first generates neural point clouds from multi-view images, +then uses point-based volume rendering to synthesize im- +ages. To transfer the learned network weight to the Point- +NeRF pipeline, we 1) replace the 2D image feature back- +bone with a pre-trained point cloud encoder to get the neural +point cloud features, 2) replace the color decoder by a pre- +trained color decoder, 3) keep the other Point-NeRF module +untouched. Since a large amount of point cloud is hard to +be directly processed by the point cloud encoder, we down- +sample the point cloud to 1%, which will decrease the ren- +dering quality but help reduce the GPU memory require- +ments. We report the PSNR results of the unmasked region +as the evaluation metric, which is directly adopted from the +original codebase2. For training Point-NeRF, we follow the +same setting as Point-NeRF. +B. Supplementary Experiments +B.1. Ablation Study +1https://github.com/autonomousvision/convolutional occupancy networks +2https://github.com/Xharlie/pointnerf + +Mask ratio +ScanNet +SUN RGB-D +0% +40.7 +37.3 +25% +40.7 +36.2 +50% +40.3 +36.9 +75% +41.7 +37.0 +90% +41.0 +36.6 +Table 6. Ablation study for mask ratio. 3D detection AP50 on +ScanNet and SUN RGB-D. +Resolution +ScanNet +SUN RGB-D +16 +40.7 +36.6 +16+32+64 +41.0 +36.6 +Table 7. +Ablation study for feature volume resolution. +3D detection AP50 on ScanNet and SUN RGB-D. +Method +IoU↑ +Normal Consistency↑ +F-Score↑ +ConvOcc [37] +0.778 +0.887 +0.906 +IAE [55] +0.757 +0.887 +0.910 +Ours +0.802 +0.893 +0.920 +Table 8. 3D scene reconstruction IoU, NC, and F-Score on SISD +dataset with PointNet++ model. +Influence on mask ratio. +In this paper, we use random +masking as one type of point cloud augmentation. We ap- +ply the same mask ratio as MaskPoint [25]. Here, we give +additional experimental results to show the influence of us- +ing different mask ratios in Table 6. For the mask ratio of +0%, we do not apply any mask strategy to the input point +cloud. +3D feature volume resolution. +As mentioned in Section +A, Ponder build a 3D feature volume with a resolution of +[16, 32, 64], which is inspired by the recent progress of +multi-resolution in 3D reconstruction. However, building +such a 3D feature volume with large resolutions requires +heavy GPU memory usage. We conduct experiments in Ta- +ble 7 to test the performance with a smaller resolution. As +shown in the table, even with a small resolution, Ponder +is still able to achieve comparable accuracy, demonstrating +the robustness to the feature volume resolution. +B.2. Transfer Learning +3D scene reconstruction +As mentioned in the paper, we +transfer the learned PointNet++ model of IAE to the 3D +reconstruction task. The results are shown in Table 8. Com- +pared with the ConvONet baseline, the IAE pre-trained +model gets a better F-Score with 0.004 but gets worse re- +sults on the IoU metric. Our method, on the other hand, +Figure 7. Label efficiency training. We show the 3d object de- +tection experiment results using limited downstream data. Our +pretrained model is capable of achieving better performance than +training from scratch using the same percentage of data or requires +fewer data to get the same detection accuracy. +gets a better reconstruction performance than both the Con- +vONet and IAE. +Label Efficiency Training. +We also do experiments to +show the performance of our method with limited label- +ing for the downstream task. Specifically, we test the la- +bel efficiency training on the 3D object detection task for +ScanNet. Following the same setting with IAE [55], we +use 20%, 40%, 60%, and 80% of ground truth annotations. +The results are shown in Figure 7. We show constantly im- +proved results over training from scratch, especially when +only 20% of the data is available. +Color information for downstream tasks. +Different +from previous works, since our pre-training model uses a +colored point cloud as the input, we also use color informa- +tion for the downstream tasks. Results are shown in Table +9. Using color as an additional point feature can help the +VoteNet baseline achieve better performance on the SUN +RGB-D dataset, but get little improvement on the ScanNet +dataset. This shows that directly concatenating point posi- +tions and colors as point features shows limited robustness +to application scenarios. By leveraging the proposed Pon- +der pre-training method, the network is well initialized to +handle the point position and color features, and achieve +better detection accuracy. +More comparisons on 3D detection. +More detection ac- +curacy comparisons are given in Table 9. Even using an in- +ferior backbone, our Ponder model is able to achieve simi- +lar detection accuracy with 9 in ScanNet and better accuracy +in SUN RGB-D. + +Label Efficiency Training +65 +60 +Detection Result +55 +50 +45 +40 +scratch +35 +ours +30 +20 +40 +60 +80 +100 +Labeled Data PercentageMethod +Detection +Pre-training +Pre-training +Pre-training +ScanNet +SUN RGB-D +Model +Type +Data +Epochs +AP50 ↑ +AP25 ↑ +AP50 ↑ +AP25 ↑ +VoteNet* +VoteNet* +- +- +- +37.6 +60.0 +33.3 +58.4 +DPCo [24] +VoteNet* +Contrast +Depth +120 +41.5 +64.2 +35.6 +59.8 +IPCo [24] +VoteNet* +Contrast +Color & Depth +120 +40.9 +63.9 +35.5 +60.2 +VoteNet (w color) +VoteNet +- +- +- +33.4 +58.8 +34.3 +58.3 +Ponder +VoteNet +Rendering +Depth +100 +40.9 +64.2 +36.1 +60.3 +Ponder +VoteNet +Rendering +Color & Depth +100 +41.0 +63.6 +36.6 +61.0 +Table 9. 3D object detection AP25 and AP50 on ScanNet and SUN RGB-D. * means a different but stronger version of VoteNet. +B.3. More application examples +As mentioned in the paper, the pre-trained Ponder model +can be directly used for surface reconstruction and image +synthesis tasks. We give more application examples in Fig- +ure 8 and Figure 9. + +Input Point Cloud +Reconstruction +Image Synthesis +Depth Synthesis +Projected Point Cloud +Figure 8. More results of application examples of Ponder on the ScanNet validation set (part 1). The input point clouds are drawn as +spheres for better clarity. + ++$: +++++ +++ +++ ++t ++++++ ++++*+Input Point Cloud +Reconstruction +Image Synthesis +Depth Synthesis +Projected Point Cloud +Figure 9. More results of application examples of Ponder on the ScanNet validation set (part 2). The input point clouds are drawn as +spheres for better clarity. + ++ ++++ ++* +++ +t +**+ +*. ++* ++4 ++ +*+++ +*+* ++*+ ++ +主 ++ ++++ +. ++# \ No newline at end of file diff --git a/WNAyT4oBgHgl3EQfV_f-/content/tmp_files/load_file.txt b/WNAyT4oBgHgl3EQfV_f-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..927ffd27ca308c4a6acef42b326eacccbfb01ea5 --- /dev/null +++ b/WNAyT4oBgHgl3EQfV_f-/content/tmp_files/load_file.txt @@ -0,0 +1,827 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf,len=826 +page_content='Ponder: Point Cloud Pre-training via Neural Rendering Di Huang1,2 Sida Peng3 Tong He2,† Xiaowei Zhou3 Wanli Ouyang2 The University of Sydney1 Shanghai AI Laboratory2 Zhejiang University3 Abstract We propose a novel approach to self-supervised learning of point cloud representations by differentiable neural ren- dering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Motivated by the fact that informative point cloud features should be able to encode rich geometry and ap- pearance cues and render realistic images, we train a point- cloud encoder within a devised point-based neural renderer by comparing the rendered images with real images on mas- sive RGB-D data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The learned point-cloud encoder can be easily integrated into various downstream tasks, including not only high-level tasks like 3D detection and segmenta- tion, but low-level tasks like 3D reconstruction and image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Extensive experiments on various tasks demon- strate the superiority of our approach compared to exist- ing pre-training methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The code will be released at https://dihuangdh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='io/ponder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Introduction We have witnessed the widespread success of supervised learning in developing vision tasks, such as image classi- fication [11, 17] and object detection [16, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' In contrast to the 2D image domain, current 3D point cloud bench- marks only maintain limited annotations, in terms of quan- tity and diversity, due to the extremely high cost of labo- rious labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Self-supervised learning (SSL) for point cloud [7,18,20,22,25,31,36,40,47,52,55,58,60,61], con- sequently, becomes one of the main driving forces and has attracted increasing attention in the 3D research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Previous SSL methods for learning effective 3D rep- resentation can be roughly categorized into two groups: contrast-based [7, 18, 20, 22, 40, 52, 61] and completion- based [25, 31, 36, 47, 55, 58, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Contrast-based methods are designed to maintain invariant representation under dif- ferent transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' To achieve this, informative sam- ples are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' In the 2D image domain, the above challenge is addressed by (1) introducing efficient posi- tive/negative sampling methods, (2) using a large batch size and storing representative samples, and (3) applying vari- †denote corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3D Object Detection 3D Semantic Segmentation 3D Scene Reconstruction Image synthesis RGB-D Neural Scene Representation Render Compare Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' This work proposes a novel point cloud pre-training method via neural rendering, named Ponder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Ponder is directly trained with RGB-D image supervision, and can be used for vari- ous applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3D object detection, 3D semantic segmenta- tion, 3d scene reconstruction, and image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' ous data augmentation policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Inspired by these works, many works [7, 18, 20, 22, 40, 52, 61] are proposed to learn geometry-invariant features on 3D point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Completion-based methods are another line of research for 3D SSL, which utilizes a pre-training task of recon- structing the masked point cloud based on partial observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' By maintaining a high masking ratio, such a simple task encourages the model to learn a holistic understand- ing of the input beyond low-level statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Although the masked autoencoders have been successfully applied for SSL in images [14] and videos [12,46], it remains challeng- ing and still in exploration due to the inherent irregularity and sparsity of the point cloud data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Different from the two groups of methods above, we pro- pose point cloud pre-training via neural rendering (Pon- der).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Our motivation is that neural rendering, one of the most amazing progress and domain-specific design in 3D vision, can be leveraged to enforce the point cloud features being able to encode rich geometry and appearance cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' As illustrated in Figure 1, we address the task of learning representative 3D features via point cloud rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' To the best of our knowledge, this is the first exploration of neural rendering for pre-training 3D point cloud models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Specif- ically, given one or a sequence of RGB-D images, we lift them to 3D space and obtain a set of colored points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Points arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='00157v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='CV] 31 Dec 2022 QRGB-D Supervision Augmented Point Cloud Point Cloud Point Cloud Supervision Encoder Decoder Encoder Neural Rendering 3D Feature Volume Contrast Augmented Point Cloud Augmented Point Cloud Contrast-based Completion-based Ponder Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Different types of point cloud pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' are then forwarded to a 3D encoder to learn the geome- try and appearance of the scene via a neural representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Provided specific parameters of the camera and the neural representation from the encoder, neural rendering is lever- aged to render the RGB and depth images in a differentiable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The network is trained to minimize the difference be- tween rendered and observed 2D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' In doing so, our approach enjoys multiple advantages: Our method is able to learn effective point cloud rep- resentation, which encodes rich geometry and appear- ance clues by leveraging neural rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Our method can be flexibly integrated into various tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' For the first time, we validate the effectiveness of the proposed pre-training method to low-level tasks like surface reconstruction and image synthesis tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The proposed method can leverage rich RGB-D im- ages for pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The easier accessibility of the RGB-D data enables the possibility of 3D pre-training on a large amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We conduct comprehensive experiments on a host of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The consistent improvements demonstrate the effectiveness of our proposed Ponder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Our approach can serve as a strong alternative to contrast-based methods and completion-based methods in 3D point cloud pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Related Work Neural rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Neural Rendering is a type of render- ing technology that uses neural networks to differentiablely render images from 3D scene representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' NeRF [30] is one of the representative neural rendering methods, which represents the scene as the neural radiance field and renders the images via volume rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Based on NeRF, there are a series of works [4,33,34,41,49,50,56,57,59] trying to im- prove the NeRF representation, including accelerate NeRF training, boost the quality of geometry, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Another type of neural rendering leverages neural point clouds as the scene representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' [2, 39] take points locations and corresponding descriptors as input, rasterize the points with z-buffer, and use a rendering network to get the final image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Later work of PointNeRF [53] renders realistic images from neural point cloud representation using a NeRF-like render- ing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Our work is inspired by the recent progress of neural rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Self-supervised learning in point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Current meth- ods can be roughly categorized into two categories: contrast-based and completion-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Inspired by the works [6,15] from the 2D image domain, PointContrast [52] is one of the pioneering works for 3D contrastive learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Similarly, it encourages the network to learn invariant 3D representation under different transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Some works [7,18,20,22,40,61] follow the pipeline by either de- vising new sampling strategies to select informative pos- itive/negative training pairs, or explore various types of data augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Another line of work is completion- based [25,31,36,55,58,60] methods, which get inspiration from Masked Autoencoders [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' PointMAE [36] proposes restoring the masked points via a set-to-set Chamfer Dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' VoxelMAE [31] instead recovers the underlying ge- ometry by distinguishing if the voxel contains points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' An- other work MaskPoint [25] pre-train point cloud encoder by performing binary classification to check if a sampled point is occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Later, IAE [55] proposes to pre-train point cloud encoder by recovering continuous 3D geometry in an implicit manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Different from the above pipelines, we propose a novel framework for point cloud pre-training via neural rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Multi-modal point cloud pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Some recent works explore the pre-training pipeline with multi-modality data of 2D images and 3D point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Pri3D [19] use 3D point cloud and multi-view images to pre-train the 2D im- age networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' CrossPoint [1] aligns the 2D image features and 3D point cloud features through a contrastive learning pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' [23] proposes a unified framework for exploring the invariances with different input data formats, including 2D images and 3D point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Different from previous methods, most of which attempt to align 2D images and 3D point clouds in the feature space, our method proposes to connect 2D and 3D in the RGB-D image domain via differentiable rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Methods An overview of our Ponder is presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Pro- vided the camera pose, 3D point clouds are obtained by pro- jecting the RGB-D images back to 3D space (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' we extract point-wise feature using a point cloud en- Multi-view RGB-D Images Point Cloud Point Cloud Encoder SDF RGB Query Point Feature (PointNet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' PointNet++ or DGCNN) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='Supervision ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='3D Feature Volume ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='Rendered ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='RGB-D 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='Hh61TZxqxlslrHuBtRwKRvoUDJu4nmNAok7wSTu9zvPHFtRKwecZpwP6IjJULBKFqp148ojhmVWXc2qNbcujsHWSVeQWpQoDmofvWHMUsjrpBJakzPcx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='P0M6pRMlnlX5qeELZhI54z1JFI278bB5Rs6sMiRhrO1TSObq742MRsZMo8BO5hHNspeL/3m9FMbPxMqSZErtvgoTCXBmOT3k6HQnKGcWkKZFjYrYWO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='qKUPbUsW4C2fvEraF3Xvqu49XNYat0UdZTiBUzgHD6hAfQhBYwiOEZXuHNQefFeXc+FqMlp9g5hj9wPn8AlTmRdA=X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The pipeline of our point cloud pre-training via neural rendering (Ponder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Given multi-view RGB-D images, we first construct the point cloud by back-projection, then use a point cloud encoder fp to extract per-point features E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' E are organized to a 3D feature volume (visualized as an image in this figure) by average pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Finally, the 3D feature volume is rendered to multi-view RGB-D images via a differentiable neural rendering, which are compared with the original input multi-view RGB-D images as the supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Point cloud encoder fp and color decoder fc are used for transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' coder (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2) and organize it to a 3D feature volume (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='3), which is used to reconstruct the neural scene representation and render images in a differentiable manner (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Constructing point cloud from RGB-D images The proposed method makes use of sequential RGB- D images {(Ii, Di)}N i=1, the camera intrinsic parameters {Ki}N i=1, and extrinsic poses {ξi}N i=1 ∈ SE(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' N is the input view number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' SE(3) refers to the Special Euclidean Group representing 3D rotations and translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The cam- era parameters can be easily obtained from SfM or SLAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We construct the point cloud X by back-projecting RGB-D images to point clouds in a unified coordinate: X = N � i π−1(Ii, Di, ξi, Ki), (1) where π−1 back-projects the RGB-D image to 3D world space using camera poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Note that different from pre- vious methods which only consider the point location, our method attributes each point with both point location and RGB color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The details of π−1 are provided in the supple- mentary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Point cloud encoder for feature extraction Given the point cloud X constructed from RGB-D im- ages, a point cloud encoder fp is used to extract per-point feature embedding E: E = fp(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' (2) The encoder fp pre-trained with the method mentioned in the Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='4 serves as a good initialization for various downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Building feature volume Once the feature extraction is done, we map the point embeddings E to a 3D sparse feature volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' To fill in the empty space, we perform average pooling, followed by a 3D CNN, to aggregate features from the nearby points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The dense 3D volume is denoted as V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Pre-training with Neural Rendering This section introduces how to reconstruct the implicit scene representation and render images differentiablely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We first give a brief introduction to neural scene representation, then illustrate how to integrate it into our point cloud pre- training pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Last, we show the differentiable render- ing formulation to render color and depth images from the neural scene representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Brief introduction of neural scene representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Neural scene representation aims to represent the scene ge- ometry and appearance through a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' In this paper, we use the Signed Distance Function (SDF), which measures the distance between a query point and the sur- face boundary, to represent the scene geometry implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' SDF is capable of representing high-quality geometry de- tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' For any query point of the scene, the neural network takes points features as input and outputs the corresponding SDF value and RGB value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' In this way, the neural network captures both the geometry and appearance information of a specific scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Following NeuS [49], the scene can be reconstructed as: s(p) = ˜fs(p), c(p, d) = ˜fc(p, d), (3) where ˜fs is the SDF decoder and ˜fc is the RGB color de- coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' ˜fs takes point location p as input, and predicts the SDF value s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' ˜fc takes point location p and viewing direc- tion d as input, and outputs the RGB color value c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Both ˜fs and ˜fc are implemented by simple MLP networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Neural scene representation from point cloud input in Ponder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' To predict a neural scene representation from the input point cloud, we change the scene formulation to take 3D feature volume V as an additional input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Specifically, given a 3D query point p and viewing direction d, the fea- ture embedding V(p) can be extracted from the processed feature volume V by trilinear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The scene is then represented as: s(p) = fs(p, V(p)), c(p, d) = fc(p, d, V(p)), (4) where V is predicted by the point cloud encoder fp and en- codes information of each scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' fs and fc are SDF and RGB decoders shared for all scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Different from Equa- tion (3), which is used for storing single-scene information in the { ˜fs, ˜fc}, the formulation in Equation (4) includes an extra input V(p) to facilitate representing the information of multiple scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Differentiable rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Given the dense 3D volume V and viewing point, we use differentiable volume render- ing to render the projected color images and depth images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' For each rendering ray with camera origin o and viewing direction d, we sample a set of ray points {p(z)|p(z) = o + zd, z ∈ [zn, zf]} along the ray, where z denotes the length of the ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Note that o and d can be calculated from paired camera parameters {(Ki, ξi)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' zn and zf denote the near and far bounds of the ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Different from previ- ous methods [30,49], we automatically determine {zn, zf} by the ray intersection with the 3D feature volume box, us- ing axis-aligned bounding boxes (AABB) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Then, the ray color and depth value can be aggregated as: ˆC = � zf zn ω(z)c(p(z), d)dz, (5) ˆD = � zf zn ω(z)zdz, (6) where the ˆC is the ray color and the ˆD is the ray depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We follow NeuS [49] to build an unbiased and occlusion- awareness weight function w(z): w(z) = T(z) · ρ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' (7) T(z) measures the accumulated transmittance from zn to z and ρ(z) is the occupied density function which are defined as: T(z) = exp(− � zf zn ρ(z)dz), (8) ρ(z) = max �− dΦh dz (s(p(z))) Φh(s(p(z))) , 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' (9) Φh(x) is the Sigmoid function Φh(x) = (1 + e−hx)−1 where h−1 is treated as a trainable parameter, h−1 ap- proaches to zero as the network training converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' In prac- tice, we use a numerically approximated version by quadra- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We make the decode networks {fs, fc} relatively smaller than [30,49] to accelerate the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Projected Points Rendered Color Reference Color Rendered Depth Reference Depth Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Rendered images by Ponder on the ScanNet validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The projected point clouds are visualized in the first column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Even though input point clouds are very sparse, our model is still capable of rendering color and depth images similar to the refer- ence images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Rendered examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The rendered color images and depth images are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' As shown in the fig- ure, even though the input point cloud is pretty sparse, our method is still capable of rendering color and depth images similar to the reference image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Pre-training loss We leverage the input {Ii, Di} to supervise neural scene representation reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The total loss function con- tains five parts, L = λcLc + λdLd + λeLe + λsLs + λfLf, (10) which are loss functions responsible for color supervision Lc, depth supervision Ld, Eikonal regularization Le, near- surface SDF supervision Ls, and free space SDF supervi- sion Lf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' These loss functions are illustrated in the follow- ing section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Color and depth loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Lc and Ld are the color loss and depth loss, which measure consistency between the ren- dered pixels and the ground-truth pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Assume that we sample Nr rays for each image and Np points for each ray, then the Lc and Ld can be written as: Lc = 1 Nr Nr � i || ˆC − C||2 2 (11) Ld = 1 Nr Nr � i || ˆD − D||2 2, (12) where C and D are the ground-truth color and depth re- spectively for each ray, ˆC and ˆD are their corresponding rendered ones in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' +**+Loss for SDF regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Le is the widely used Eikonal loss [13] for SDF regularization: Le = 1 NrNp Nr,Np � i,j (|∇s(pi,j)| − 1)2, (13) where ∇s(pi,j) denotes the gradient of SDF s at location pi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Since SDF is a distance measure, Le encourages this distance to have a unit norm gradient at the query point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Near-surface and free space loss for SDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' To stabilize the training and improve the reconstruction performance, similar to iSDF [35] and GO-Surf [48], we add additional approximate SDF supervision to help the SDF estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Specifically, for near-surface points, the difference between rendered depth and ground-truth depth can be viewed as the pseudo-SDF ground-truth supervision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' for points far from the surface, a free space loss is used to regularize the irreg- ular SDF value additionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' To calculate the approximate SDF supervision, we first define an indicator b(z) for each sampled ray point with ray length z and corresponding GT depth D: b(z) = D − z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' (14) b(z) can be viewed as the approximate SDF value, which is credible only when b(z) is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Let t be a human-defined threshold, which is set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='05 in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' For sampled ray points that satisfy b(z) ≤ t, we leverage the near-surface SDF loss to constrain the SDF prediction s(zi,j): Ls = 1 NrNp Nr,Np � i,j |s(zi,j) − b(zi,j)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' (15) For the remaining sampled ray points, we use a free space loss: Lf = 1 NrNp Nr,Np � i,j max(0, e−α·s(zi,j)−1, s(zi,j)−b(zi,j)), (16) where α is set as 5 following the same with [35, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Note that due to the noisy depth images, we only apply Ls and Lf on the rays that have valid depth values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' In our experiments, we follow a similar loss of weight with GO-Surf [48], which sets λc as 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0, λd as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0, λs as 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0, and λf as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We observe that the Eikonal term in our method can easily lead to over-smooth reconstructions, thus we use a small weight of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='01 for the Eikonal loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Pre-training Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We use ScanNet [10] RGB-D images as our pre-training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' ScanNet is a widely used real-world in- door dataset, which contains more than 1500 indoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Each scene is carefully scanned by an RGB-D camera, lead- ing to about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='5 million RGB-D frames in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We follow the same train/val split with VoteNet [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Data preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' During pre-training, a mini-batch of batch size 8 includes point clouds from 8 scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The point cloud of a scene, serving as the input of the point cloud en- coder in our approach, is back-projected from the 5 RGB-D frames of the video for the scene with an interval of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The 5 frames are also used as the supervision of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We augment the point cloud by ran- dom sampling, normalization, and random masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' First, we randomly down-sample the point cloud to 20,000 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Then, the point cloud is normalized into a 3D unit cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Fi- nally, we apply the same masking strategy as used in Mask Point [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Specifically, we use FPS to split the point cloud into 2,048 groups, each group containing 64 points, then mask the point groups with a mask ratio of 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We train the proposed pipeline for 100 epochs using an AdamW optimizer [29] with a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The learning rate is initialized as 1e- 4 with Exponential scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' For the rendering process, we randomly choose 128 rays for each image and sample 128 points for each ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' More implementation details can be found in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Transfer Learning In contrast to previous methods, our approach is able to encode rich geometry and appearance cues into the point cloud representations via neural rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' These strengths make it flexible to be applied to various tasks, including not only 3D semantic segmentation and 3D detection tasks but also low-level surface reconstruction and image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='1 High-level 3D Tasks 3D object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' For transfer learning on 3D ob- ject detection task, we use VoteNet [38] as the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' VoteNet leverage a voting mechanism to generate object centers, which are used for 3D bounding box proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Two datasets are applied to verify the effectiveness of our method: ScanNet [10] and SUN RGB-D [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Differ- ent from ScanNet, which contains fully reconstructed 3D scenes, SUN RGB-D is a single-view RGB-D dataset with 3D bounding box annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' It has 10,335 RGB-D images for 37 object categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' For pre-training, we use Point- Net++ as the point cloud encoder fp, which is identical to the backbone used in VoteNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We pre-train the point cloud encoder on the ScanNet dataset and transfer the weight as the VoteNet initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Following [38], we use average precision with 3D detection IoU threshold 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='25 and thresh- old 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='5 as the evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Method Detection Pre-training Pre-training Pre-training ScanNet SUN RGB-D Model Type Data Epochs AP50 ↑ AP25 ↑ AP50 ↑ AP25 ↑ 3DETR [32] 3DETR 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='5 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='7 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0 Point-BERT [58] 3DETR Completion 3D Model 300 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='3 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0 MaskPoint [25] 3DETR Completion Depth 300 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='6 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='4 VoteNet [38] VoteNet 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='5 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='9 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='7 STRL [20] VoteNet Contrast Depth 100 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='4 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2 RandomRooms [40] VoteNet Contrast Synthesis 300 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='3 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='4 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2 PointContrast [52] VoteNet Contrast 3D Model 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='8 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='5 PC-FractalDB [54] VoteNet Contrast Synthesis 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='3 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='9 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='9 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='4 DepthContrast [61] VoteNet Contrast Depth 1000 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='1 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='4 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='4 IAE [55] VoteNet Completion 3D Model 1000 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='8 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='5 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='4 Ponder VoteNet Rendering Depth 100 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='9 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='3 Ponder VoteNet Rendering Color & Depth 100 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='6 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3D object detection AP25 and AP50 on ScanNet and SUN RGB-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' VoteNet [38] and 3DETR [32] are two baseline 3D object de- tection models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The DepthContrast [61] and Point-BERT [58] results are adopted from IAE [55] and MaskPoint [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Ponder outperforms both VoteNet-based and 3DETR-based point cloud pre-training methods with fewer training epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The 3D detection results are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Our method improves the baseline of VoteNet without pre- training by a large margin, boosting AP50 by 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='5% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='7% for ScanNet and SUN RGB-D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' IAE [55] is a pre-training method that represents the inherent 3D ge- ometry in a continuous manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Our learned point cloud representation achieves higher accuracy because it is able to recover both the geometry and appearance of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The AP50 and AP25 of our method are higher than that of IAE by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='1% on ScanNet, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Mask- Point [25] is another method aiming to learn a continuous surface by classifying if the query point is occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' How- ever, its performance can be constrained due to the noisy labeling of the query point occupancy value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' As presented in Table 1, even with an inferior backbone (PointNet++ vs 3DETR), our method is able to achieve better accuracy with fewer pre-training epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3D semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3D semantic segmentation is another fundamental scene understanding task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Follow- ing [43,47,55], we choose DGCNN [51] as our baseline for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' DGCNN applies a dynamic graph CNN as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' For pre-training, we use DGCNN as the point cloud encoder fp, and pre-train the model on ScanNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We validate the effectiveness of our method by transfer- ring the weights to Stanford Large-Scale3D Indoor Spaces (S3DIS) [3] dataset, which is an indoor 3D understanding dataset containing 6 large-scale indoor scenes with point se- mantic annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Following the same setting of [51], we use the overall accuracy (OA) mean IoU(mIoU) on points as the evaluation metric, and report the average evaluation results across six folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Table 2 shows the quantitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Compared with the DGCNN baseline, the proposed method boost the seg- mentation performance by a large margin, boosting OA and mIoU for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='1% and 5%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Jigsaw and OcCo use ShapeNet as the pre-train dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Although they get improvements compared with the baseline, the limited scale of training data constrains the transferring ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' IAE achieves significant improvements by lever- aging the large-scale dataset and an implicit reconstruc- tion manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Compared with IAE, the proposed approach achieves a higher semantic segmentation performance with the DGCNN backbone (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='3% for OA and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='4% for mIoU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Besides, IAE requires a large amount of 3D mesh for supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Our approach, in contrast, only requires RGB-D images as the supervision, which is much cheaper and easy to fetch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2 Low-level 3D Tasks Low-level 3D tasks like scene reconstruction and image synthesis are getting increasing attention due to their wide applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' However, most of them are trained from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' How to pre-train a model with a good initialization is desperately needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We are the first pre-training work to demonstrate a strong transferring ability to such low-level 3D tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3D scene reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3D scene reconstruction task aims to recover the scene geometry, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' mesh, from the point cloud input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We choose ConvONet [37] as the baseline model, whose architecture are widely adopted in [9,26,56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Following the same setting as ConvONet, we conduct experiments on the Synthetic Indoor Scene Dataset (SISD) [37], which is a synthetic dataset and contains 5000 scenes with multiple ShapeNet [5] objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We pre-train the PointNet encoder, which is the same as the original Con- Method OA↑ mIoU↑ DGCNN [51] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='1 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='1 Jigsaw [43] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='4 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='6 OcCo [47] 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='1 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='5 IAE [55] 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='9 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='7 Ponder 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='1 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3D semantic segmentation OA and mIoU on S3DIS dataset with DGCNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Ponder outperforms previous state-of- the-art models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Method Encoder IoU↑ Normal Consistency↑ F-Score↑ ConvONet [37] PointNet 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='915 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='964 Ponder PointNet 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='917 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='965 ConvONet PointNet++ 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='887 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='906 Ponder PointNet++ 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='893 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='920 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3D scene reconstruction IoU, NC, and F-Score on SISD dataset with PointNet and PointNet++ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' For both PointNet and PointNet++, Ponder is able to boost the reconstruction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Comparison of image synthesis from point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Compared with training from scratch, our Ponder model is able to converge faster and achieve better image synthesis results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' vONet implementation, and test the reconstruction quality on the SISD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Additionally, to compare with another self-supervised learning method IAE [55], we add extra ex- periments using VoteNet-style PointNet++ as the encoder of ConvONet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Following [37], we use Volumetric IoU, Nor- mal Consistency, and F-Score [45] with the threshold value of 1% as the evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The results are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Compared to the baseline ConvONet model with PointNet, the proposed ap- proach is able to improve the reconstruction quality (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='8% for IoU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' By replacing the encoder of ConvONet from PointNet to PointNet++, ours is able to achieve more accu- racy improvement (+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='4% for IoU and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='014 for F-Score).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Our method also gets better reconstruction results than IAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Check our supplementary materials for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Image synthesis from point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We also validate the effectiveness of our method on another low-level task of im- age synthesis from point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We use Point-NeRF [53] as the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Point-NeRF uses neural 3D point clouds with associated neural features to render images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' It can be used both for a generalizable setting for various scenes and a single-scene fitting setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' In our experiments, we mainly focus on the generalizable setting of Point-NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We re- Supervision ScanNet SUN RGB-D Depth 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='9 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='1 Color 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='8 Color+RGB 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='6 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Ablation study for supervision type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3D detection AP50 on ScanNet and SUN RGB-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Combining color supervision and depth supervision can lead to better detection performance than using a single type of supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' place the 2D image features of Point-NeRF with point fea- tures extracted by a DGCNN network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Following the same setting with PointNeRF, we use DTU [21] as the evaluation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' DTU dataset is a multiple-view stereo dataset con- taining 80 scenes with paired images and camera poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We transfer both the DGCNN encoder and color decoder as the weight initialization of Point-NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We use PSNR as the metric for synthesized image quality evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The results are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' By leveraging the pre- trained weights of our method, the image synthesis model is able to converge faster with fewer training steps and achieve better final image quality than training from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Ablation study In this section, we do two ablation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' First, we show the effectiveness of using different 2D supervi- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Then, we test how the view number affects the final performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' For both experiments, we use the 3D object detection task as the transfer learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Influence on Rendering Targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The rendering part of our method contains two items: RGB color image and depth image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We study the influence of each item with the trans- ferring task of 3D detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The results are presented in Ta- ble 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Combining depth and color images for reconstruction shows the best detection results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' In addition, using depth re- construction presents better performance than color recon- struction for 3D detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Image Synthesis 20 18 PSNR 16 14 12 scratch ours 10 100 200 500 2000 5000 Train StepsInput Point Cloud Projected Point Cloud Reconstruction Image Synthesis Reference Image Depth Synthesis Reference Depth Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Direct applications of Ponder on the ScanNet validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The proposed Ponder model can be directly used for various applications, such as 3D reconstruction and image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The input point clouds are drawn as spheres for better clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Number of input RGB-D view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Our method utilizes N RGB-D images, where N is the input view number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We study the influence of N and conduct experiments on 3D detection, as shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Using multi-view super- vision helps to reduce single-view ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Similar ob- servations are also found in the multi-view reconstruction task [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Compared with the single view, multiple views achieve higher accuracy, boosting AP50 by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='9% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2% for ScanNet and Sun RGB-D datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Other applications In previous sections, we show that the proposed pipeline can be used for transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' In this section, we show that the pre-trained model from our pipeline Ponder itself can also be directly used for surface reconstruction and im- age synthesis from sparse point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3D reconstruction from sparse point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The learned model has the capability to recover the scene sur- face from sparse point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Specifically, after learning the neural scene representation, we query the SDF value in the 3D space and leverage the Marching Cubes [28] to ex- tract the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We show the reconstruction results in Fig- ure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The results show that even though the input is sparse point clouds from complex scenes, our method is able to recover high-fidelity meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Image synthesis from sparse point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Another in- teresting experiment to explore is that our pipeline is able to render realistic images from sparse point cloud input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' As shown in Figure 6, our method is able to recover similar color images with the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Also, the recovered depth may even look better compared with the ground-truth depth image which has irregular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' View number ScanNet SUN RGB-D 1 view 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='1 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='4 3 views 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='8 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0 5 views 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='6 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Ablation study for view number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3D detection AP50 on ScanNet and SUN RGB-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Using multi-view supervision for point cloud pre-training can achieve better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Conclusion In this paper, we show that differentiable neural render- ing is a powerful tool for point cloud representation learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The proposed pre-training pipeline, Ponder, is able to encode rich geometry and appearance cues into the point cloud representation via neural rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' For the first time, our model can be transferred to not only high-level 3D per- ception tasks but also 3D low-level tasks, like 3D recon- struction and image synthesis from point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Also, the learned Ponder model can be directly used for 3D recon- struction and image synthesis from sparse point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Several directions could be explored in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' First, there are various types of neural rendering, which could also be leveraged for point cloud representation learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Second, other 3D domain-specific designs could be integrated into point cloud pre-training pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Third, exploring the proposed pre-training pipeline Ponder on a larger 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 1, 2 [61] Zaiwei Zhang, Rohit Girdhar, Armand Joulin, and Ishan Misra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Self-supervised pretraining of 3d features on any point-cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 1, 2, 6 Ponder: Point Cloud Pre-training via Neural Rendering Supplementary Material A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Implementation Details In this section, we give more implementation details of our Ponder model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Our code will be released upon ac- ceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Pre-training Details 3D feature volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' In our experiments, we build a hi- erarchical feature volume V with a resolution of [16, 32, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Building a 3D hierarchical feature volume has been wildly used for recovering detailed 3D geometry, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' [8,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' After processing the 3D feature volume with a 3D CNN, we use trilinear interpolation to get the feature of the query point p, denoted as V(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We use the drop-in replacement of grid sampler from [48] to accelerate the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Ray sampling strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Similar to [30, 49], we sample twice for each rendering ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' First, we uniformly sample coarse points between the near bound zn and far bound zf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Then, we use importance sampling with the coarse proba- bility estimation to sample fine points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Folowing [49], the coarse probability is calculated based on Φh(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' By this sampling strategy, our method can automatically determine sample locations and can collect more points near the sur- face, which makes the training process more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Back projection Here we give details of the back projec- tion function π−1 to get point clouds from depth images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Let K be camera intrinsic parameters, ξ = [R|t] be camera extrinsic parameters, where R is the rotation matrix and t is the translation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Xuv is the projected point location and Xw is the point location in the 3D world coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Then, according to the pinhole camera model: sXuv = K(RXw + t), (17) where s is the depth value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' After expanding the Xuv and Xw: s � � u v 1 � � = K(R � � X Y Z � � + t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' (18) Then, the 3D point location can be calculated as follows: � � X Y Z � � = R−1(K−1s � � u v 1 � � − t) (19) The above Equation 19 is the back-projection equation π−1 used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Training Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The Ponder model is trained with 8 NVIDIA A100 GPUs for 96 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Transfer Learning Details 3D scene reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' ConvONet [37] reconstructs scene geometry from the point cloud input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' It follows a two-step manner, which first encodes the point cloud into a 3D feature volume or multiple feature planes, then de- codes the occupancy probability for each query point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' To test the transfer learning ability of our point cloud encoder, we directly replace the point cloud encoder of ConvONet, without any other modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We choose the highest per- forming configuration of ConvONet as the baseline setting, which uses a 3D feature volume with a resolution of 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' For the training of ConvONet, we follow the same training setting as the released code1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Image synthesis from point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Point-NeRF [53] renders images from neural point cloud representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' It first generates neural point clouds from multi-view images, then uses point-based volume rendering to synthesize im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' To transfer the learned network weight to the Point- NeRF pipeline, we 1) replace the 2D image feature back- bone with a pre-trained point cloud encoder to get the neural point cloud features, 2) replace the color decoder by a pre- trained color decoder, 3) keep the other Point-NeRF module untouched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Since a large amount of point cloud is hard to be directly processed by the point cloud encoder, we down- sample the point cloud to 1%, which will decrease the ren- dering quality but help reduce the GPU memory require- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We report the PSNR results of the unmasked region as the evaluation metric, which is directly adopted from the original codebase2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' For training Point-NeRF, we follow the same setting as Point-NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Supplementary Experiments B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Ablation Study 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='com/autonomousvision/convolutional occupancy networks 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='com/Xharlie/pointnerf Mask ratio ScanNet SUN RGB-D 0% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='7 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='3 25% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='7 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2 50% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='9 75% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='7 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0 90% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='6 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Ablation study for mask ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3D detection AP50 on ScanNet and SUN RGB-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Resolution ScanNet SUN RGB-D 16 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='7 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='6 16+32+64 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='6 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Ablation study for feature volume resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3D detection AP50 on ScanNet and SUN RGB-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Method IoU↑ Normal Consistency↑ F-Score↑ ConvOcc [37] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='778 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='887 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='906 IAE [55] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='757 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='887 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='910 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='802 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='893 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='920 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3D scene reconstruction IoU, NC, and F-Score on SISD dataset with PointNet++ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Influence on mask ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' In this paper, we use random masking as one type of point cloud augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We ap- ply the same mask ratio as MaskPoint [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Here, we give additional experimental results to show the influence of us- ing different mask ratios in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' For the mask ratio of 0%, we do not apply any mask strategy to the input point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3D feature volume resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' As mentioned in Section A, Ponder build a 3D feature volume with a resolution of [16, 32, 64], which is inspired by the recent progress of multi-resolution in 3D reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' However, building such a 3D feature volume with large resolutions requires heavy GPU memory usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We conduct experiments in Ta- ble 7 to test the performance with a smaller resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' As shown in the table, even with a small resolution, Ponder is still able to achieve comparable accuracy, demonstrating the robustness to the feature volume resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Transfer Learning 3D scene reconstruction As mentioned in the paper, we transfer the learned PointNet++ model of IAE to the 3D reconstruction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The results are shown in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Com- pared with the ConvONet baseline, the IAE pre-trained model gets a better F-Score with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='004 but gets worse re- sults on the IoU metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Our method, on the other hand, Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Label efficiency training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We show the 3d object de- tection experiment results using limited downstream data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Our pretrained model is capable of achieving better performance than training from scratch using the same percentage of data or requires fewer data to get the same detection accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' gets a better reconstruction performance than both the Con- vONet and IAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Label Efficiency Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We also do experiments to show the performance of our method with limited label- ing for the downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Specifically, we test the la- bel efficiency training on the 3D object detection task for ScanNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Following the same setting with IAE [55], we use 20%, 40%, 60%, and 80% of ground truth annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The results are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We show constantly im- proved results over training from scratch, especially when only 20% of the data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Color information for downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Different from previous works, since our pre-training model uses a colored point cloud as the input, we also use color informa- tion for the downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Results are shown in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Using color as an additional point feature can help the VoteNet baseline achieve better performance on the SUN RGB-D dataset, but get little improvement on the ScanNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' This shows that directly concatenating point posi- tions and colors as point features shows limited robustness to application scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' By leveraging the proposed Pon- der pre-training method, the network is well initialized to handle the point position and color features, and achieve better detection accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' More comparisons on 3D detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' More detection ac- curacy comparisons are given in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Even using an in- ferior backbone, our Ponder model is able to achieve simi- lar detection accuracy with 9 in ScanNet and better accuracy in SUN RGB-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Label Efficiency Training 65 60 Detection Result 55 50 45 40 scratch 35 ours 30 20 40 60 80 100 Labeled Data PercentageMethod Detection Pre-training Pre-training Pre-training ScanNet SUN RGB-D Model Type Data Epochs AP50 ↑ AP25 ↑ AP50 ↑ AP25 ↑ VoteNet* VoteNet* 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='6 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='4 DPCo [24] VoteNet* Contrast Depth 120 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='5 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='6 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='8 IPCo [24] VoteNet* Contrast Color & Depth 120 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='9 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='9 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2 VoteNet (w color) VoteNet 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='4 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='8 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='3 Ponder VoteNet Rendering Depth 100 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='9 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='2 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='3 Ponder VoteNet Rendering Color & Depth 100 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='6 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='0 Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' 3D object detection AP25 and AP50 on ScanNet and SUN RGB-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' * means a different but stronger version of VoteNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' More application examples As mentioned in the paper, the pre-trained Ponder model can be directly used for surface reconstruction and image synthesis tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' We give more application examples in Fig- ure 8 and Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' Input Point Cloud Reconstruction Image Synthesis Depth Synthesis Projected Point Cloud Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' More results of application examples of Ponder on the ScanNet validation set (part 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The input point clouds are drawn as spheres for better clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' +$: ++++ ++ ++ +t +++++ +++*+Input Point Cloud Reconstruction Image Synthesis Depth Synthesis Projected Point Cloud Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' More results of application examples of Ponder on the ScanNet validation set (part 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' The input point clouds are drawn as spheres for better clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' + +++ +* ++ t **+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' +* +4 + +++ +* +*+ + 主 + +++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} +page_content=' +#' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfV_f-/content/2301.00157v1.pdf'} diff --git 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E. J. Newman1, 2 +1Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA +2Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48109, USA +We study core-periphery structure in networks using inference methods based on a flexible network +model that allows for traditional onion-like cores within cores, but also for hierarchical tree-like +structures and more general non-nested types of structure. We propose an efficient Monte Carlo +scheme for fitting the model to observed networks and report results for a selection of real-world data +sets. Among other things, we observe an empirical distinction between networks showing traditional +core-periphery structure with a dense core weakly connected to a sparse periphery, and an alternative +structure in which the core is strongly connected both within itself and to the periphery. Networks +vary in whether they are better represented by one type of structure or the other. We also observe +structures that are a hybrid between core-periphery structure and community structure, in which +networks have a set of non-overlapping cores that correspond roughly to communities, surrounded +by a single undifferentiated periphery. Computer code implementing our methods is available. +I. +INTRODUCTION +Networks are widely used as a compact and convenient +mathematical representation of the connections between +the elements of a complex system, such as data con- +nections on the Internet, citations between papers, so- +cial contacts among people or animals, synaptic connec- +tions between brain cells, and biological and biochemi- +cal networks of many kinds [1, 2]. A significant amount +of effort has been devoted in recent years to analyz- +ing and understanding large-scale structure in such net- +works, especially community structure [3, 4], but also +nested [3, 5] and overlapping [6, 7] communities and strat- +ification [8, 9], as well as the related issues of embedding +and graph representation learning [10]. In this paper we +focus on a less well-studied form of large-scale structure, +core-periphery structure, in which a network is divided +into a densely connected core of nodes surrounded by +a sparser periphery [11–13]. +The observation of core- +periphery structure communicates different information +about a network from community structure. Where com- +munity structure deals with the identification of groups +or types of nodes, core-periphery structure focuses on +their roles and centrality. +Core-periphery structure is +integral to understanding the link between node posi- +tion and function in networks, for instance in the Inter- +net [14, 15], neuroscience [16], and economics [17]. +A range of heuristic methods have been proposed in the +past for detecting core-periphery structure in networks. +In the simplest case the challenge is to take unlabelled +network data and assign each node of the network in some +automated fashion to either the core or the periphery. In +more complex cases one may attempt to infer a onion-like +sequence of deeper and deeper cores within cores. +The problem, however, is not entirely well posed, since +we have not precisely defined what “core” and “periph- +ery” mean. +Various researchers have chosen to define +them in various ways and there is, as a result, a cor- +responding spectrum of algorithmic approaches. +Per- +haps the oldest approach is the k-core decomposition, in +which nodes are recursively removed from a network in +order of increasing degree and the sequence in which they +are removed defines a sliding scale from core to periph- +ery [18]. This method essentially equates the core with +high-degree nodes. Another well-established method is +that of Borgatti and Everett [11], who defined a quality +function akin to the well-known modularity function for +community detection [19]. Borgatti and Everett’s func- +tion takes as input a network and a putative division +into core and periphery and returns a score that indi- +cates whether the division is a “good” one in a certain +sense. Then by maximizing the function over all possi- +ble divisions one can find the “best” core-periphery de- +composition. +Variants of the same idea have been ex- +plored by Rombach et al. [13], as well as by Kojaku and +Masuda [20] who proposed a multi-group version. Cu- +curingu et al. [21] have explored several methods for find- +ing core-periphery structure based on counting geodesic +paths and on spectral network properties. Other meth- +ods use node-level properties of the network such as the +clustering coefficient [12] or centrality measures [22]. +Stochastic block models, commonly used in commu- +nity detection [23, 24], have also been applied to core- +periphery structure. In these models, one assumes that +nodes are divided into types (e.g., core and periphery) +and that the probability ωrs of an edge between a pair +of nodes depends on the types r and s of the nodes. +Zhang et al. [25] studied a two-group version of this +model with ω11 > ω12 > ω22, which generates net- +works with classic core-periphery structure, then fitted +the model to empirical network data to detect structure. +Gallagher et al. [26] took a similar approach but with +more than two groups and multiple nested cores such +that ωrs = f[max(r, s)] for some function f[r]. +In this paper we also take a model-fitting approach to +the study of core-periphery structure, but we formulate +a more general model that includes the classic two-group +structure but also allows for more flexible structures as +well. In particular, we consider a hierarchical model with +any number of groups, the number being determined by +the network structure using Bayesian model selection, +arXiv:2301.03630v1 [cs.SI] 9 Jan 2023 + +2 +Group 0 +Group 1 +Group 2 +(a) Nested groups +Group 0 (all nodes) +Group 1 +Group 2 +Group 3 +Group 4 +Group 5 +(b) Tree-like hierarchy +Group 0 (all nodes) +Group 1 +Group 2 +Group 3 +Group 4 +Group 5 +Group 6 +(c) General core-periphery structure +FIG. 1: The model proposed here is capable of capturing a range of different types of structure, including +(a) traditional nested groups, (b) tree-like branching structures, and (c) complex structures of overlapping groups. +and we specifically allow for the possibility that the hi- +erarchy may not be perfectly nested and moreover that +there maybe multiple cores and multiple peripheries at +any level in the hierarchy. These features allow our model +to capture variants of core-periphery structure that oc- +cur in real-world networks but are not captured by tra- +ditional analyses. In a multilevel hierarchy, for instance, +an inner core may not be perfectly contained within an +outer one. In a network with community structure the +individual communities may each have a separate core of +their own, or there may be several cores within a single +periphery. By fitting our proposed model to network data +we show how it can reveal subtle structure and, in the +process, encounter some new formations within networks +that have not previously received significant attention. +II. +A HIERARCHICAL MODEL OF +NETWORK STRUCTURE +One can think of conventional core-periphery structure +as a hierarchy. In the simplest case the hierarchy has just +two levels, an outer periphery and an inner core. In more +elaborate cases there can be an onion-like succession of +levels, each one nested inside the next, from the outer- +most to the innermost—see Fig. 1a. This, however, is +not the most general kind of nested structure. A more +general structure can have any number of first-level cores +within a single periphery, and any number of second-level +cores within those first-level ones, and so forth (Fig. 1b). +A structure like this can be captured with a tree or den- +drogram, and some attempts have been made to fit den- +drograms to networks [5]. The work presented here goes +still further by allowing for the possibility that the hier- +archy is not perfectly nested. In our model, we envisage a +hierarchical series of cores, but we allow these cores to be +placed in any way around the network, with no require- +ment that they be nested within one another (Fig. 1c). +In practice, if two cores do not overlap then they do not +interact and hence their relative order in the hierarchy +has no effect, but if they overlap then the higher ranked +one takes precedence over the lower in a manner we will +describe. The net result is a hierarchical model that gen- +eralizes both the conventional onion layers and the den- +drogram and, as we will see, captures a wide range of +possible structures. We then fit this model to network +data to infer core-periphery structure in real-world net- +works. In detail the procedure is as follows. +Consider an unweighted, undirected network of n +nodes, which can belong to any of k groups labeled by +r = 0 . . . k − 1. By contrast with traditional community +structure models we allow nodes to belong to any num- +ber of groups simultaneously, up to and including all of +them. In addition, every node always belongs to group 0, +which acts as a sort of default or base group. Thus each +node belongs to group 0 plus some selection of the other +groups 1 . . . k − 1. +(We label the first group 0 rather +than 1 to remind ourselves of its special status.) +We define a set of indicator variables gr +u such that +gr +u = 1 if node u belongs to group r and 0 otherwise. +Then we define a set of probabilities ωr, one for each +group including group 0, and place edges between node +pairs independently with probability ωr, where r is the +highest common group that both nodes of the pair belong +to, meaning the one with the highest number. For exam- +ple, if one node belongs to groups 0, 1, 2 and another +belongs to 0, 1, 3, then there is an edge between them +with probability ω1. +Figure 1c illustrates the behavior of the resulting net- +work: the groups form “patches” that lie one on top of +another and the topmost visible patch takes precedence +for each node pair. For instance, if all nodes belong to +group 0 only, then every pair of nodes has equal proba- +bility ω0 of being connected, which gives us a standard +random graph. But if we assign some subset of the nodes +to group 1 then for any pair of nodes that are both in +group 1 the probability of an edge becomes ω1 and over- +rides the previous ω0. +Similarly any pair of nodes as- +signed to group 2, including nodes already in groups 0 +and 1, have probability ω2 of connection, overriding ω0 +and ω1, and so forth. The end result is a model that +starts out as a random graph but then adds variation + +3 +and detail to the network wherever it is needed to cap- +ture local structural features. This approach has some +similarities to those of Kojaku and Masuda [20] and Gal- +lagher et al. [26], but it differs crucially in that it does +not force the groups to be either strictly nested within +each other or nonoverlapping. Given the definition of the +model, our goal is now to fit it to observed network data +to infer the best choice of groups gr +u for each node. +Suppose we have a network of n nodes, represented by +its n × n adjacency matrix A with elements auv = 1 if +there is an edge between nodes u and v and 0 otherwise. +Then the probability of observing a particular network if +it was generated from our model with given k and given +group memberships g is +P(A|ω, k, g) = +� +u 0 we first choose a size nr uniformly at +random between 0 and n, each value thus having prob- +ability 1/(n + 1). Then we choose uniformly at random +one of the +� n +nr +� +ways to assign nr nodes to the group, +each choice thus having probability 1/ +� n +nr +� +. Hence for all +groups the total probability of an assignment is +P(g|k) = +k−1 +� +r=1 +1 +(n + 1) +� n +nr +� = +k−1 +� +r=1 +nr!(n − nr)! +(n + 1)! +. +(6) +B. +Prior on number of groups +In traditional core-periphery structure calculations one +considers the existence of a single core and a single +periphery, and this is the approach taken for instance +in [25]. If our goal, however, is to find multiple groups +of unknown number, including multiple or overlapping +cores, then we need to allow k to vary, which means +choosing a prior on k. Here we adopt the approach taken +in [29, 30] and use a Poisson distribution with mean 1: +P(k) = +e−1 +(k − 1)!. +(7) +Note that group 0 always exists, so the distribution of the +number of groups is effectively a distribution over k − 1, +which is why we have 1/(k − 1)! in the denominator. +With this choice, we can now write +P(g, k|A) = P(k)P(g|A, k) = P(k)P(g|k)P(A|g, k) +P(A) +∝ +1 +(k − 1)! +k−1 +� +r=1 +nr!(n − nr)! +(n + 1)! +k−1 +� +r=0 +mr!(tr − mr)! +(tr + 1)! +. +(8) + +4 +Again, we can sample from this probability to generate +a selection of values k and group assignments g that are +representative of the network. In the following section +we describe the algorithm we use to achieve this. +III. +SAMPLING FROM THE POSTERIOR +DISTRIBUTION +We sample from the posterior distribution (8) using +a Markov-chain Monte Carlo method. We describe the +method first for the simpler case where the number of +groups k is fixed, then for the more complicated case of +varying k. +A. +Monte Carlo algorithm for the case of fixed k +For the case of fixed k our Monte Carlo scheme is as +follows. +1. We choose a group s uniformly at random from +1 . . . k − 1. +2. With equal probability 1 +2 we propose to either re- +move a node from group s or add a node to it. If +we are removing, the node to be removed is cho- +sen uniformly at random from those currently in +the group. If there are no nodes in the group, we +do nothing and move on to the next Monte Carlo +step. If we are adding, the node to be added is cho- +sen uniformly at random from those currently not +in the group. If the group is full—all n nodes are +already members—we do nothing and move on to +the next step. +3. The +proposed +move +is +accepted +with +the +Metropolis-Hastings style acceptance probability +α(g → g′) = min +� +1, P(A|g′, k) +P(A|g, k) +� +, +(9) +where g′ represents the group assignments after the +addition or removal. If the move is accepted, the +chosen node is added or removed as proposed. If +the move is not accepted, the group assignments g +remain unchanged on this step. +4. Repeat from step 1. +In the limit where this algorithm tends to an equilib- +rium distribution of states, that distribution will be the +one given in Eq. (5). To demonstrate this, it suffices to +prove two results: first that the algorithm is ergodic and +second that it satisfies detailed balance. Ergodicity re- +quires that every state of the system be accessible from +every other by a finite sequence of moves. This is triv- +ially true in the present case, since the membership of +any group can be set to anything we like in at most n +moves by first removing any nodes we don’t want and +then adding in those we do. +Detailed balance is a little more complicated. Detailed +balance requires that in equilibrium the average rate of +moves g → g′ equals the average rate g′ → g, which +means +P(g|A, k)P(g → g′) = P(g′|A, k)P(g′ → g), +(10) +where P(g → g′) is the probability of making the transi- +tion g → g′. This probability can be written as +P(g → g′) = π(g → g′)α(g → g′), +(11) +where π(g → g′) is the probability of proposing the move +and α(g → g′) is the probability of accepting it as in +Eq. (9). Then Eq. (10) can be written as +P(g′|A, k) +P(g|A, k) = π(g → g′) α(g → g′) +π(g′ → g) α(g′ → g). +(12) +We can show that this condition is satisfied by the pro- +posed Monte Carlo algorithm as follows. +From Eq. (5) we have +P(g′|A, k) +P(g|A, k) = P(A|g′, k)P(g′|k) +P(A|g, k)P(g|k) , +(13) +while from Eq. (9) the ratio of the two acceptance prob- +abilities is +α(g → g′) +α(g′ → g) = P(A|g′, k) +P(A|g, k) . +(14) +Substituting +(13) and (14) into (12), +a factor of +P(A|g′, k)/P(A|g, k) cancels and we are left with +P(g′|k) +P(g|k) = π(g → g′) +π(g′ → g). +(15) +If our Monte Carlo algorithm satisfies this condition, then +it satisfies detailed balance. +Using Eq. (6), the left-hand side can be written as +P(g′|k) +P(g|k) = +k−1 +� +r=1 +n′ +r!(n − n′ +r)! +nr!(n − nr)! , +(16) +where nr is the number of nodes in group r before the +move and n′ +r is the number afterwards. Suppose the par- +ticular move we are considering g → g′ is one that adds +a node to group s. Then n′ +s = ns + 1, while n′ +r = nr for +all other groups, so (16) simplifies to +P(g′|k) +P(g|k) = (ns + 1)!(n − ns − 1)! +ns!(n − ns)! += ns + 1 +n − ns +. +(17) +For the right-hand side of Eq. (15), for the same move +that adds a node to group s, the proposal probability is +π(g → g′) = +1 +k − 1 × 1 +2 × +1 +n − ns += +1 +2(k − 1)(n − ns). +(18) + +5 +Here the factor 1/(k − 1) is the probability of choosing +the particular group r out of all k − 1 possibilities, the +factor 1 +2 is the probability of choosing to add a node, and +the factor 1/(n − ns) is the probability of choosing the +particular node to be added from the n−ns possibilities. +Meanwhile, for the reverse move g′ → g, which involves +removing the same node from group s again, the proposal +probability is +π(g′ → g) = +1 +k − 1 × 1 +2 × +1 +ns + 1 = +1 +2(k − 1)(ns + 1), +(19) +since there are now ns + 1 nodes in the group. The ratio +of the two proposal probabilities is thus +π(g → g′) +π(g′ → g) = 1/2(k − 1)(n − ns) +1/2(k − 1)(ns + 1) = ns + 1 +n − ns +, +(20) +which agrees with Eq. (17) and hence Eq. (15) is satis- +fied and detailed balance is obeyed in this instance. The +proof for the case where we remove a node from group s +follows the same lines and leads to the same conclusion: +the algorithm satisfies detailed balance and hence sam- +ples correctly from the target distribution P(g|A, k) in +equilibrium. +B. +Algorithm for varying k +When k is allowed to vary the algorithm is more com- +plex, involving two types of moves that each take us from +a combined state (g, k) to a state (g′, k′), as follows. +Type 1: In a move of type 1 we choose a group s uni- +formly at random from 1 . . . k − 1. With probability 1 +2 +we add a new node to the group chosen uniformly from +the set of nodes that do not currently belong; otherwise, +we remove an existing node from the group, chosen uni- +formly from those in the group. If we choose to add a +node but group s is already full then we do nothing and +move on to the next Monte Carlo step. If we choose to +remove a node and group s is already empty then the +entire group is deleted and the number of groups k de- +creases by one, with the labels of all groups above s also +decreasing by one so that they still run to a maximum +of k − 1. +Type 2: In a move of type 2 we choose a group index s +uniformly at random from 1 . . . k. We increase by one the +labels of all groups s and greater (if there are any), create +a new empty group with label s, and increase the value +of k by one. +With these definitions, the complete algorithm is now +as follows: +1. With probability 1 − 1/2k(n + 1) propose a move +of type 1. +1a) If k = 1 do nothing, since this implies all nodes +are in group 0 only, so there are no moves to +be made and there is no change of state on +this Monte Carlo step. +1b) Otherwise when k > 1 choose a random move +of type 1. +2. Else, with probability 1/2k(n+1), choose a random +move of type 2. +3. Accept the proposed move with probability +α(g, k → g, k) = min +� +1, P(A|g′, k′) +P(A|g, k) +� +. +(21) +Accepted moves are performed as proposed. If the +move is not accepted the state of the system re- +mains unchanged. +4. Repeat from step 1. +This algorithm again satisfies the condition of ergod- +icity trivially: we can reach any state with any number +of groups in a finite number of moves by first removing +all nodes from all groups except group 0, then removing +the groups themselves, then adding back the appropri- +ate number of groups and filling them with the desired +nodes. The algorithm also satisfies the condition of de- +tailed balance, which for this algorithm takes the form +P(g′, k′|A) +P(g, k|A) = π(g, k → g′, k′) α(g, k → g′, k′) +π(g′, k′ → g, k) α(g′, k′ → g, k). +(22) +The left-hand side can be written as +P(g′, k′|A) +P(g, k|A) = P(g′, k′)P(A|g′, k′) +P(g, k)P(A|g, k) , +(23) +and the ratio of acceptance probabilities is +α(g, k → g′, k′) +α(g′, k′ → g, k) = P(A|g′, k′) +P(A|g, k) . +(24) +Substituting from (23) and (24) into (22), a factor of +P(A|g′, k′)/P(A|g, k) cancels and our detailed balance +condition reduces to +P(g′, k′) +P(g, k) = π(g, k → g′, k′) +π(g′, k′ → g, k). +(25) +From Eqs. (6) and (7) the left-hand side is +P(g′, k′) +P(g, k) = (k − 1)! �k−1 +r=1 n′ +r!(n − n′ +r)!/(n + 1)! +(k′ − 1)! �k′−1 +r=1 nr!(n − nr)!/(n + 1)! +. (26) +Consider first the case where we propose a move of type 1 +that adds a node to group s. Then k′ = k and n′ +s = ns+1, +and n′ +r = nr for all other groups r, so Eq. (26) becomes +P(g′, k′) +P(g, k) = ns + 1 +n − ns +. +(27) +The probability of proposing such a move is +π(g, k → g′, k′) += +� +1 − +1 +2k(n + 1) +� +× +1 +k − 1 × 1 +2 × +1 +n − ns += +� +1 − +1 +2k(n + 1) +� +1 +2(k − 1)(n − ns), +(28) + +6 +while the probability of proposing the reverse move is +π(g′, k′ → g, k) = +� +1 − +1 +2k(n + 1) +� +1 +2(k − 1)(ns + 1). +(29) +Thus the ratio of the two is +π(g, k → g′, k′) +π(g′, k′ → g, k) = ns + 1 +n − ns +. +(30) +Between Eqs. (27) and (30), our detailed balance condi- +tion (25) is now satisfied. By a similar argument we can +show that detailed balance is also satisfied when a node +is removed from a group. +Now consider a move of type 2, which creates a new +empty group with a random label s. For such a move +Eq. (26) becomes +P(g′, k′) +P(g, k) = (k − 1)! �k +r=1 n′ +r!(n − n′ +r)!/(n + 1)! +k! �k−1 +r=1 nr!(n − nr)!/(n + 1)! += +1 +k(n + 1) , +(31) +where all factors inside the products have canceled except +for those pertaining to the new group, which gives us the +factor of 1/(n + 1). +The proposal probability for this move is equal to the +probability that we decide to do a move of type 2 times +the probability that we choose to add a new group with +a particular label s out of the k possibilities, giving +π(g, k → g′, k′) = +1 +2k(n + 1) × 1 +k = +1 +2k2(n + 1). +(32) +The reverse move on the other hand occurs when we per- +form a move of type 1 and choose group s from the k pos- +sibilities, then attempt to remove a node only to discover +that the group is already empty, causing us to delete the +entire group. The proposal probability for this move is +π(g′, k′ → g, k) = +� +1 − +1 +2(k + 1)(n + 1) +� +× 1 +k × 1 +2 += 1 +2k +� +1 − +1 +2(k + 1)(n + 1) +� +. +(33) +Now the ratio of the two probabilities is +π(g, k → g′, k′) +π(g′, k′ → g, k) = 4k(k + 1)(n + 1)/(2(k + 1)(n + 1) − 1) +2k2(n + 1) += +2(k + 1)/k +2(k + 1)(n + 1) − 1 += +1 +k(n + 1) + O(1/n2). +(34) +Here we assume n is large and hence that terms of +order 1/n2 can be neglected, making Eq. (34) equal +to Eq. (31), and hence our detailed balance condition, +Eq. (25), is satisfied. +This completes the proof of correctness of our algo- +rithms. +In the following sections we apply these algo- +rithms to fit our model to a variety of networks in order +to study core-periphery structure. +IV. +EXAMPLE APPLICATIONS +In this section we give example applications of our +methods to a selection of real-world networks, revealing a +range of behaviors and structures of interest in the core- +periphery divisions of these systems. +A. +Traditional two-group core-periphery structure +For our first set of examples, we perform calculations +in which the number of groups is fixed at k = 2, which +corresponds to the traditional two-group core-periphery +structure, as studied by many previous authors. Figure 2 +shows examples of such structure found in four different +networks. For each network the structure shown is the +highest-probability structure found during a single run +of our algorithm with 109 Monte Carlo steps and in each +case the algorithm finds clear core-periphery divisions, as +highlighted by the colors. +A number of interesting features emerge in these ex- +amples. First, we note that, as our model is defined, it is +arguably the edges that belong to groups, not the nodes. +As we have said, a node can belong to any number of +groups, but an edge only belongs to one: the proper- +ties of each edge are determined solely by the highest- +numbered group to which its two nodes both belong and +in this sense the edge belongs to this group only. In Fig. 2 +we have colored the edges according to the group they be- +long to and this provides a clear and useful visualization. +(The same trick will also be useful in Section IV B when +we study divisions with larger numbers of groups.) +All the images in Fig. 2 use the same color scheme: +group 0 is in yellow and group 1 is in blue. The figure +reveals that there are two distinctly different types of +core-periphery structure, one where the core is group 0 +and one where it is group 1. Recall that the probabil- +ity ωr of connection between two nodes depends on their +highest common group r, meaning in this case that edges +between nodes that are both in group 1 have probabil- +ity ω1 while all others have probability ω0. With this in +mind take a look at the figure. +Figures 2(a) and (b) show results for a network of air- +line routes [31] and a network of associations among a +group of terrorists [32] respectively. In both of these cases +the core found in the network is represented by group 1 +(in blue) and the periphery by group 0 (in yellow), with +ω1 > ω0. This implies that there is a high probability of +edges within the core (blue edges) and a lower probabil- +ity both in the periphery and also between the core and +the periphery (yellow edges). +Conversely, in Figs. 2(c) and (d), which represent the +Internet at the autonomous system level [33] and a net- +work of political weblogs [34], the groups are reversed, +with the core being group 0 and the periphery being +group 1, and ω1 < ω0. +In this “inside-out” type of +structure there is a high probability of connections both +within the core and between the core and periphery (yel- + +7 +(a) Airline routes among European airports [31] +(b) A network of associations among terrorists involved in +the 2004 Madrid train bombing [32] +(c) Network representation of the Internet in November 1997 +at the autonomous system level [33] +(d) A network of hyperlinks among a set of US political +blogs [34] +FIG. 2: Two types of two-group core-periphery structure distinguished by the hierarchical model. Panels (a) and (b) +have a core that is densely connected within itself but only sparsely connected to the periphery. This is the +traditional definition of core-periphery structure. Panels (c) and (d) on the other hand show a kind of “inside-out” +structure in which the core is strongly connected within itself and strongly connected to the periphery. +low edges), and a lower probability in the periphery (blue +edges). +These two types of core-periphery structure represent +quite different circumstances. In the first, the core is iso- +lated from the periphery in the sense that it is densely +connected only within itself and sparsely connected to +everything else. +In the second, the core is strongly +connected everywhere, both to itself and to others and +dominates the connectivity of the network. +The lat- +ter (“inside-out”) structure is particularly interesting be- +cause it deviates from the traditional definition of core- +periphery structure as formulated for instance by Bor- +gatti and Everett [11], who assumed an isolated core. +Our method naturally and automatically distinguishes + +8 +FIG. 3: Political books network with a periphery and +two cores corresponding to left and right leaning books. +between the two types of structure. +The two types make some sense in the present case. For +the airline route network, for instance, the core broadly +represents airline hubs and the periphery represents re- +gional airports. One expects strong connections between +hubs—almost all pairs of hubs have direct flights—but +one expects only weak connections to the outlying air- +ports, many of which only fly to a single hub. Conversely, +in the weblog network, for example, the core represents +the most influential blogs, ones which most members of +the community link to, so we expect connections to be +strong not only within the core but also between the core +and the periphery. +B. +Structure with an arbitrary number of groups +Now let us look at what happens when we allow the +number of groups to vary, taking whatever value is nec- +essary to best fit the structure of the network. +Here +again we find some interesting features. As a first exam- +ple, Fig. 3 shows a copurchasing network of books. The +nodes in this network represent 105 popular books on US +politics and the edges represent frequent copurchase on +Amazon.com, i.e., purchase by the same buyers. +This +network, which has been studied previously by a number +of authors [35, 36], is known to show clear community +structure in which the network divides into communi- +ties of left- and right-leaning books. Our core-periphery +analysis, as indicated by the colors in the figure, finds +three groups: two cores and a single periphery. The two +cores correspond to the innermost members of the left- +and right-leaning communities while the periphery cap- +tures the remainder of the network. Thus, the algorithm +has found the political divide between left and right but +also finds a large group of peripheral books that, at least +in this analysis, are well represented as a homogeneous +mass, suggesting that they are not strongly connected to +either side of the political aisle. +Figure 4 shows a similar finding for another well stud- +ied example of community structure, a network of com- +petition between US college teams in the sport of Amer- +ican football [3]. College football teams are divided into +a number of groups or “conferences,” and most games +are played between teams in the same conference, so the +FIG. 4: American Football network where each of the +cliques are connected to each other via the periphery. +network of games played, as analyzed here, has strong +community structure which can easily be discovered with +a range of community detection algorithms. Again, how- +ever, our core-periphery analysis returns a more subtle +picture, as shown in the figure. Our algorithm finds a sep- +arate core for each conference, accurately dividing most +teams into the 11 conferences in the network. A small +number of teams—many of them independents who be- +long to no conference—are not assigned to any core, and +all inter-conference games are assigned to the periph- +ery. +This makes good sense: it tells us that the con- +ferences constitute a clear set of separate groups in the +network, while inter-conference play and non-conference +teams constitute a single periphery. +This is an accu- +rate description of the network and a more economi- +cal one than the standard community structure division, +as found for instance using the stochastic block model, +which also assigns a separate community for each con- +ference but in addition assigns a separate probability for +inter-conference play between every single pair of confer- +ences, rather than recognizing that a single periphery is +an adequate and more parsimonious description. +In these last two examples our algorithm has found a +hybrid of core-periphery structure and community struc- +ture. While this is illuminating for these particular exam- +ples, it is important to realize that this is not inevitable, +and that the algorithm will return other structures where +appropriate. Figure 5 shows an example. The network +in this figure is a famous one from the social networks +literature, a network of interactions observed by Free- +man [37] between a group of people windsurfing off the +California coast in 1986. This network is known to have + +9 +(a) Traditional community structure +(b) Core-periphery structure +FIG. 5: Structure found in the network of windsurfers. +a clear two-group community structure which is easily +found by community detection—see Fig. 5a. When an- +alyzed using the methods of this paper we also find two +groups, but they are not the same: now we find core and +periphery but no clear division between the communities, +suggesting that connections within the core may be just +as important as divisions between the two communities. +V. +CONCLUSIONS +In this paper we have proposed a hierarchical model of +core-periphery structure in networks and a Monte Carlo +scheme for fitting it to observed network data. Apply- +ing these methods to a variety of real-world networks +we find a number of interesting patterns. The method +is able to capture traditional two-group core-periphery +structure consisting of a dense core weakly connected to +a sparse periphery. In some networks, however, we find +that a better fit is given by a novel “inside-out” struc- +ture in which the core is connected strongly both within +itself and to the periphery. Various networks are better +represented by one or other of the two types of structure +and the distinction between the two could offer a more +nuanced view of structure and function in these networks. +We have also investigated cases where there are more +than two groups in the network, generalizing the tra- +ditional core-periphery structure (as other authors have +also done). For this we use a Monte Carlo scheme that +allows the number of groups to vary freely, automatically +choosing the number that best fits the network in ques- +tion. In some cases, we find a structure akin to a hybrid +between core-periphery structure and community struc- +ture in which there is a separate core in each of several +communities plus a single periphery surrounding all of +them. In other cases, we find pure core-periphery struc- +ture without any communities. +There are a number of possible directions for further +research using these methods. First, we have looked here +at only the highest probability structures found by our +algorithms but in principle the algorithms return a com- +plete sample of high-probability structures drawn from +the posterior distribution of the model and it would be +interesting to study the range of structures within such +a sample. +Are they all closely similar, so that a sin- +gle consensus structure can well represent them all, or is +there significant variation between structures, and if so +of what kind? Second, one could examine generalizations +of the method to broader classes of networks, such as di- +rected and weighted networks and multiplex networks. +Another interesting question is whether there exists a +natural “degree-corrected” version of the model akin to +the degree-corrected stochastic block model of [24]. The +model proposed here is not degree corrected, which could +cause issues with networks that have a very broad degree +distribution. 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Journal of Social and Biologi- +cal Structures 11, 415–425 (1988). + diff --git a/XtE2T4oBgHgl3EQfEAYx/content/tmp_files/load_file.txt b/XtE2T4oBgHgl3EQfEAYx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9dff34b7dfbe06967c3017503aeb328c2e55b3ae --- /dev/null +++ b/XtE2T4oBgHgl3EQfEAYx/content/tmp_files/load_file.txt @@ -0,0 +1,640 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf,len=639 +page_content='Hierarchical core-periphery structure in networks Austin Polanco1 and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Newman1, 2 1Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA 2Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48109, USA We study core-periphery structure in networks using inference methods based on a flexible network model that allows for traditional onion-like cores within cores, but also for hierarchical tree-like structures and more general non-nested types of structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' We propose an efficient Monte Carlo scheme for fitting the model to observed networks and report results for a selection of real-world data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Among other things, we observe an empirical distinction between networks showing traditional core-periphery structure with a dense core weakly connected to a sparse periphery, and an alternative structure in which the core is strongly connected both within itself and to the periphery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Networks vary in whether they are better represented by one type of structure or the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' We also observe structures that are a hybrid between core-periphery structure and community structure, in which networks have a set of non-overlapping cores that correspond roughly to communities, surrounded by a single undifferentiated periphery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Computer code implementing our methods is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' INTRODUCTION Networks are widely used as a compact and convenient mathematical representation of the connections between the elements of a complex system, such as data con- nections on the Internet, citations between papers, so- cial contacts among people or animals, synaptic connec- tions between brain cells, and biological and biochemi- cal networks of many kinds [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' A significant amount of effort has been devoted in recent years to analyz- ing and understanding large-scale structure in such net- works, especially community structure [3, 4], but also nested [3, 5] and overlapping [6, 7] communities and strat- ification [8, 9], as well as the related issues of embedding and graph representation learning [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' In this paper we focus on a less well-studied form of large-scale structure, core-periphery structure, in which a network is divided into a densely connected core of nodes surrounded by a sparser periphery [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' The observation of core- periphery structure communicates different information about a network from community structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Where com- munity structure deals with the identification of groups or types of nodes, core-periphery structure focuses on their roles and centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Core-periphery structure is integral to understanding the link between node posi- tion and function in networks, for instance in the Inter- net [14, 15], neuroscience [16], and economics [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' A range of heuristic methods have been proposed in the past for detecting core-periphery structure in networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' In the simplest case the challenge is to take unlabelled network data and assign each node of the network in some automated fashion to either the core or the periphery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' In more complex cases one may attempt to infer a onion-like sequence of deeper and deeper cores within cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' The problem, however, is not entirely well posed, since we have not precisely defined what “core” and “periph- ery” mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Various researchers have chosen to define them in various ways and there is, as a result, a cor- responding spectrum of algorithmic approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Per- haps the oldest approach is the k-core decomposition, in which nodes are recursively removed from a network in order of increasing degree and the sequence in which they are removed defines a sliding scale from core to periph- ery [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' This method essentially equates the core with high-degree nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Another well-established method is that of Borgatti and Everett [11], who defined a quality function akin to the well-known modularity function for community detection [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Borgatti and Everett’s func- tion takes as input a network and a putative division into core and periphery and returns a score that indi- cates whether the division is a “good” one in a certain sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Then by maximizing the function over all possi- ble divisions one can find the “best” core-periphery de- composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Variants of the same idea have been ex- plored by Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' [13], as well as by Kojaku and Masuda [20] who proposed a multi-group version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Cu- curingu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' [21] have explored several methods for find- ing core-periphery structure based on counting geodesic paths and on spectral network properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Other meth- ods use node-level properties of the network such as the clustering coefficient [12] or centrality measures [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Stochastic block models, commonly used in commu- nity detection [23, 24], have also been applied to core- periphery structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' In these models, one assumes that nodes are divided into types (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=', core and periphery) and that the probability ωrs of an edge between a pair of nodes depends on the types r and s of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' [25] studied a two-group version of this model with ω11 > ω12 > ω22, which generates net- works with classic core-periphery structure, then fitted the model to empirical network data to detect structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Gallagher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' [26] took a similar approach but with more than two groups and multiple nested cores such that ωrs = f[max(r, s)] for some function f[r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' In this paper we also take a model-fitting approach to the study of core-periphery structure, but we formulate a more general model that includes the classic two-group structure but also allows for more flexible structures as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' In particular, we consider a hierarchical model with any number of groups, the number being determined by the network structure using Bayesian model selection, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content='03630v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content='SI] 9 Jan 2023 2 Group 0 Group 1 Group 2 (a) Nested groups Group 0 (all nodes) Group 1 Group 2 Group 3 Group 4 Group 5 (b) Tree-like hierarchy Group 0 (all nodes) Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 (c) General core-periphery structure FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' 1: The model proposed here is capable of capturing a range of different types of structure, including (a) traditional nested groups, (b) tree-like branching structures, and (c) complex structures of overlapping groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' and we specifically allow for the possibility that the hi- erarchy may not be perfectly nested and moreover that there maybe multiple cores and multiple peripheries at any level in the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' These features allow our model to capture variants of core-periphery structure that oc- cur in real-world networks but are not captured by tra- ditional analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' In a multilevel hierarchy, for instance, an inner core may not be perfectly contained within an outer one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' In a network with community structure the individual communities may each have a separate core of their own, or there may be several cores within a single periphery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' By fitting our proposed model to network data we show how it can reveal subtle structure and, in the process, encounter some new formations within networks that have not previously received significant attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' A HIERARCHICAL MODEL OF NETWORK STRUCTURE One can think of conventional core-periphery structure as a hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' In the simplest case the hierarchy has just two levels, an outer periphery and an inner core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' In more elaborate cases there can be an onion-like succession of levels, each one nested inside the next, from the outer- most to the innermost—see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' This, however, is not the most general kind of nested structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' A more general structure can have any number of first-level cores within a single periphery, and any number of second-level cores within those first-level ones, and so forth (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' A structure like this can be captured with a tree or den- drogram, and some attempts have been made to fit den- drograms to networks [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' The work presented here goes still further by allowing for the possibility that the hier- archy is not perfectly nested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' In our model, we envisage a hierarchical series of cores, but we allow these cores to be placed in any way around the network, with no require- ment that they be nested within one another (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' In practice, if two cores do not overlap then they do not interact and hence their relative order in the hierarchy has no effect, but if they overlap then the higher ranked one takes precedence over the lower in a manner we will describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' The net result is a hierarchical model that gen- eralizes both the conventional onion layers and the den- drogram and, as we will see, captures a wide range of possible structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' We then fit this model to network data to infer core-periphery structure in real-world net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' In detail the procedure is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Consider an unweighted, undirected network of n nodes, which can belong to any of k groups labeled by r = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' By contrast with traditional community structure models we allow nodes to belong to any num- ber of groups simultaneously, up to and including all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' In addition, every node always belongs to group 0, which acts as a sort of default or base group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Thus each node belongs to group 0 plus some selection of the other groups 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' (We label the first group 0 rather than 1 to remind ourselves of its special status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=') We define a set of indicator variables gr u such that gr u = 1 if node u belongs to group r and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Then we define a set of probabilities ωr, one for each group including group 0, and place edges between node pairs independently with probability ωr, where r is the highest common group that both nodes of the pair belong to, meaning the one with the highest number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' For exam- ple, if one node belongs to groups 0, 1, 2 and another belongs to 0, 1, 3, then there is an edge between them with probability ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Figure 1c illustrates the behavior of the resulting net- work: the groups form “patches” that lie one on top of another and the topmost visible patch takes precedence for each node pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' For instance, if all nodes belong to group 0 only, then every pair of nodes has equal proba- bility ω0 of being connected, which gives us a standard random graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' But if we assign some subset of the nodes to group 1 then for any pair of nodes that are both in group 1 the probability of an edge becomes ω1 and over- rides the previous ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Similarly any pair of nodes as- signed to group 2, including nodes already in groups 0 and 1, have probability ω2 of connection, overriding ω0 and ω1, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' The end result is a model that starts out as a random graph but then adds variation 3 and detail to the network wherever it is needed to cap- ture local structural features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' This approach has some similarities to those of Kojaku and Masuda [20] and Gal- lagher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' [26], but it differs crucially in that it does not force the groups to be either strictly nested within each other or nonoverlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Given the definition of the model, our goal is now to fit it to observed network data to infer the best choice of groups gr u for each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Suppose we have a network of n nodes, represented by its n × n adjacency matrix A with elements auv = 1 if there is an edge between nodes u and v and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfEAYx/content/2301.03630v1.pdf'} +page_content=' Then the probability of observing a particular network if it was generated from our model with given k and given group memberships g is P(A|ω, k, g) = � u 𝑇C), +(1.1) +where 𝛽 is the stretched index. In the case of 𝛽 = 1.0, the equation (1.1) gives a critical slowing down of +normal ferroelectrics without random fields. In the case of 𝛽 > 1.0, the slowing down of relaxation time +is suppressed and/or stretched by an increase of the strength of random fields. In PZN–7PT, it is found +that the value of 𝛽 = 3.0 gives a good reproduction of slowing down [16]. +In some ferroelectrics, the mechanism of the ferroelectricity is not simple. The coexistence of both +mechanisms or the crossover from displacive to order-disorder nature has been reported. In LiNbO3, +two-stage process involving a displacive transition in the Nb–O cages and an order-disorder transition in +the Li–O planes was reported at the Curie temperature [17]. The unified model theory describing both the +“order-disorder” and “displacive” ferroelectric phase transitions was proposed by introducing the model +pseudospin-phonon Hamiltonian [18]. For such phase transitions, the study of the lowest frequency soft +optic modes by Raman scattering or infrared spectroscopy is also necessary. +2. Ferroelectrics with tungsten-bronze structure +Ferroelectricity has been observed in various kinds of organic and inorganic materials. Regarding +inorganic ferroelectrics, the oxygen octahedra ferroelectrics are the most popular. One of this family is the +ferroelectrics with tungsten-bronze structure. It is technologically important in the field of telecommuni- +cations due to its superior electro-optical, photorefractive, and nonlinear optical properties such as second +harmonic generation, and its resistance to optical damage is high. The ferroelectricity was reported at +first in lead metaniobate, PbNb2O6 with 𝑇C = 570℃ [19]. Its piezoelectric constant is the same order of +magnitude as that of barium titanate, BaTiO3 with 𝑇C = 120℃. The useful nonlinear coefficients and low +optical damage were reported in barium sodium niobate, Ba2NaNb5O15 (BNN), with 𝑇C = 560℃ [20]. +Nowadays, a lot of tungsten-bronze type ferroelectrics are known [2]. +Figure 1 shows the projection of tungsten bronze structure on the 𝑐-plane. In BNN, the A1 and A2 sites +are fully filled by Na and Ba, respectively, and there is no vacancy. The B1 and B2 sites are fully occupied +by Nb, while the C sites are vacant. BNN is called a filled tungsten-bronze structure with no charge +disorder at A1 and A2 sites. BNN undergoes successive phase transitions at 560, 300, and −163℃ [21]. +Figure 1. (Colour online) Projection of tungsten bronze structure on the 𝑐-plane. Tetragonal and or- +thorhombic unit cells are shown by dotted and solid lines, respectively. +43702-2 + +B +2 +B +C +X区 +Oxygen +octahedron +b +0 +b +T +T +p +0Ferroelectric instability of barium sodium niobate +The higher one is associated with the ferroelectric Curie temperature at 560℃, and its crystal symmetry +changes from prototypic 4/𝑚𝑚𝑚 to ferroelectric tetragonal 4𝑚𝑚 with a spontaneous polarization 𝑃3 +along the 𝑐-axis. The symmetry changes from tetragonal to incommensurate (IC) orthorhombic 𝑚𝑚𝑚 +systems at 300℃, the 𝑎 and 𝑏 axes are rotated for 45◦ along the 𝑐-axis as shown in figure 1. The +modulation direction of the IC wave vectors is along 𝑎 and 𝑏 axes of the orthorhombic coordinate [22]. +The lowest temperature phase transition at −163℃ is the reentrant ferroelastic phase transition into the +tetragonal 4𝑚𝑚 phase. The pressure induced reentrant ferroelastic phase transition was also observed +at 2.2 GPa and at room temperature [23, 24]. Recently, a new type of the IC phase transition was proposed +by Ishibashi [25]. This new type of phase transition is referred to as type III, and it is characterized by +the parabolic splitting of the doubly degenerate modes at the Brillouin zone boundary. The related +macroscopic change in the IC phase transition was studied by Brillouin scattering [26]. Regarding a +ferroelectric instability, the accurate measurement of low-frequency polaritons was performed on the +optical phonon branch of A1(𝑧) symmetry. However, any evidence of displacive nature was not found +down to 18 cm−1 [27]. +Since the 𝑇C of Ba2NaTa5O15 (BNT) is −233℃, the high tunability of the 𝑇C of the temperature +width of about 800℃ was reported for Ba2NaNb5(1−𝑥)Ta5𝑥O15 (BNNT), and this is technologically +important [28, 29]. The ferroelectric phase transition of the BNNT single crystals with 𝑥 = 0.57 at +𝑇C = 115℃ was studied by broadband dielectric spectroscopy up to 4 GHz. The order-disorder nature +of the proper ferroelectric phase transition was observed, and its origin is attributed to the anharmonic +motion of the Nb (Ta) atoms in a double well potential of oxygen octahedra [30]. +Up to the present, no soft optic mode was observed in the tungsten-bronze ferroelectrics by vibrational +spectroscopy. Recent theoretical studies reported the local pseudo-Jahn–Teller effect (PJTE) in transition +metal B ion center of ABO3 perovskite crystals. The vibronic coupling between the ground and excited +electronic states of the local BO6 center results in dipolar distortions, leading to an eight-well adiabatic +potential energy surface [31]. Such a situation may also occur in tungsten bronze ferroelectrics and the +order-disorder nature of ferroelectricity may exist. Therefore, the order-disorder nature of a ferroelectric +phase transition of BNN has been examined by the broadband Brillouin scattering spectroscopy, which is a +powerful tool to observe a critical slowing down in the vicinity of an order-disorder type phase transition +temperature. In this paper, we review the broadband Brillouin scattering study on the ferroelectric +instability of a BNN crystal [32]. +3. Broadband Brillouin scattering and ferroelectric instability +Vibrational spectroscopy i.e., infrared spectroscopy and Raman scattering observes the vibrational +modes of atoms, molecules, and crystal lattice. In the vibrational study of inorganic ferroelectric crystals, +it is possible to observe not only the internal modes of octahedra or tetrahedra but also the external modes +such as a soft optic mode. The spectral resolution of vibrational spectroscopy is usually 1 cm−1 = 30 GHz +or more, and it is sufficient to detect the change of mode frequency related to a phase transition. The +resolution of 1 cm−1 is sufficient to measure the temperature dependence of a ferroelectric soft optic +mode. However, in the study of a ferroelectric phase transition of order-disorder type, the resolution +of 1 cm−1 is not sufficient to measure the critical slowing down of the relaxation time towards the Curie +temperature [33]. +Polarization fluctuations related to a ferroelectric instability are detected as a broad CP in an inelastic +scattering spectrum. The colorless and transparent BNN crystal studied was grown by Czochralski +method in Tamagawa factory, NEC, Japan. The (100) plate with the size of 2.65 × 2.25 × 0.68 mm +with two optically polished surfaces was used for Brillouin scattering measurements. Brillouin scattering +spectra were measured at the backward scattering geometry using a 3 + 3 tandem multi-pass Fabry– +Perot interferometer and a conventional photon counting system. A single frequency green YAG laser +(𝜆 = 532 nm) with power of 100 mW was used as an exciting source. The light spot size at a sample +surface was about 10 𝜇m using the optical microscope (BX-60) [33]. The temperature of a sample was +controlled by the heating stage of a T1500 (high T Linkam) from room temperature up to 750℃. All +the Brillouin scattering spectra were measured in the condition that the free spectral range (FSR) and +the scan range are 300 and 600 GHz, respectively. An intense polarized CP of BNN was observed in the +43702-3 + +S. Kojima +500 +400 +300 +200 +100 +0 +Intensity (arb. unit) +-400 +-200 +0 +200 +400 +Frequency shift (GHz) +600 +oC +BNN a-plate +FSR = 300 GHz +VV +VH +Paraelectric phase +Figure 2. (Colour online) Broadband VV and VH Brillouin scattering spectra of a BNN crystal at 600℃ +observed by 𝑎(𝑐𝑐) ¯𝑎 and 𝑎(𝑐𝑏) ¯𝑎 back scattering geometry, respectively. +vicinity of the Curie temperature, 𝑇C = 560℃, in the broadband Brillouin scattering spectra as shown +in figure 2 [34]. In the polarized VV spectrum observed at 𝑎(𝑐𝑐) ¯𝑎, backward scattering geometry shows +an intense broad CP with A1(𝑧) symmetry, while in the depolarized VH spectrum observed at 𝑎(𝑐𝑏) ¯𝑎, +backward scattering geometry does not show an intense CP with B2 symmetry. Therefore, the polarization +fluctuations along a ferroelectric 𝑐-axis are the origin of an intense broad CP. +4. Critical slowing down on a ferroelectric phase transition of barium +sodium niobate +For the detailed analysis of the width of a CP, the temperature dependence of broadband Brillouin +scattering spectra of a BNN crystal was measured at the backward scattering geometry with the free +spectral range of 300 GHz as shown in figure 3. +1000 +800 +600 +400 +200 +Intensity (arb. unit) +-400 +-200 +0 +200 +400 +Frequency shift (GHz) +BNN a-plate +FSR = 300 GHz +600 +oC +580 +oC +700 +oC +Figure 3. (Colour online) Broadband VV Brillouin scattering spectra of a BNN crystal in a paraelectric +phase. +Under the assumption of a single Debye relaxation process, the relaxation time 𝜏CP was determined +43702-4 + +Ferroelectric instability of barium sodium niobate +by the relation π × (CP width) = 𝜏−1 +CP. The relaxation process related to the order-disorder nature of +a ferroelectric phase transition has been observed as a CP with a zero frequency shift in an inelastic +scattering spectrum. In the order-disorder phase transition, the relaxation time 𝜏 of the fluctuations of the +order parameters increases toward the phase transition point and was called the critical slowing down. +The relaxation time determined from the CP width shows a critical slowing down in the vicinity of +𝑇C = 560℃ as shown in figure 4 [34]. The temperature dependence of the relaxation time is given by the +following equation of the case of 𝛽 = 1.0 in equation (1.1) for a first order phase transition: +1 +𝜏CP += 1 +𝜏0 ++ 1 +𝜏1 +�𝑇 − 𝑇1 +𝑇1 +� +, +(𝑇 > 𝑇C > 𝑇1). +(4.1) +For example, in the ferroelectric phase transition at 𝑇C += 500 K of the relaxor ferroelectric +0.70Pb(Sc1/2Nb1/2)O3–0.30PbTiO3 with the perovskite structure, the values of the fitting parameters +are 𝜏0 = 14 ps and 𝜏1 = 0.47 ps, and 𝑇1 = 500 K [35]. The temperature dependences of 𝑇/𝐼CP of a BNN +crystal are shown in figure 5. In BNN, the fitting parameters of 1/𝜏 are 𝜏0 = 1.29 ps, 𝜏1 = 0.73 ps, and +𝑇1 = 555℃. The intensity of a CP 𝐼CP obeys the following equation in a paraelectric phase [36]: +𝑇 +𝐼CP +∝ +������ +∞ +∫ +0 +𝜒′′ (𝜔) +𝜔 +d𝜔 +������ +−1 +∝ +1 +𝜒′(0) = 𝑇 − 𝑇1 +𝐶 +, +(𝑇 > 𝑇C > 𝑇1). +(4.2) +Here, for the first order phase transition, 𝑇C >T1, because the ferroelectric phase transition of BNN is the +first order. In the ferroelectric phase transition at 𝑇C = 500 K of the 0.70Pb(Sc1/2Nb1/2)O3–0.30PbTiO3, +the Curie–Weiss law also holds for 𝐼CP/𝑇 above 𝑇C [35]. +1.2x10 +12 +1.1 +1.0 +0.9 +0.8 +0.7 +1/τCP (GHz) +700 +600 +500 +400 +Temperature (oC) +BNN +TC=560 +oC +Figure 4. (Colour online) Temperature dependence of the inverse relaxation time. The dotted line is the +fitted line by the equation (4.1) above 𝑇C = 560℃. +The experimental results of the critical slowing down of relaxation time and the Curie–Weiss behavior +of the CP intensity indicate the order-disorder nature of a ferroelectric phase transition of BNN. In the +study of the order-disorder phase transition, Brillouin scattering is a powerful tool to detect the critical +slowing down [37]. +43702-5 + +S. Kojima +3.0x10 +-3 +2.5 +2.0 +1.5 +1.0 +T/ICP +700 +600 +500 +400 +Temperature (oC) +BNN +TC=560 +oC +Figure 5. (Colour online) Temperature dependence of temperature divided by the intensity of a central +peak. The dotted line is the fitted line by the equation (4.2) above 𝑇C = 560℃. +5. Conclusions +For the study of the lattice instability of ferroelectrics, vibrational spectroscopy is a powerful tool to +discuss not only displacive but also order-disorder nature. This paper reviews the experimental studies on +the ferroelectric instability of a ferroelectric phase transition of barium sodium niobate (BNN) crystals +with tungsten-bronze structure. BNN is one of well-known optical crystals for electro-optic and nonlinear +optic applications. It shows a uniaxial ferroelectricity with a spontaneous polarization along the tetragonal +𝑐-axis. In the vicinity of the Curie temperature, 𝑇C = 560℃, an intense central peak (CP) was observed +by the broadband Brillouin scattering experiment. The CP has a strong polarization dependence, which +originates from the polarization fluctuations along the ferroelectric 𝑐-axis. The CP intensity shows +a maximum at 𝑇C. The relaxation time determined by the CP width shows a critical slowing down +towards 𝑇C. The temperature dependence of the CP intensity shows the Curie–Weiss behavior. These +experimental results are the evidence of the order-disorder nature of the ferroelectric instability of BNN. +Acknowledgements +Author thanks to Prof. J. Grigas, Prof. J. Banys, Prof. M. Maczka for the collaboration and S. Ohta, +Y. Christy, K. Matsumoto, K. Suzuki, and M. Aftebuzzamann for the discussion and experiments. +Funding +This research was funded in part by JSPS KAKENHI, Grant No. JP17K05030. +References +1. Valasek J., Phys. Rev., 1921, 17, No. 4, 475–481, doi:10.1103/PhysRev.17.475. +2. Xu Y., Ferroelectric Materials and Their Applications, North-Holland, Amsterdam, 1991. +3. Blinc R., Advanced Ferroelectricity, Oxford University Press, New York, 2011. +43702-6 + +Ferroelectric instability of barium sodium niobate +4. Yamada Y., Fujii Y., Hatta I., J. Phys. Soc. Jpn., 1968, 24, No. 5, 1053–1058, doi:10.1143/JPSJ.24.1053. +5. Hill R. M., Ichiki S. K., Phys. Rev., 1962, 128, No. 3, 1140, doi:10.1103/PhysRev.128.1140. +6. Nakamura E., Hosoya M., J. Phys. Soc. Jpn., 1967, 23, No. 4, 844–847, doi:10.1143/JPSJ.23.844. +7. Ohta R., Zushi J., Ariizumi T., Kojima S., Appl. Phys. Lett., 2011, 98, No. 9, 092909, doi:10.1063/1.3560345. +8. Rahaman M. M., Imai T., Miyazu J., Kobayashi J., Tsukada S., Helal M. A., Kojima S., J. Appl. 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Kojima S., Aftabuzzaman M., Dec J., Kleemann W., In: Proceedings of the Conference “IV Lithuanian– +Ukrainian–Polish meeting on physics of ferroelectricity”, Palanga, Lithuania, 2016. +33. Kojima S., Jpn. J. Appl. Phys., 2010, 49, 07HA01, doi:10.1143/JJAP.49.07HA01. +34. Ota S., Matsumoto K., Suzuki K., Kojima S., IOP Conf. Ser.: Mater. Sci. Eng., 2014, 54, 012018, +doi:10.1088/1757-899X/54/1/012018. +35. Kojima S., Tsukada S., Hidaka Y., Bokov A. A., Ye Z. G., J. Appl. Phys., 2011, 109, No. 8, 084114, +doi:10.1063/1.3581025. +36. Hays W., Loudon R., Scattering of Light by Crystals, Dover Publishing, New York, 1978. +37. Kojima S., Materials, 2022, 15, No. 10, 3518, doi:10.3390/ma15103518. +43702-7 + +S. Kojima +Дослiдження сегнетоелектричної нестiйкостi в нiобатi +барiю-натрiю методами широкосмугового розсiювання +Брiллюена +С. Коджiма +Вiддiлення матерiалознавчих наук, Унiверситет Цукуби, Цукуба, Iбаракi 305-8573, Японiя +Нiобат барiю-натрiю (BNN) зi структурою вольфрамової бронзи є одним з добре вiдомих оптичних криста- +лiв, якi використовуються для електрооптичних дослiджень та у нелiйнiйнiй оптицi. У данiй роботi розгля- +дається сегнетоелектрична нестiйкiсть в кристалах BNN. BNN є одновiсним сегнетоелектриком, в якому +спонтанна поляризацiя напрямлена вздовж тетрагональної осi 𝑐. У лiтературi немає згадок про спостере- +ження оптичної м’якої моди, вiдповiдальної за сегнетоелектричний фазовий перехiд у цьому кристалi. В +околi температури Кюрi 𝑇C = 560°C в спектрах широкосмугового розсiювання Брiллюена спостерiгається +iнтенсивний центральний пiк, пов’язаний з флуктуацiями поляризацiї удовж осi 𝑐. Час релаксацiї, який +визначається шириною центрального пiка, виявляє критичне сповiльнення при наближеннi до 𝑇C. Цей +факт свiдчить про те, що сегнетоелектрична нестiйкiсть у BNN-сполуках є типу “лад-безлад”. +Ключовi слова: розсiювання Брiллюена, сегнетоелектрик, лад-безлад, центральний пiк, нiобат +барiю-натрiю +43702-8 + diff --git a/YNAzT4oBgHgl3EQfYvyp/content/tmp_files/load_file.txt b/YNAzT4oBgHgl3EQfYvyp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..57476a6e19f0b7a35fd4f2c4c5a6490415b66538 --- /dev/null +++ b/YNAzT4oBgHgl3EQfYvyp/content/tmp_files/load_file.txt @@ -0,0 +1,615 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf,len=614 +page_content='Condensed Matter Physics, 2022, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' 25, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' 4, 43702: 1–8 DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='5488/CMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='43702 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='icmp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='lviv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='ua/journal Broadband Brillouin scattering study of ferroelectric instability of barium sodium niobate S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Kojima∗ Division of Materials Science, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan Received July 18, 2022 The barium sodium niobate (BNN) with tungsten-bronze structure is one of well-known optical crystals for electro-optic and nonlinear optic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' This paper reviews the ferroelectric instability of BNN crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' BNN is a uniaxial ferroelectric with a spontaneous polarization along the tetragonal 𝑐-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' There is no report on the observation of an optical soft mode responsible for a ferroelectric phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In the vicinity of the Curie temperature, 𝑇C = 560°C, an intense central peak (CP) related to the polarization fluctuations along the 𝑐-axis was observed by the broadband Brillouin scattering experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The relaxation time determined by the CP width shows the critical slowing down towards 𝑇C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' This fact indicates that the ferroelectric instability of BNN is an order-disorder type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Key words: Brillouin scattering, ferroelectric, order-disorder, central peak, barium sodium niobate 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Introduction Ferroelectricity is defined by the existence of switchable spontaneous polarization by an external electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Ferroelectric phenomenon was identified for the first time in 1920 by Valasek on the study of Rochelle salt [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The microscopic origin of ferroelectricity has two typical cases, namely, the displacive type and order-disorder type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In the displacive type, an infrared active soft optic mode exists in a paraelectric phase, and the freezing of a soft mode displacement induces a spontaneous polarization [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The softening of a soft mode frequency has been observed by far-infrared spectroscopy, Raman scattering, and neutron inelastic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' On the other hand, in the order-disorder type, the relaxation time of the polarization fluctuations of polar molecules along a ferroelectric axis diverges at the Curie temperature, and the aligned polar molecules induce a spontaneous polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Rochelle salt belongs to this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The critical slowing down towards a Curie temperature has been observed by dielectric spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In NaNO2, Hatta observed the divergence of the relaxation time of the flipping motion of each NO2 ion toward a Curie temperature due to the thermodynamical slowing down process of the correlated fluctuation of polarization [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The critical slowing-down of the polarization relaxation process was also observed in triglycine sulfate [5] and Ca2Sr(C2H5CO2)6 above the Curie temperature [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Another method to observe the critical slowing down is the low-frequency inelastic light scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The polarization fluctuations along a ferroelectric axis are observed as a central peak (CP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In the vicinity of a Curie temperature, the divergence of CP intensity and the narrowing of CP width are observed for an order-disorder phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In a K(Ta0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='68Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='32)O3 crystal, the relaxation time determined by the CP width clearly shows a critical slowing down towards the Curie temperature, 𝑇C = 258 K, indicating an order-disorder feature of the ferroelectric phase transition [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Up to the present, the critical slowing down has been studied by the observation of a CP in ferroelectric phase transitions of 12 mol% KF substituted BaTiO3 [9], LiTaO3 [10], KNN [11], MAPbCl3 [12], BaTi2O5 [13], and K2MgWO2(PO4)2 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' For the ferroelectric phase transitions of relaxor ferroelectrics, the diffusive nature was observed in the critical slowing down [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='70Pb(Zn1/3Nb2/3)O3–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='30PbTiO3 (PZN–7PT), the slowing down ∗e-mail: kojima@ims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='tsukuba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='jp This work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='0 International License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' 43702-1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='01341v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='mtrl-sci] 3 Jan 2023 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Kojima was suppressed below the intermediate temperature 𝑇∗ and the typical critical slowing down was not observed near 𝑇C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The local transition from dynamic to static PNRs at 𝑇∗ stops the farther slowing down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' To describe such a suppressed slowing down by random fields, the empirical equation of the stretched slowing down was used in the vicinity of 𝑇C as given by the following equation, 1 𝜏CP = 1 𝜏0 + 1 𝜏1 �𝑇 − 𝑇C 𝑇C �𝛽 , (𝑇 > 𝑇C), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='1) where 𝛽 is the stretched index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In the case of 𝛽 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='0, the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='1) gives a critical slowing down of normal ferroelectrics without random fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In the case of 𝛽 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='0, the slowing down of relaxation time is suppressed and/or stretched by an increase of the strength of random fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In PZN–7PT, it is found that the value of 𝛽 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='0 gives a good reproduction of slowing down [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In some ferroelectrics, the mechanism of the ferroelectricity is not simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The coexistence of both mechanisms or the crossover from displacive to order-disorder nature has been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In LiNbO3, two-stage process involving a displacive transition in the Nb–O cages and an order-disorder transition in the Li–O planes was reported at the Curie temperature [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The unified model theory describing both the “order-disorder” and “displacive” ferroelectric phase transitions was proposed by introducing the model pseudospin-phonon Hamiltonian [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' For such phase transitions, the study of the lowest frequency soft optic modes by Raman scattering or infrared spectroscopy is also necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Ferroelectrics with tungsten-bronze structure Ferroelectricity has been observed in various kinds of organic and inorganic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Regarding inorganic ferroelectrics, the oxygen octahedra ferroelectrics are the most popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' One of this family is the ferroelectrics with tungsten-bronze structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' It is technologically important in the field of telecommuni- cations due to its superior electro-optical, photorefractive, and nonlinear optical properties such as second harmonic generation, and its resistance to optical damage is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The ferroelectricity was reported at first in lead metaniobate, PbNb2O6 with 𝑇C = 570℃ [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Its piezoelectric constant is the same order of magnitude as that of barium titanate, BaTiO3 with 𝑇C = 120℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The useful nonlinear coefficients and low optical damage were reported in barium sodium niobate, Ba2NaNb5O15 (BNN), with 𝑇C = 560℃ [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Nowadays, a lot of tungsten-bronze type ferroelectrics are known [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Figure 1 shows the projection of tungsten bronze structure on the 𝑐-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In BNN, the A1 and A2 sites are fully filled by Na and Ba, respectively, and there is no vacancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The B1 and B2 sites are fully occupied by Nb, while the C sites are vacant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' BNN is called a filled tungsten-bronze structure with no charge disorder at A1 and A2 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' BNN undergoes successive phase transitions at 560, 300, and −163℃ [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' (Colour online) Projection of tungsten bronze structure on the 𝑐-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Tetragonal and or- thorhombic unit cells are shown by dotted and solid lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' 43702-2 B 2 B C X区 Oxygen octahedron b 0 b T T p 0Ferroelectric instability of barium sodium niobate The higher one is associated with the ferroelectric Curie temperature at 560℃, and its crystal symmetry changes from prototypic 4/𝑚𝑚𝑚 to ferroelectric tetragonal 4𝑚𝑚 with a spontaneous polarization 𝑃3 along the 𝑐-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The symmetry changes from tetragonal to incommensurate (IC) orthorhombic 𝑚𝑚𝑚 systems at 300℃, the 𝑎 and 𝑏 axes are rotated for 45◦ along the 𝑐-axis as shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The modulation direction of the IC wave vectors is along 𝑎 and 𝑏 axes of the orthorhombic coordinate [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The lowest temperature phase transition at −163℃ is the reentrant ferroelastic phase transition into the tetragonal 4𝑚𝑚 phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The pressure induced reentrant ferroelastic phase transition was also observed at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='2 GPa and at room temperature [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Recently, a new type of the IC phase transition was proposed by Ishibashi [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' This new type of phase transition is referred to as type III, and it is characterized by the parabolic splitting of the doubly degenerate modes at the Brillouin zone boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The related macroscopic change in the IC phase transition was studied by Brillouin scattering [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Regarding a ferroelectric instability, the accurate measurement of low-frequency polaritons was performed on the optical phonon branch of A1(𝑧) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' However, any evidence of displacive nature was not found down to 18 cm−1 [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Since the 𝑇C of Ba2NaTa5O15 (BNT) is −233℃, the high tunability of the 𝑇C of the temperature width of about 800℃ was reported for Ba2NaNb5(1−𝑥)Ta5𝑥O15 (BNNT), and this is technologically important [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The ferroelectric phase transition of the BNNT single crystals with 𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='57 at 𝑇C = 115℃ was studied by broadband dielectric spectroscopy up to 4 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The order-disorder nature of the proper ferroelectric phase transition was observed, and its origin is attributed to the anharmonic motion of the Nb (Ta) atoms in a double well potential of oxygen octahedra [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Up to the present, no soft optic mode was observed in the tungsten-bronze ferroelectrics by vibrational spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Recent theoretical studies reported the local pseudo-Jahn–Teller effect (PJTE) in transition metal B ion center of ABO3 perovskite crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The vibronic coupling between the ground and excited electronic states of the local BO6 center results in dipolar distortions, leading to an eight-well adiabatic potential energy surface [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Such a situation may also occur in tungsten bronze ferroelectrics and the order-disorder nature of ferroelectricity may exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Therefore, the order-disorder nature of a ferroelectric phase transition of BNN has been examined by the broadband Brillouin scattering spectroscopy, which is a powerful tool to observe a critical slowing down in the vicinity of an order-disorder type phase transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In this paper, we review the broadband Brillouin scattering study on the ferroelectric instability of a BNN crystal [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Broadband Brillouin scattering and ferroelectric instability Vibrational spectroscopy i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=', infrared spectroscopy and Raman scattering observes the vibrational modes of atoms, molecules, and crystal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In the vibrational study of inorganic ferroelectric crystals, it is possible to observe not only the internal modes of octahedra or tetrahedra but also the external modes such as a soft optic mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The spectral resolution of vibrational spectroscopy is usually 1 cm−1 = 30 GHz or more, and it is sufficient to detect the change of mode frequency related to a phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The resolution of 1 cm−1 is sufficient to measure the temperature dependence of a ferroelectric soft optic mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' However, in the study of a ferroelectric phase transition of order-disorder type, the resolution of 1 cm−1 is not sufficient to measure the critical slowing down of the relaxation time towards the Curie temperature [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Polarization fluctuations related to a ferroelectric instability are detected as a broad CP in an inelastic scattering spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The colorless and transparent BNN crystal studied was grown by Czochralski method in Tamagawa factory, NEC, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The (100) plate with the size of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='65 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='25 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='68 mm with two optically polished surfaces was used for Brillouin scattering measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Brillouin scattering spectra were measured at the backward scattering geometry using a 3 + 3 tandem multi-pass Fabry– Perot interferometer and a conventional photon counting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' A single frequency green YAG laser (𝜆 = 532 nm) with power of 100 mW was used as an exciting source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The light spot size at a sample surface was about 10 𝜇m using the optical microscope (BX-60) [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The temperature of a sample was controlled by the heating stage of a T1500 (high T Linkam) from room temperature up to 750℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' All the Brillouin scattering spectra were measured in the condition that the free spectral range (FSR) and the scan range are 300 and 600 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' An intense polarized CP of BNN was observed in the 43702-3 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Kojima 500 400 300 200 100 0 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' unit) 400 200 0 200 400 Frequency shift (GHz) 600 oC BNN a-plate FSR = 300 GHz VV VH Paraelectric phase Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' (Colour online) Broadband VV and VH Brillouin scattering spectra of a BNN crystal at 600℃ observed by 𝑎(𝑐𝑐) ¯𝑎 and 𝑎(𝑐𝑏) ¯𝑎 back scattering geometry, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' vicinity of the Curie temperature, 𝑇C = 560℃, in the broadband Brillouin scattering spectra as shown in figure 2 [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In the polarized VV spectrum observed at 𝑎(𝑐𝑐) ¯𝑎, backward scattering geometry shows an intense broad CP with A1(𝑧) symmetry, while in the depolarized VH spectrum observed at 𝑎(𝑐𝑏) ¯𝑎, backward scattering geometry does not show an intense CP with B2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Therefore, the polarization fluctuations along a ferroelectric 𝑐-axis are the origin of an intense broad CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Critical slowing down on a ferroelectric phase transition of barium sodium niobate For the detailed analysis of the width of a CP, the temperature dependence of broadband Brillouin scattering spectra of a BNN crystal was measured at the backward scattering geometry with the free spectral range of 300 GHz as shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' 1000 800 600 400 200 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' unit) 400 200 0 200 400 Frequency shift (GHz) BNN a-plate FSR = 300 GHz 600 oC 580 oC 700 oC Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' (Colour online) Broadband VV Brillouin scattering spectra of a BNN crystal in a paraelectric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Under the assumption of a single Debye relaxation process, the relaxation time 𝜏CP was determined 43702-4 Ferroelectric instability of barium sodium niobate by the relation π × (CP width) = 𝜏−1 CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The relaxation process related to the order-disorder nature of a ferroelectric phase transition has been observed as a CP with a zero frequency shift in an inelastic scattering spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In the order-disorder phase transition, the relaxation time 𝜏 of the fluctuations of the order parameters increases toward the phase transition point and was called the critical slowing down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The relaxation time determined from the CP width shows a critical slowing down in the vicinity of 𝑇C = 560℃ as shown in figure 4 [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The temperature dependence of the relaxation time is given by the following equation of the case of 𝛽 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='0 in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='1) for a first order phase transition: 1 𝜏CP = 1 𝜏0 + 1 𝜏1 �𝑇 − 𝑇1 𝑇1 � , (𝑇 > 𝑇C > 𝑇1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='1) For example, in the ferroelectric phase transition at 𝑇C = 500 K of the relaxor ferroelectric 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='70Pb(Sc1/2Nb1/2)O3–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='30PbTiO3 with the perovskite structure, the values of the fitting parameters are 𝜏0 = 14 ps and 𝜏1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='47 ps, and 𝑇1 = 500 K [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The temperature dependences of 𝑇/𝐼CP of a BNN crystal are shown in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In BNN, the fitting parameters of 1/𝜏 are 𝜏0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='29 ps, 𝜏1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='73 ps, and 𝑇1 = 555℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The intensity of a CP 𝐼CP obeys the following equation in a paraelectric phase [36]: 𝑇 𝐼CP ∝ ������ ∞ ∫ 0 𝜒′′ (𝜔) 𝜔 d𝜔 ������ −1 ∝ 1 𝜒′(0) = 𝑇 − 𝑇1 𝐶 , (𝑇 > 𝑇C > 𝑇1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='2) Here, for the first order phase transition, 𝑇C >T1, because the ferroelectric phase transition of BNN is the first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In the ferroelectric phase transition at 𝑇C = 500 K of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='70Pb(Sc1/2Nb1/2)O3–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='30PbTiO3, the Curie–Weiss law also holds for 𝐼CP/𝑇 above 𝑇C [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='2x10 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='7 1/τCP (GHz) 700 600 500 400 Temperature (oC) BNN TC=560 oC Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' (Colour online) Temperature dependence of the inverse relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The dotted line is the fitted line by the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='1) above 𝑇C = 560℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The experimental results of the critical slowing down of relaxation time and the Curie–Weiss behavior of the CP intensity indicate the order-disorder nature of a ferroelectric phase transition of BNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In the study of the order-disorder phase transition, Brillouin scattering is a powerful tool to detect the critical slowing down [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' 43702-5 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Kojima 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='0x10 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='0 T/ICP 700 600 500 400 Temperature (oC) BNN TC=560 oC Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' (Colour online) Temperature dependence of temperature divided by the intensity of a central peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The dotted line is the fitted line by the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content='2) above 𝑇C = 560℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Conclusions For the study of the lattice instability of ferroelectrics, vibrational spectroscopy is a powerful tool to discuss not only displacive but also order-disorder nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' This paper reviews the experimental studies on the ferroelectric instability of a ferroelectric phase transition of barium sodium niobate (BNN) crystals with tungsten-bronze structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' BNN is one of well-known optical crystals for electro-optic and nonlinear optic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' It shows a uniaxial ferroelectricity with a spontaneous polarization along the tetragonal 𝑐-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' In the vicinity of the Curie temperature, 𝑇C = 560℃, an intense central peak (CP) was observed by the broadband Brillouin scattering experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The CP has a strong polarization dependence, which originates from the polarization fluctuations along the ferroelectric 𝑐-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The CP intensity shows a maximum at 𝑇C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The relaxation time determined by the CP width shows a critical slowing down towards 𝑇C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' The temperature dependence of the CP intensity shows the Curie–Weiss behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' These experimental results are the evidence of the order-disorder nature of the ferroelectric instability of BNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Acknowledgements Author thanks to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Grigas, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Banys, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Maczka for the collaboration and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Ohta, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Christy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Matsumoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Suzuki, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Aftebuzzamann for the discussion and experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Funding This research was funded in part by JSPS KAKENHI, Grant No.' metadata={'source': 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+page_content=' Kojima Дослiдження сегнетоелектричної нестiйкостi в нiобатi барiю-натрiю методами широкосмугового розсiювання Брiллюена С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Коджiма Вiддiлення матерiалознавчих наук, Унiверситет Цукуби, Цукуба, Iбаракi 305-8573, Японiя Нiобат барiю-натрiю (BNN) зi структурою вольфрамової бронзи є одним з добре вiдомих оптичних криста- лiв, якi використовуються для електрооптичних дослiджень та у нелiйнiйнiй оптицi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' У данiй роботi розгля- дається сегнетоелектрична нестiйкiсть в кристалах BNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' BNN є одновiсним сегнетоелектриком, в якому спонтанна поляризацiя напрямлена вздовж тетрагональної осi 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' У лiтературi немає згадок про спостере- ження оптичної м’якої моди, вiдповiдальної за сегнетоелектричний фазовий перехiд у цьому кристалi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' В околi температури Кюрi 𝑇C = 560°C в спектрах широкосмугового розсiювання Брiллюена спостерiгається iнтенсивний центральний пiк, пов’язаний з флуктуацiями поляризацiї удовж осi 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Час релаксацiї, який визначається шириною центрального пiка, виявляє критичне сповiльнення при наближеннi до 𝑇C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Цей факт свiдчить про те, що сегнетоелектрична нестiйкiсть у BNN-сполуках є типу “лад-безлад”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} +page_content=' Ключовi слова: розсiювання Брiллюена, сегнетоелектрик, лад-безлад, центральний пiк, нiобат барiю-натрiю 43702-8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQfYvyp/content/2301.01341v1.pdf'} diff --git a/YdAzT4oBgHgl3EQf1_7C/content/2301.01809v1.pdf b/YdAzT4oBgHgl3EQf1_7C/content/2301.01809v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e62079cf8c8c2a9af07274bdab9862e9ba4a0fe2 --- /dev/null +++ b/YdAzT4oBgHgl3EQf1_7C/content/2301.01809v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ef569cf5a24fb056a5cdedc0fb167c3eafff0d30b763439301df2a36f7ea71d5 +size 302974 diff --git a/YdAzT4oBgHgl3EQf1_7C/vector_store/index.faiss b/YdAzT4oBgHgl3EQf1_7C/vector_store/index.faiss new file mode 100644 index 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b/ZdE3T4oBgHgl3EQfcgrr/content/tmp_files/2301.04527v1.pdf.txt @@ -0,0 +1,2771 @@ +Fast and Reliable Jackknife and Bootstrap +Methods for Cluster-Robust Inference∗ +James G. MacKinnon† +Queen’s University +mackinno@queensu.ca +Morten Ørregaard Nielsen +Aarhus University +mon@econ.au.dk +Matthew D. Webb +Carleton University +matt.webb@carleton.ca +January 12, 2023 +Abstract +We provide computationally attractive methods to obtain jackknife-based cluster-robust +variance matrix estimators (CRVEs) for linear regression models estimated by least squares. +We also propose several new variants of the wild cluster bootstrap, which involve these +CRVEs, jackknife-based bootstrap data-generating processes, or both. Extensive simula- +tion experiments suggest that the new methods can provide much more reliable inferences +than existing ones in cases where the latter are not trustworthy, such as when the number +of clusters is small and/or cluster sizes vary substantially. Three empirical examples illus- +trate the new methods. +Keywords: clustered data, grouped data, cluster-robust variance estimator, CRVE, clus- +ter sizes, wild cluster bootstrap +JEL Codes: C10, C12, C21, C23. +∗We are grateful to David Drukker, Alexander Fischer, David Roodman, the Co-Editor, Francis Vella, an +anonymous referee, and seminar participants at Aarhus University, Carleton University, University of Toronto, +and New York Camp Econometrics 2022 for helpful comments and suggestions. MacKinnon and Webb thank the +Social Sciences and Humanities Research Council of Canada (SSHRC grants 435-2016-0871 and 435-2021-0396) +for financial support. Nielsen thanks the Danish National Research Foundation for financial support (DNRF +Chair grant number DNRF154). +†Corresponding author. +Address: Department of Economics, 94 University Avenue, Queen’s University, +Kingston, Ontario K7L 3N6, Canada. Email: mackinno@queensu.ca. Tel. 613-533-2293. Fax 613-533-6668. +1 +arXiv:2301.04527v1 [econ.EM] 11 Jan 2023 + +1 +Introduction +In applications of linear regression models to many fields of economics and other disciplines, it is +common to divide the sample into disjoint clusters and employ a cluster-robust variance matrix +estimator (or CRVE) for inference. These estimators are based on the assumption that the +disturbances of the regression model are uncorrelated across clusters, but they allow for arbitrary +patterns of dependence and heteroskedasticity within each cluster. The literature on cluster- +robust inference has grown rapidly in recent years. Cameron and Miller (2015) is a classic survey +article. Conley, Gonçalves and Hansen (2018) surveys a broader class of methods for dependent +data. MacKinnon, Nielsen and Webb (2022a) provides a guide that explores the implications of +key theoretical results for empirical practice, with an emphasis on bootstrap methods. +There are several CRVEs for ordinary least squares (OLS) estimates of linear regression mod- +els; see Section 2. However, mainly for computational reasons, almost all empirical work to date +has made use of the simplest one, usually known as CV1, which is the default in Stata. Cluster- +robust tests and confidence intervals based on CV1 may or may not yield reliable inferences. +Whether they do so depends primarily on the number of clusters G and how homogeneous these +are. When all clusters are roughly equal in size and approximately balanced, asymptotic infer- +ence based on CV1 seems to be fairly reliable whenever G is at least moderately large (say 50 or +more). However, even when G is very large, cluster-robust t-tests and Wald tests are at risk of +severe over-rejection, and cluster-robust confidence intervals are at risk of severe under-coverage +in at least two situations. The first is when one or a few clusters are much larger than the rest, +and the second is when the only “treated” observations belong to just a few clusters; Djogbe- +nou, MacKinnon and Nielsen (2019) discusses the first case, and MacKinnon and Webb (2017, +2018) discuss the second. +Alternatives to CV1 have been known since Bell and McCaffrey (2002), but computational +difficulties have kept them from widespread use. The first contribution of this paper, which +is discussed in Section 3, is to provide a fast method for computing jackknife-based CRVEs, +of which the simplest is generally known as CV3. By explicitly using the cluster jackknife for +computation, our method makes it feasible to employ CV3 for inference even in very large samples +with very large clusters. +Because CV3 standard errors used to be hard to compute, there has been very little work +comparing the finite-sample performance of t-tests based on CV3 with those of similar procedures +based on CV1; a partial exception is Niccodemi and Wansbeek (2022). The second contribution +of this paper is to compare the finite-sample properties of these tests, and also ones based on +CV2, by simulation; see Section 6. In concurrent work that cites our simulations, Hansen (2022) +provides important theoretical results which suggest that asymptotic inference based on CV3 is +generally more reliable, and more conservative, than asymptotic inference based on CV1. +Existing bootstrap methods for cluster-robust inference are all based on CV1. +The best +2 + +known of these (and until now the best performing one) seems to be the wild cluster restricted +(or WCR) bootstrap proposed in Cameron, Gelbach and Miller (2008). There is also a closely +related procedure called the wild cluster unrestricted (or WCU) bootstrap, which generally does +not work quite as well. The asymptotic validity of these procedures is proved in Djogbenou et al. +(2019), which also analyzes their higher-order asymptotic properties. Until a few years ago, the +WCR and WCU bootstraps were computationally expensive for large samples, but that is no +longer the case. Roodman, MacKinnon, Nielsen and Webb (2019) describes a remarkably ef- +ficient implementation in the Stata package boottest, and MacKinnon (2022) discusses other +methods for fast computation. The boottest routines are now available as a Julia package +which can be also be called from R, Python, and Stata. The package fwildclusterboot im- +plements the boottest method natively in R (Fischer and Roodman, 2022). +The third contribution of this paper is to propose several new variants of the wild cluster +bootstrap. One modification simply replaces CV1 by CV3. The other, which requires some new +results, involves modifying the bootstrap data-generating process, or DGP. Modern treatments +of the wild cluster bootstrap, such as MacKinnon et al. (2022a), express the bootstrap DGP +as a function of the empirical scores. We show how to make the bootstrap DGP more closely +resemble the (unknown) true DGP by transforming the residuals before forming the scores. The +transformation we propose is based on the jackknife. Accordingly, it does not actually require +any calculations that explicitly involve residuals. This makes it very fast when the number of +clusters is small relative to the sample size, even when the latter is extremely large. +The next section establishes notation and briefly reviews the literature on asymptotic cluster- +robust inference for the linear regression model. Section 3 then provides a new computational +method for CV3, which is conceptually simple and extremely fast in many cases, as we demon- +strate in Section 4. Next, Section 5 discusses several ways of modifying the wild cluster boot- +strap. Simulation results in Section 6 suggest that our new versions of the WCR and WCU +bootstraps perform better, sometimes very much better, than the original ones. This is partic- +ularly true when cluster sizes vary greatly. One modified version of the WCR bootstrap that +uses transformed scores seems to work especially well in most settings. Section 7 presents three +empirical examples in which our methods are likely to be more reliable than existing ones. Sec- +tion 8 concludes with a brief discussion of the methods that we recommend in practice. +2 +The Linear Regression Model with Clustering +Consider the linear regression model yi = x⊤ +i β+ui. If we divide the data into G disjoint clusters, +where the allocation of observations to clusters is assumed to be known, this can be written as +yg = Xgβ + ug, +g = 1, . . . , G. +(1) +3 + +The g th cluster has Ng observations, and the total sample size is N = �G +g=1 Ng. In (1), Xg is an +Ng × k matrix of regressors, β is a k-vector of coefficients, yg is an Ng-vector of observations on +the regressand, and ug is an Ng-vector of disturbances (or error terms). Stacking the yg yields +the N-vector y, stacking the Xg yields the N × k matrix X, and stacking the ug yields the N- +vector u, so that (1) can be rewritten as y = Xβ + u. +The OLS estimator of β is +ˆβ = (X⊤X)−1X⊤y = β0 + (X⊤X)−1X⊤u, +(2) +where the second equality depends on the assumption that the data are actually generated by +(1) with true value β0. Thus, if sg = X⊤ +g ug is the score vector for the g th cluster, +ˆβ − β0 = (X⊤X)−1 +G +� +g=1 +X⊤ +g ug = +� G +� +g=1 +X⊤ +g Xg +�−1 +G +� +g=1 +sg. +(3) +Obtaining valid inferences evidently requires assumptions about the score vectors. For a correctly +specified model, E(sg) = 0 for all g. We further assume that +E(sgs⊤ +g ) = Σg +and +E(sgs⊤ +g′) = 0, +g, g′ = 1, . . . , G, +g′ ̸= g, +(4) +where Σg is the symmetric, positive semidefinite variance matrix of the scores for the g th cluster. +The second assumption in (4) is crucial. It states that the scores for every cluster are uncorrelated +with the scores for every other cluster. +From the rightmost expression in (3), we see that the distribution of ˆβ depends on the +disturbance subvectors ug only through the distribution of the score vectors sg. +It follows +immediately that an estimator of Var( ˆβ) should be based on the usual sandwich formula, +(X⊤X)−1 +� G +� +g=1 +Σg +� +(X⊤X)−1. +(5) +Every CRVE replaces the Σg in (5) by functions of the Xg and the residual subvectors ˆug. There +is more than one way to do this. Since Σg is the expectation of sgs⊤ +g , the simplest approach is +just to replace it by ˆsg ˆs⊤ +g , where ˆsg = X⊤ +g ˆug is the empirical score vector for the g th cluster. If +in addition we multiply by a correction for degrees of freedom, we obtain +CV1: +ˆV1( ˆβ) = +G(N − 1) +(G − 1)(N − k)(X⊤X)−1 +� G +� +g=1 +ˆsg ˆs⊤ +g +� +(X⊤X)−1. +(6) +This is by far the most widely-used CRVE in practice, and it is the default in Stata. The leading +scalar is chosen so that, when G = N, ˆV1( ˆβ) reduces to the familiar HC1 estimator (MacKinnon +and White, 1985) that is robust only to heteroskedasticity of unknown form. +Inference about β is typically based on cluster-robust t-statistics and Wald statistics. If βj +denotes the j th element of β and β0j is its value under the null hypothesis, then the appropriate +4 + +t-statistic is +tj = +ˆβj − β0j +se1(ˆβj) +, +(7) +where ˆβj is the OLS estimate, and se1(ˆβj) is the square root of the j th diagonal element of ˆV1( ˆβ). +Under extremely strong assumptions (Bester, Conley and Hansen, 2011), it can be shown that tj +asymptotically follows the t(G − 1) distribution. Conventional “asymptotic” inference is based +on this distribution. +We should expect inferences based on CV1 to be reliable if the sum of the sg, suitably +normalized, is well approximated by a multivariate normal distribution with mean zero, and if +the sg are well approximated by the ˆsg. But asymptotic inference can be misleading when either +or both of these approximations is poor; see Djogbenou et al. (2019) and MacKinnon et al. +(2022a). Whether or not the first approximation is a good one depends on the model and the +data, and there is not much the investigator can do about it. But the second approximation +can, in principle, be improved by using modified empirical score vectors instead of the ˆsg. +Two CRVEs based on this idea, usually known as CV2 and CV3, were proposed (under +different names) in Bell and McCaffrey (2002). These are the cluster analogs of the hetero- +skedasticity-consistent variance matrix estimators HC2 and HC3 proposed in MacKinnon and +White (1985). All of these estimators are designed to compensate, in different ways, for the +shrinkage and intra-cluster correlation of the residuals induced by least squares. +The CV2 variance matrix is +CV2: +ˆV2( ˆβ) = (X⊤X)−1 +� +G +� +g=1 +`sg `s⊤ +g +� +(X⊤X)−1, +(8) +where the modified score vectors `sg are defined as +`sg = X⊤ +g M −1/2 +gg +ˆug. +(9) +Here Mgg = INg − Xg(X⊤X)−1X⊤ +g is the g th diagonal block of the projection matrix MX, +which satisfies ˆu = MXu, and M −1/2 +gg +is the symmetric square root of its inverse. The CV2 +estimator has been recommended in Imbens and Kolesár (2016) and Pustejovsky and Tipton +(2018). +Following Bell and McCaffrey (2002), these papers provide methods for computing +critical values based on t and F distributions with computed degrees of freedom. +The CV3 variance matrix is very similar to CV2, but, as we explain in Section 3, it is based +on the jackknife. The usual definition is +CV3: +ˆV3( ˆβ) = G − 1 +G +(X⊤X)−1 +� +G +� +g=1 +´sg ´s⊤ +g +� +(X⊤X)−1, +(10) +5 + +where now the modified score vectors ´sg are defined as +´sg = X⊤ +g M −1 +gg ˆug. +(11) +The rescaling factor (G − 1)/G in (10) is the analog of the factor (N − 1)/N that occurs in +jackknife variance matrix estimators at the individual level. This factor implicitly assumes that +all clusters are the same size and perfectly balanced, with disturbances that are independent +and homoskedastic; an alternative is proposed in Niccodemi and Wansbeek (2022). +Although (8) and (10) look simple enough, computing either CV2 or CV3 has until recently +been extremely expensive, or even computationally infeasible, when any of the Ng are large. The +problem is that, before computing (11), we apparently need to rescale the residual vector ˆug for +each cluster. This involves storing and inverting the Ng ×Ng matrix Mgg. Before computing (9), +we also need to compute the symmetric square roots of the Mgg, and this requires calculating +their eigenvalues and eigenvectors. Of course, when all clusters are very small, this is not difficult. +When G = N, CV2 reduces to HC2, and CV3 reduces to HC3, both of which can be computed +very quickly. +Niccodemi et al. (2020) has recently proposed a method that is much faster for large clusters. +Versions of this method apply to both CV2 and CV3. Instead of rescaling the residual vectors, +it calculates the score vectors `sg or ´sg directly using equations that do not involve any Ng × Ng +matrices. A revised version of this method, which appears to be new, works as follows. First, +form the k × k matrices +Ag = (X⊤X)−1/2X⊤ +g Xg(X⊤X)−1/2, +g = 1, . . . , G. +(12) +Then, for (8), calculate the rescaled score vectors +`sg = (X⊤X)1/2(Ik − Ag)−1/2(X⊤X)−1/2ˆsg, +g = 1, . . . , G, +(13) +and, for (10), calculate the rescaled score vectors +´sg = (X⊤X)1/2(Ik − Ag)−1(X⊤X)−1/2ˆsg, +g = 1, . . . , G. +(14) +These rescaled score vectors are used in (8) and (10) as before. Unless all the clusters are very +small, computing CV2 and CV3 using (13) and (14) is much faster than computing them using +(9) and (11); see Section 4. +In the case of CV3, however, an even faster and more intuitive method is available. This +jackknife-based method, which we discuss in the next section, can be extremely fast when N is +large and G is much smaller than N, so that at least some clusters are large; see Section 4. +6 + +3 +Jackknife Variance Matrix Estimators +The jackknife is a simple method for reducing bias and estimating standard errors by omitting +observations sequentially. +Tukey (1958) suggested using the jackknife to estimate standard +errors, and Miller (1974) is a classic reference. The key idea of the cluster jackknife is to compute +G sets of parameter estimates, each of which omits one cluster at a time. In this section, we use +it to compute two closely related CRVEs in an efficient fashion. +The OLS estimates of β when each cluster is omitted in turn are +ˆβ(g) = (X⊤X − X⊤ +g Xg)−1(X⊤y − X⊤ +g yg), +g = 1, . . . , G. +(15) +It is easy to obtain the ˆβ(g) in a computationally efficient manner. We start by calculating the +cluster-level matrices and vectors +X⊤ +g Xg +and +X⊤ +g yg, +g = 1, . . . , G. +(16) +Unless G is very large, this involves very little cost beyond that of computing ˆβ, because we +can use the quantities in (16) to construct X⊤X and X⊤y and then use (2) to obtain ˆβ. For +typical values of k, it should then be reasonably inexpensive to calculate ˆβ(g) for every cluster +using (15). The main cost, beyond that of computing ˆβ, is that we need to calculate the inverse +of a k × k matrix for each of the ˆβ(g). +The cluster jackknife estimator of Var( ˆβ) is the cluster analog of the usual jackknife variance +matrix estimator given in Efron (1981), among others. It is defined as +CV3J: +ˆV3J( ˆβ) = G − 1 +G +G +� +g=1 +( ˆβ(g) − ¯β)( ˆβ(g) − ¯β)⊤, +(17) +where ¯β = G−1 �G +g=1 ˆβ(g) is the sample average of the ˆβ(g). Notice that (17) calculates the +variance matrix around ¯β. Centering around ¯β is common in jackknife variance estimation, but +it is also common to center around ˆβ, as in Bell and McCaffrey (2002). +There is a very close relationship between ˆV3J( ˆβ) and ˆV3( ˆβ). In fact, +ˆV3( ˆβ) = G − 1 +G +G +� +g=1 +( ˆβ(g) − ˆβ)( ˆβ(g) − ˆβ)⊤, +(18) +which is just (17) with ¯β replaced by ˆβ. This follows from (10) and (11) because +(X⊤X)−1´sg = (X⊤X)−1X⊤ +g M −1 +gg ˆug = ˆβ − ˆβ(g). +(19) +Note that the summation in (18) is unchanged if ˆβ(g) − ˆβ is replaced by ˆβ − ˆβ(g). +Although the second equality in (19) is not new, it will turn out to be very useful in Section 5, +7 + +and so we now prove it. The middle expression in (19) can be written as +(X⊤X)−1X⊤ +g M −1 +gg yg − (X⊤X)−1X⊤ +g M −1 +gg Xg(X⊤X)−1X⊤y. +(20) +Using the updating formula +(X⊤X − X⊤ +g Xg)−1 = (X⊤X)−1 + (X⊤X)−1X⊤ +g M −1 +gg Xg(X⊤X)−1, +(21) +ˆβ(g) can be written as the sum of four terms, the first of which is just ˆβ. Thus the right-hand +side of (19) can be written as +(X⊤X)−1X⊤ +g M −1 +gg Xg(X⊤X)−1X⊤ +g yg + (X⊤X)−1X⊤ +g yg +− (X⊤X)−1X⊤ +g M −1 +gg Xg(X⊤X)−1X⊤y. +(22) +The last term in (22) is identical to the last term in (20). The first two terms in (22) can be +rewritten as +(X⊤X)−1X⊤ +g M −1 +gg Pggyg + (X⊤X)−1X⊤ +g yg, +where Pgg = Xg(X⊤X)−1X⊤ +g is the g th diagonal block of the matrix PX = I − MX, so that +Pgg = I − Mgg. Inserting this straightforwardly yields the result that +(X⊤X)−1X⊤ +g M −1 +gg Pggyg + (X⊤X)−1X⊤ +g yg += (X⊤X)−1X⊤ +g M −1 +gg (I − Mgg)yg + (X⊤X)−1X⊤ +g yg = (X⊤X)−1X⊤ +g M −1 +gg yg. +(23) +The right-hand side of (23) is the first term in (20), which proves the second equality in (19). +When Ng = 1 for all g, ˆV3J( ˆβ) is numerically equal to the original HC3 estimator proposed in +MacKinnon and White (1985). The modern version of HC3, which uses ˆβ instead of ¯β and omits +the factor of N/(N − 1), is due to Davidson and MacKinnon (1993, Chapter 16). +Both cluster jackknife estimators may be used to compute cluster-robust t-statistics. Since +there are G terms in the summation, it is natural to compare these with quantiles of the t(G−1) +distribution, as usual. These procedures should almost always be more conservative than t-tests +based on CV1 (Hansen, 2022). We expect CV3 and CV3J to be very similar in most cases. This +issue will be investigated in Section 6.1, where we conclude that it is reasonable to focus on CV3. +Both CV3 and CV3J have been available in Stata for some years by using the options +“vce(jackknife,mse)” and “vce(jackknife)”, respectively. However, the implementations +discussed here are much more efficient when G is not very small, and are available in Stata and +R packages, both named summclust; see MacKinnon, Nielsen and Webb (2022c) and Fischer +(2022). Both packages also calculate a number of summary statistics that may be used to assess +the reliability of cluster-robust inference as described in MacKinnon, Nielsen and Webb (2022b). +There may be cases in which the matrix X⊤X − X⊤ +g Xg in (15) is singular for one or more +values of g. If so, then at least some elements of ˆβ(g) cannot be identified. This can happen in +otherwise well-specified models when there are cluster-level fixed effects, and in that case the +8 + +solution is simply to partial them out before running the regression. In other cases where a +singularity occurs, there are two possible courses of action. The first is to modify (17) and (18) +so that the summation is taken only over values of g for which ˆβ(g) can be estimated, and G is +replaced by the number of clusters for which that is the case (this is the approach followed in +the native Stata implementations; see also Section 7.3). When there are only a few problematic +clusters, this approach may be attractive. But since ˆβ and ¯β would then be based on different +samples, it seems likely that CV3J and CV3 may differ more than they would usually do, which +suggests that it may be safer to use the former. +The second course of action is to replace the inverse in (15) by a generalized inverse. In +practice, this means that coefficients that cannot be identified are replaced by zeros. When the +elements of ˆβ(g) that are of primary interest can always be identified, this approach may be +attractive, especially when there are many problematic clusters, as in the example of Section 7.3. +4 +Speed of Computation +The CV3 estimator can be challenging to compute. Following Bell and McCaffrey (2002), it is +natural to employ what we call the “residual method” based on (10) and (11). To compute the +modified score vector ´sg for the g th cluster, this method uses the Ng-vector of residuals ˆug and +the Ng × Ng matrix M −1 +gg . Unless every Ng is small, storing and inverting the Mgg matrices +is computationally expensive. Indeed, for even moderately large values of the Ng, this can be +effectively impossible. +A much faster method, recently proposed in Niccodemi et al. (2020) and revised modestly +in Section 2, uses (14) to obtain the modified score vectors ´sg. Since it operates directly on the +score vectors ˆsg, we call it the “score method.” An even faster approach, discussed in Section 3, +computes the ˆβ(g) using (15) and then calculates their variance matrix as (18). For obvious +reasons, we refer to this as the “jackknife method.” +To compare timings for the residual, score, and jackknife methods, we generate two datasets +with N = 1, 048, 576 = 220 observations. In one case, there are 20 regressors, and in the other +case there are 40. The observations are divided into G equal-sized clusters, where G varies from +16 to 512K and K denotes 1024 = 210. Thus the cluster size M = N/G varies from 2 to 64K. +Figure 1 shows the time in seconds, on a log2 scale, for each of the three methods and the two +datasets as a function of cluster sizes M = N/G, which vary from 2 to 64K. These times include +the time required to compute the OLS estimates. For both the jackknife and score methods, +there is considerable overlap between the computations needed for the OLS estimates and the +ones needed for CV3. Thus, for large clusters, the cost of computing the OLS estimates and CV3 +together using one or both of these methods was sometimes less than the cost of computing the +OLS estimates alone. This is probably because of cache congestion, which seems to be alleviated +by forming X⊤X on a cluster-by-cluster basis. For large clusters, the speed of all methods could +9 + +Figure 1: Timings for three ways to compute CV3 +2 +4 +8 +16 +32 +64 +128 256 512 +1K +2K +4K +8K 16K 32K 64K +1/32 +1/8 +1/2 +2 +8 +32 +128 +512 +2K +8K +32K +................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. +....................................................................................... +Residual method +k = 20 +...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +....................................................................................... +k = 40 +....................................................................................................................................... +................ +Score method +k = 20 +........................................................................................................................................ +................ +k = 40 +................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +...................................................................................................................... +Jackknife method +k = 20 +................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +...................................................................................................................... +k = 40 +M +Time in Seconds +Notes: The sample size is N = 1, 048, 576 = 1024K, where K = 1024 = 210. The number of clusters varies from +16 to 512K. All clusters have M = N/G observations, so that cluster sizes vary from 2 to 64K. The number +of regressors k is either 20 or 40. Times required to compute ˆβ are included; see text. All computations were +performed in Fortran using one core of an Intel i9-13900K processor. +almost certainly be increased by using a fast BLAS implementation. However, in the interest of +programming ease, we have not done this. The jackknife and score methods are already very fast. +In Figure 1, the residual method works well for very small values of M. It is always the +fastest method for M ≤ 4. We did not perform any timings for M = 1, where CV3 reduces to +HC3, because we would have needed a different program that eliminated the loops within each +cluster to obtain optimal results. But the residual method is certainly the fastest one for this +case. However, its cost rises very rapidly as M increases. Results for this method are only shown +for M ≤ 4096, because using it for larger values would have been prohibitively costly. For the +largest values of M, the cost of the residual method is almost the same for k = 20 and k = 40, +because it is dominated by the computations needed to form and invert the Mgg matrices. +In contrast, both the score and jackknife methods become faster as M increases and G +consequently decreases, except that, when k = 40, they are both a bit slower for M = 64K +than for M = 32K. This probably occurs because of cache congestion. The jackknife method is +always quicker than the score method. For small values of M, it seems to be faster by a factor +of about 12 when k = 20 and by a factor of about 26 when k = 40. However, the advantage of +the jackknife method gradually diminishes as M increases. When M = 64K, so that there are +10 + +only 16 clusters, the jackknife method is only slightly faster. +It is easy to see that the jackknife method will have a big advantage over the residual method +whenever cluster sizes vary much, even if most of them are very small. Imagine a sample with, +say, 1000 equal-sized clusters and M = 5. For such a sample, the residual and jackknife methods +will perform about the same. Suppose we then merge 100 of the tiny clusters into one large +cluster with 500 observations. Doing this will reduce the cost of the jackknife method slightly, +but it will greatly increase the cost of the residual method. Indeed, when there is even a single +very large cluster, the latter inevitably becomes extremely slow. +Based on these results, the jackknife method for computing CV3 is clearly the procedure of +choice unless all clusters are tiny (say, Ng ≤ 5 for all g). For datasets with large clusters, an +efficient implementation of this method (such as the one provided by the summclust package +mentioned in Section 3), can compute both the OLS estimates and the CV3 variance matrix in +roughly the same amount of time as a reasonably fast program for the OLS estimates alone. +5 +New Versions of the Wild Cluster Bootstrap +The existing WCR bootstrap is based on CV1 standard errors and the restricted empirical score +vectors defined in (25) below. Henceforth, we will refer to this as the classic WCR bootstrap, +or WCR-C. It often works well, but not always. We therefore propose three new versions of +the WCR bootstrap, along with three corresponding versions of the WCU bootstrap. These are +based on two distinct modifications. One involves replacing CV1 by CV3. The other involves +modifying the scores used in the bootstrap DGP, in the hope that the modified bootstrap DGP +will provide a better approximation to the unknown process that actually generated the data. +We first discuss the bootstrap DGPs for the classic wild cluster bootstraps, WCR-C and +WCU-C, expressing them in terms of scores instead of observations. This approach is intuitive +and computationally attractive (Roodman et al., 2019; MacKinnon, 2022). In terms of the G +score vectors, a generic wild cluster bootstrap DGP is +s∗b +g = v∗b +g ¨sg, +g = 1, . . . , G, +b = 1, . . . , B, +(24) +where b indexes bootstrap samples, v∗b +g is a random variate with mean 0 and variance 1, and +the ¨sg are empirical score vectors to be discussed below. In most cases, it seems to be best +to generate the v∗b +g using the Rademacher distribution, which takes the values 1 and −1 with +equal probabilities (Davidson and Flachaire, 2008; Djogbenou et al., 2019). However, since the +number of possible Rademacher bootstrap samples that are distinct from the original sample +is only 2G − 1, it is better to use a distribution with more mass points, such as the six-point +distribution proposed in Webb (2022), when G is less than about 12. +The vector ¨sg in (24) is an empirical score vector for the g th cluster. +For the WCU-C +bootstrap, it is simply the unrestricted empirical score vector ˆsg = X⊤ +g ˆug. For the WCR-C +11 + +bootstrap, it is the restricted empirical score vector ˜sg defined as +˜sg = X⊤ +g yg − X⊤ +g Xg ˜β, +g = 1, . . . , G, +(25) +where ˜β is the vector of OLS estimates under the null hypothesis. Like ˆβ, ˜sg is a k-vector, even +though some elements of ˜β may equal zero or satisfy other linear restrictions. The bootstrap +DGP (24) looks very much like the one for the wild score cluster bootstrap for nonlinear models +proposed in Kline and Santos (2012). In the context of (1), however, it is just a different way of +writing the bootstrap DGP for the wild cluster bootstrap. +In order to calculate a bootstrap P value or a bootstrap confidence interval, we need to +compute B bootstrap test statistics indexed by b. These depend only on the bootstrap scores +in (24) and the matrix (X⊤X)−1. For each bootstrap sample, we use s∗b +g to obtain a bootstrap +estimate, not of β itself, but of the vector δ = β − ¨β, where ¨β = ˜β for the WCR-C bootstrap +and ¨β = ˆβ for the WCU-C bootstrap. This estimate is simply +ˆδ∗b = (X⊤X)−1 +G +� +g=1 +s∗b +g = (X⊤X)−1s∗b, +(26) +where s∗b = �G +g=1 s∗b +g . When v∗b +g = 1 for all g, the bootstrap sample is the same as the original +sample. In this very special case, ˆδ∗b = 0 for the WCU-C bootstrap, and ˆδ∗b = ˆβ − ˜β for the +WCR-C bootstrap. +If we are testing the hypothesis that βj = 0, where βj is an element of β, then we just need +to multiply the j th row of (X⊤X)−1 by s∗b in order to obtain ˆδ∗b +j , the j th element of δ∗b. The +bootstrap t-statistic is then equal to +t∗b +j = +ˆδ∗b +j +se(ˆδ∗b +j ) +, +(27) +where se(·) denotes the standard error formula used to obtain tj, the original t-statistic. We +automatically get the correct numerator, which is ˆβ∗b +j +for the WCR-C bootstrap, since ¨β = ˜β, +and ˆβ∗b +j − ˆβj for the WCU-C bootstrap, since ¨β = ˆβ. As usual, a symmetric bootstrap P value +is then given by +P ∗ +S(tj) = 1 +B +� +I +� +|t∗b +j | > |tj| +� +, +(28) +where I(·) denotes the indicator function. The bootstrap P value in (28) is simply the fraction of +the bootstrap samples for which |t∗b +j | is more extreme than |tj|. The value of B should be chosen +so that α(B + 1) is an integer, where α is the level of the test (Racine and MacKinnon, 2007). +It is common to use B = 999, but B = 9,999 and (when feasible) B = 99,999 are better choices. +In the classic versions of the wild cluster bootstrap, the standard error formula in (27) is +se1(·), the square root of the j th diagonal element of CV1. But the results in Section 3 make +it equally feasible to use standard errors based on CV3, even in large samples. This gives us +new versions of both the WCR and WCU bootstraps, which we will refer to as WCR-V and +12 + +WCU-V, because only the variance matrices have changed. The bootstrap standard errors can +be calculated without computing an entire variance matrix for each bootstrap sample. +For +example, the CV3 standard error of ˆδ∗b +j is just +se3(ˆδ∗b +j ) = +� +�G − 1 +G +G +� +g=1 +�ˆδ∗b +j(g) − ˆδ∗b +j +�2 +� +� +1/2 +, +(29) +where ˆδ∗b +j(g) is the j th element of the vector +ˆδ∗b +(g) = (X⊤X − X⊤ +g Xg)−1(s∗b − s∗b +g ). +(30) +Only ˆδ∗b +j +and the ˆδ∗b +j(g) need to be computed for each bootstrap sample. In (26) and (30), the +first terms are invariant across bootstrap samples and only need to be computed once. +We now have two versions of the WCR bootstrap, WCR-C and WCR-V, and two versions of +the WCU bootstrap, WCU-C and WCR-V. The two WCR bootstraps use the bootstrap DGP +(24) with ¨sg = ˜sg, and the two WCU bootstraps use the bootstrap DGP (24) with ¨sg = ˆsg. The +“C” and “V” versions calculate both the actual and bootstrap test statistics using se1(·) and +se3(·), respectively. These bootstrap methods use the restricted or unrestricted empirical scores +in their raw form. But empirical scores differ from true scores, because residuals differ from +disturbances. It therefore seems attractive to replace the empirical score vectors by modified +score vectors that implicitly rescale the residuals on a cluster-by-cluster basis. This is analogous +to methods discussed in Davidson and Flachaire (2008) and MacKinnon (2013) for the ordinary +wild bootstrap. However, quite a lot more algebra is needed. +We first consider the WCU bootstrap, since this case is slightly easier to deal with. +In +principle, we could simply replace the vectors ¨sg in (24) with the modified empirical score vectors +´sg defined in (11). However, using (11) is expensive, or even computationally infeasible, for large +clusters. But the result (19) lets us compute ´sg very rapidly as +´sg = X⊤X +� ˆβ − ˆβ(g)� +, +g = 1, . . . , G. +(31) +For large clusters, using (14) to compute the ´sg is much faster than using (11), but using (31) +is faster still; see Section 4. This yields two new bootstrap methods, which we will refer to as +WCU-S and WCU-B, respectively. The WCU-S bootstrap (S for score) employs the modified +score vectors ´sg instead of ˆsg, but it uses the familiar se1(·) standard error. +The WCU-B +bootstrap (B for both) employs both the modified score vectors and the se3(·) standard error. +Finding the analogous versions of the WCR bootstrap takes a bit more work. We need to +specify a restricted wild bootstrap DGP based on modified score vectors. Suppose the restrictions +have the usual linear form, Rβ = r, for a given matrix R and a given vector r. We can write +this equivalently in terms of free parameters, φ, as β = Hφ + h for a given matrix H and a +13 + +given vector h. Then the modified score vectors are +˙sg = X⊤ +g +˜ +M −1 +gg (yg − Xg ˜β), +(32) +which are the analogs of the ´sg from (11). Here ˜ +Mgg is the g th diagonal block of the projection +matrix ˜ +M = I− ˜ +X( ˜ +X⊤ ˜ +X)−1 ˜ +X⊤, where ˜ +X = XH. However, evaluating (32) is computationally +infeasible when the clusters are not all small. We need to replace (32) by something that is +feasible for any sample size. +The first step is to compute the restricted estimates ˜β = H ˜φ + h. Here ˜y = y − Xh and +˜φ = ( ˜ +X⊤ ˜ +X)−1 ˜ +X⊤ ˜y. The corresponding estimates when each cluster is omitted in turn are +˜β(g) = H ˜φ(g) + h, where +˜φ(g) = ( ˜ +X⊤ ˜ +X − ˜ +X⊤ +g ˜ +Xg)−1( ˜ +X⊤ ˜y − ˜ +X⊤ +g ˜yg), +g = 1, . . . , G. +(33) +Then it can be shown that +˙sg = X⊤ +g ˜yg − X⊤ +g ˜ +Xg ˜φ(g), +g = 1, . . . , G. +(34) +To see that (32) and (34) are equal, note that the right-hand side of (34) is +X⊤ +g +� +˜yg − ˜ +Xg( ˜ +X⊤ ˜ +X − ˜ +X⊤ +g ˜ +Xg)−1( ˜ +X⊤ ˜y − ˜ +X⊤ +g ˜yg) +� += X⊤ +g +� +˜yg − ˜ +Xg +� +( ˜ +X⊤ ˜ +X)−1 + ( ˜ +X⊤ ˜ +X)−1 ˜ +X⊤ +g +˜ +M −1 +gg ˜ +Xg( ˜ +X⊤ ˜ +X)−1� +( ˜ +X⊤ ˜y − ˜ +X⊤ +g ˜yg) +� +, +where the equality uses the updating formula (21) applied to ˜ +X, ˜ +Xg, and +˜ +M −1 +gg . Then we use +the fact that ˜φ = ( ˜ +X⊤ ˜ +X)−1 ˜ +X⊤ ˜y together with the relation ˜ +Xg( ˜ +X⊤ ˜ +X)−1 ˜ +X⊤ +g = ˜Pgg = I − ˜ +Mgg +to rewrite the last expression as +X⊤ +g +� +˜yg − ˜ +Xg ˜φ − (I − ˜ +Mgg) ˜ +M −1 +gg ˜ +Xg ˜φ + (I − ˜ +Mgg)˜yg + (I − ˜ +Mgg) ˜ +M −1 +gg (I − ˜ +Mgg)˜yg +� += X⊤ +g +˜ +M −1 +gg (˜yg − ˜ +Xg ˜φ). +(35) +Replacing ˜yg by yg − Xgh and ˜ +Xg by XgH, and using the fact that H ˜φ = ˜β − h, the right- +hand side of (35) equals (32). +An important special case is the restriction that βk = 0. This is obtained by setting R = +(0, . . . , 0, 1) and r = 0, or, equivalently, H = (Ik−1, 0)⊤ and h = 0. In this case, we find that +˜ +X = X1, which contains the first k − 1 columns of X, and ˜φ = ˜β1 = (X⊤ +1 X1)−1X⊤ +1 y. The +corresponding estimates when each cluster is omitted in turn are +˜β(g) +1 += (X⊤ +1X1 − X⊤ +1gX1g)−1(X⊤ +1 y − X⊤ +1gyg), +g = 1, . . . , G, +(36) +where X1g contains the first k − 1 columns of Xg. Then (34) reduces to +˙sg = X⊤ +g yg − X⊤ +g X1g ˜β(g) +1 , +g = 1, . . . , G. +(37) +14 + +Exactly the same arguments that led to (34) can be applied to the modified unrestricted +empirical scores, giving us +´sg = X⊤ +g yg − X⊤ +g Xg ˆβ(g), +g = 1, . . . , G. +(38) +Either (31) or (38) can be used to compute the ´sg, and both are computationally attractive. +However, in situations where both ˙sg and ´sg need to be computed, (38) may offer some pro- +gramming advantages relative to (31) due to its similarity to (34). +It may seem puzzling that the scalar factors in (6) and (10) do not appear in the bootstrap +DGPs that correspond to them. The reason is that rescaling all the bootstrap scores by the same +factor has no impact on the resulting bootstrap t-statistics. From (26) and (30), it is easy to see +that multiplying all the s∗b +g by a scalar C simply makes ˆδ∗b and all the ˆδ∗b +(g) larger by a factor +of C. But this also makes the empirical scores for every bootstrap sample larger by the same +factor. Therefore, from (6), (8), and (10), the variance matrices become larger by a factor of C 2 +and the standard errors by a factor of C. The factors of C in the numerator and denominator +of t∗b +j cancel out, leaving the bootstrap t-statistics unchanged. +However, if we chose not to studentize the test statistic, it would make sense to multiply +the right-hand side of (24) by the square root of G(N − 1)/((G − 1)(N − k)) for methods that +use CV1 and by the square root of (G − 1)/G for methods that use CV3. Doing this should +improve the correspondence between the bootstrap DGP and the unknown process that actually +generated the data. An unstudentized test statistic for βj = 0 is just ˆβj, and its bootstrap analog +would be ˆδ∗b +j , which equals ˆβ∗b +j +for WCR and ˆβ∗b +j − ˆβj for WCU. The usual theory of higher- +order refinements for the bootstrap suggests that it is generally better to studentize (Hall, 1992). +However, there may be cases in which unstudentized test statistics are of interest (Canay, Santos +and Shaikh, 2021). Nevertheless, since we have eight studentized bootstrap methods to study, +we do not consider unstudentized ones further. +To generate the transformed scores needed for the WCR/WCU-S and WCR/WCU-B boot- +straps, (31) and (38) must be used for all G clusters. In the event that ˆβ(g) and ˜β(g) cannot be +calculated for, say, cluster h, we have two choices. The simplest is to replace the inverses in (15) +and (36) by generalized inverses. Alternatively, we could use ˆsh instead of ´sh and ˜sh instead of +˙sh, along with the transformed scores for the remaining clusters. The latter would be appro- +priate if we have chosen to omit the problematic clusters when computing the cluster-jackknife +variance matrix; see the discussion at the end of Section 3. +Table 1 provides a convenient summary of the eight wild cluster bootstrap methods that +we have discussed. Conceptually, they differ along two dimensions. The horizontal dimension +represents the way in which the standard errors for both the actual and bootstrap test statistics +are calculated. The vertical dimension represents the score vectors used in the four versions of +the bootstrap DGP (24). Note that the boottest and fwildclusterboot packages now provide +15 + +Table 1: Eight versions of the wild cluster bootstrap +Standard errors based on +Scores in bootstrap DGP (24) +CV1 +CV3 +Null hypothesis imposed +˜sg defined in (25) +WCR-C +WCR-V +˙sg defined in (37) +WCR-S +WCR-B +Null hypothesis not imposed +ˆsg = X⊤ +g ˆug +WCU-C +WCU-V +´sg defined in (31) or (38) +WCU-S +WCU-B +Notes: WCR-C and WCU-C are the classic versions of the wild cluster restricted and wild +cluster unrestricted bootstraps. WCR-S and WCU-S employ transformed scores with the +usual CV1 variance matrix. WCR-V and WCU-V employ the usual scores with the CV3 +variance matrix. WCR-B and WCU-B employ both transformed scores and CV3. +fast implementations of the WCR/WCU-S bootstraps as well as the classic ones. This is possible +because, in contrast to the WCR/WCU-V and WCR/WCU-B bootstraps, the former do not +involve any jackknife calculations for the bootstrap samples. Once the transformed scores have +been computed, the fast bootstrap algorithm proposed in Roodman et al. (2019) applies directly +to the WCR/WCU-S bootstraps. +It seems highly likely that all the methods discussed in this section are asymptotically valid, in +the sense that, under suitable regularity conditions, the rejection frequencies for any test converge +to the nominal level of the test as G → ∞. Formal proofs could be obtained by modifying the +arguments in Djogbenou et al. (2019). For the WCU bootstrap methods, the key fact is that +the modified empirical score vectors ´sg computed using (31) or (38) are asymptotically equal to +the ordinary empirical score vectors ˆsg. For the WCR bootstrap methods, the key fact is that +the modified restricted empirical score vectors ˙sg defined in (34) are asymptotically equal to the +ordinary restricted empirical score vectors ˜sg in (25). +6 +Monte Carlo Simulations +Previous simulation results in MacKinnon and Webb (2017, 2018), Brewer et al. (2018), Djog- +benou et al. (2019), MacKinnon (2022), and several other papers have shown that the reliability +of both bootstrap and asymptotic methods for cluster-robust inference depends heavily on the +number of clusters, the extent to which cluster sizes vary, and (in the case of treatment effects) +both the number of treated clusters and their sizes. Many of our experiments therefore focus on +these features. +The model we consider is +ygi = β1 + +k +� +j=2 +βjXjgi + ugi, +g = 1, . . . , G, +i = 1, . . . , Ng, +(39) +16 + +where the ugi are generated by a normal random-effects model with intra-cluster correlation ρ. +The way in which the k −1 non-constant regressors are generated varies across the experiments. +The hypothesis to be tested is that βk = 0. +In most of our experiments, there are N = 400G observations, which are divided among the +G clusters using the formula +Ng = +� +N +exp(γg/G) +�G +j=1 exp(γj/G) +� +, +g = 1, . . . , G − 1, +(40) +where [x] means the integer part of x. The value of NG is then set to N − �G−1 +g=1 Ng. The key +parameter here is γ, which determines how uneven the cluster sizes are. When γ = 0 and N/G +is an integer, (40) implies that Ng = N/G for all g. For γ > 0, cluster sizes vary more and +more as γ increases. The largest value of γ that we use is 4. In that case, when G = 24 and +N = 9600, the largest cluster (1513 observations) is about 47 times as large as the smallest +cluster (32 observations). In contrast, when γ = 2, the largest cluster (899 observations) is just +under seven times as large as the smallest (130 observations). +The sample sizes that we employ are unusually large for experiments of this type. Since +cluster-robust inference is often used with samples that have hundreds of thousands or even +millions of observations, we want our results to apply to such cases. In preliminary experiments, +we found that the results tended to change slightly, but systematically, as small values of N/G +were increased. Results for N/G > 400 are very similar to ones for N/G = 400, so we use 400 +in all the experiments based on (40). Because the bootstrap samples are generated using scores, +the cost of the experiments increases much less than proportionally with N/G. +All experiments use 400,000 replications. This number is so large that experimental ran- +domness is negligible. The most important determinant of computational cost is k, the number +of regressors. As can be seen from (24) and (34) or (38), generating each bootstrap sample in- +volves O(k2G) operations. So does calculating the test statistics using either CV1 or CV3. Thus +the experiments can be somewhat costly when k is large. Nevertheless, many of our experiments +involve k ≥ 10. We do this because results in MacKinnon (2022) suggest that the performance +of many methods of inference deteriorates as k increases. Previous Monte Carlo experiments, +which often use k ≤ 3, may therefore have tended to give too optimistic a picture. +It might seem that substantial savings could be achieved by partialing out all regressors +except the one(s) of interest prior to performing the bootstrap. However, this trick only works +in certain special cases. For methods based on the jackknife, it is easy to see the problem. If we +were to partial out some of the regressors prior to computing the delete-one-cluster estimates in +(15), then the computed ˆβ(g) would depend on the values of the partialed-out regressors for the +full sample, including those in the g th cluster which was supposed to be deleted. Consequently, +the values of the delete-one-cluster estimates would be incorrect if we partialed out any regressor +17 + +that affects more than one cluster (such as industry-level fixed effects with firm-level clustering). +An important exception is when the regressors that are partialed out are cluster fixed effects +or fixed effects at a finer level (such as firm-level fixed effects with industry-level clustering), +because each of them affects only some or all of the observations within a single cluster. In fact, +it is essential to partial out fixed effects of this type, because the coefficient on any regressor that +is non-zero only for the g th cluster cannot be identified from a sample which omits that cluster. +6.1 +Test Size +The experiments in this subsection deal with rejection frequencies under the null hypothesis. We +consider both asymptotic tests based on the t(G−1) distribution and the wild cluster bootstrap +tests listed in Table 1. +Figure 2 focuses on variation in cluster sizes. In these experiments, there are always 9600 +observations, 24 clusters, and 10 regressors. Cluster sizes vary according to (40). Regressors 2 +through k − 1 in (39) are normally distributed according to a random-effects model that yields +intra-cluster correlations of 0.50. The test regressor either follows the same normal distribution +as the others (in the three panels on the left), or a χ2(1) distribution (in the three panels on +the right). In the latter case, it is obtained by squaring a normally distributed random variable +that is generated by the same random-effects model as the other regressors. The disturbances +are also generated by such a model, but with ρ = 0.10. We focus on rejection frequencies for a +test that βk = 0 at the 5% level. +The results for asymptotic tests, based on the t(23) distribution and shown in Panels (a) +and (b), are striking. Note that a square-root transformation has been applied to the vertical axis +to prevent these panels from being too tall. Tests based on CV1 over-reject substantially. The +extent of the over-rejection increases with γ, and, except for γ = 4, it is more severe in Panel (b) +than in Panel (a). A regressor that follows the χ2(1) distribution necessarily has some extreme +values. These become points of high leverage, which makes inference more difficult in Panel (b). +Although tests based on CV2 always reject considerably less often than ones based on CV1, +they also over-reject significantly and to an extent that increases with γ. In contrast, tests +based on CV3 and CV3J either under-reject slightly all the time, in Panel (a), or under-reject +very slightly for larger values of γ, in Panel (b). The results for CV3 and CV3J are extremely +similar. The latter always rejects more often than the former, because the difference between +(17) and (18) is the positive semi-definite matrix ((G − 1)/G)( ˆβ − ¯β)( ˆβ − ¯β)⊤. Since CV3 tends +to under-reject slightly in Figure 2, it might seem that CV3J is to be preferred. However, as we +shall see, there are also many cases in which CV3 over-rejects, and CV3J therefore over-rejects +slightly more. In practice, it would be perfectly reasonable to report either CV3 or CV3J. We +never encountered a case in which it made any real difference. +The results for the WCR bootstrap tests, shown in Panels (c) and (d), are surprising. In +18 + +Figure 2: Rejection frequencies as a function of γ +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +............... +0.20 +............... +0.15 +............... +0.10 +............... +0.07 +............... +0.05 +............... +0.04 +........................................................................................................................................................................................................................................................................................................................................................... +...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +CV1 +............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +CV2 +......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +CV3 +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +CV3J +γ +Rej. Freq. +(a) N(0, 1) Regressor, t(23) Distribution +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +............... +0.20 +............... +0.15 +............... +0.10 +............... +0.07 +............... +0.05 +............... +0.04 +........................................................................................................................................................................................................................................................................................................................................................... +............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +CV1 +.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. +CV2 +.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +CV3 +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +CV3J +γ +Rej. Freq. +(b) χ2(1) Regressor, t(23) Distribution +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +........................................................................................................................................................................................................................................................................................................................................................... +............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +..............................................................................................WCR-C +..................................................................... +.............. WCR-B +......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +......................................................................................................................WCR-V +............................................................................................................................................................................................................................................................................................................. +............................................................... WCR-S +γ +Rej. Freq. +(c) N(0, 1) Regressor, WCR Bootstraps +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +........................................................................................................................................................................................................................................................................................................................................................... +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +..............................................................................................WCR-C +..................................................................... +.............. WCR-B +......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +......................................................................................................................WCR-V +............................................................................................................................................................................................................................................................................................................. +............................................................... WCR-S +γ +Rej. Freq. +(d) χ2(1) Regressor, WCR Bootstraps +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +0.11 +0.12 +0.13 +........................................................................................................................................................................................................................................................................................................................................................... +.......................................................................... +.............. WCU-C +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +..............................................................................................WCU-B +......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +......................................................................................................................WCU-V +............................................................................................................................................................................................................................................................................................................. +............................................................... WCU-S +γ +Rej. Freq. +(e) N(0, 1) Regressor, WCU Bootstraps +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +0.11 +0.12 +0.13 +........................................................................................................................................................................................................................................................................................................................................................... +.............................................................................. +.............. WCU-C +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +..............................................................................................WCU-B +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +......................................................................................................................WCU-V +............................................................................................................................................................................................................................................................................................................. +............................................................... WCU-S +γ +Rej. Freq. +(f) χ2(1) Regressor, WCU Bootstraps +Notes: The vertical axes show rejection frequencies for tests of βk = 0 in (39) at the .05 level. Results are +based on 400,000 replications, with B = 399 bootstrap samples. There are 24 clusters, 9600 observations, and +10 regressors, with ρ = 0.10. The extent to which cluster sizes vary increases with γ; see (40). +the past, WCR-C has been the only variant of the WCR bootstrap, and numerous Monte Carlo +experiments have suggested that it is the procedure of choice. But WCR-B performs notably +better than WCR-C for every value of γ, and both WCR-V and WCR-S perform better still. +Note that, although these two procedures perform almost the same here, this is not true in +general. Oddly, all the WCR procedures perform better in Panel (d), where the test regressor is +highly skewed, than they do in Panel (c), where it is Gaussian. The rather mediocre performance +of WCR-C must be due, at least in part, to the fact that k = 10, which is a larger number than +19 + +Figure 3: Rejection frequencies as a function of k +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +.............. +0.20 +.............. +0.15 +.............. +0.10 +.............. +0.07 +.............. +0.05 +.............. +0.04 +.............................................................................................................................................................................................................................................................................................................................................................................. +............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. +CV1 +...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +CV2 +.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +CV3 +....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +CV3J +k +Rej. Freq. +(a) N(0, 1) Regressor, t(23) Distribution +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +.............. +0.20 +.............. +0.15 +.............. +0.10 +.............. +0.07 +.............. +0.05 +.............. +0.04 +.............................................................................................................................................................................................................................................................................................................................................................................. +.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +CV1 +................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +CV2 +..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +CV3 +...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +CV3J +k +Rej. Freq. +(b) χ2(1) Regressor, t(23) Distribution +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +.............................................................................................................................................................................................................................................................................................................................................................................. +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +........................................................................................ WCR-C +.............................................................................. +............. WCR-B +.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. +.............................................................................................................. WCR-V +.................................................................................................................................................................................................................................................................................................................................................. +......................................................... WCR-S +k +Rej. Freq. +(c) N(0, 1) Regressor, WCR Bootstraps +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +.............................................................................................................................................................................................................................................................................................................................................................................. +.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +........................................................................................ WCR-C +............................................................................. +............. WCR-B +.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +.............................................................................................................. WCR-V +................................................................................................................................................................................................................................................................................................................................................. +......................................................... WCR-S +k +Rej. Freq. +(d) χ2(1) Regressor, WCR Bootstraps +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +0.11 +0.12 +.............................................................................................................................................................................................................................................................................................................................................................................. +................................................................................ +............. WCU-C +................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. +........................................................................................ WCU-B +.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. +.............................................................................................................. WCU-V +................................................................................................................................................................................................................................................................................................................................................. +......................................................... WCU-S +k +Rej. Freq. +(e) N(0, 1) Regressor, WCU Bootstraps +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +0.11 +0.12 +.............................................................................................................................................................................................................................................................................................................................................................................. +................................................................................... +............. WCU-C +................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. +........................................................................................ WCU-B +.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. +.............................................................................................................. WCU-V +................................................................................................................................................................................................................................................................................................................................................. +......................................................... WCU-S +k +Rej. Freq. +(f) χ2(1) Regressor, WCU Bootstraps +Notes: The vertical axes show rejection frequencies for tests of βk = 0 in (39) at the .05 level. Results are based +on 400,000 replications, with γ = 2, ρ = 0.10, and B = 399 bootstrap samples. There are 24 clusters, 9600 +observations, and k regressors, where k varies from 2 to 20 by 2. +has been used in most previous experiments; see Figure 3 below. +Some of the results for the WCU bootstrap tests, shown in Panels (e) and (f), are also sur- +prising. It is not a surprise that WCU-C rejects more often than WCR-C or that its perfor- +mance is much worse in Panel (f) than in Panel (e). However, the fact that the other three +WCU procedures over-reject much less often than WCU-C may well be surprising. In both pan- +els, WCU-B is clearly the procedure of choice. WCU-V and WCU-S perform much better than +20 + +WCU-C, but worse than WCU-B. In Panels (c) and (d), the differences between WCU-V and +WCU-S are small, but larger than the differences between WCR-V and WCR-S. +Figure 3 is similar to Figure 2, but the number of regressors k is now on the horizontal axis, +and γ = 2. In Panels (a) and (b), CV1 over-rejects to an increasing extent as k increases. So +does CV2, although it always over-rejects considerably less than CV1. In contrast, CV3 and CV3J +over-reject modestly for small values of k and under-reject modestly for large ones. +Panels (c) and (d) look a lot like the same panels in Figure 2, even though what is on the +horizontal axis is different. WCR-C performs quite well for very small values of k, but it over- +rejects more and more severely as k increases. WCR-B performs much better than WCR-C, +but WCR-V and WCR-S perform even better. In Panel (d), where the test regressor is highly +skewed, they both perform extremely well for all values of k. +Panels (e) and (f) also look a lot like the same panels in Figure 2. WCU-C performs quite +poorly, over-rejecting more and more severely as k increases. In contrast, WCU-B performs quite +well in Panel (e) and fairly well in Panel (f), and there is no tendency for its performance to +deteriorate as k increases. As before, the two other bootstrap methods generally perform much +better than WCU-C but slightly worse than WCU-B. +In the next set of experiments, we focus on what happens as G increases. Figure 4 shows +rejection frequencies as functions of G, which varies from 12 to 84 by 6, and implicitly also N, +since N = 400G. In these experiments, γ = 2 and k = 10. We report results for only five +methods, instead of twelve. We omit CV1 and CV2, because they never perform very well, and +CV3J because it is almost identical to CV3. Among the restricted bootstrap methods, we report +WCR-C, because it was until now the procedure of choice. We also report WCR-S and WCR-B, +but we do not report WCR-V, because it yields results nearly identical to those of WCR-S and +is harder to compute. Among the unrestricted bootstrap methods, we report only WCU-B, +because it always seems to outperform the other WCU methods. +In Panel (a), using CV3 with the t(G−1) distribution under-rejects quite noticeably for very +small values of G, but it performs extremely well for G ≥ 30. The bootstrap methods always +over-reject, with WCR-C always the worst of them. For G ≥ 42, however, all the bootstrap +methods perform very well, with WCR-S the winner by a tiny margin. +Panel (b) is more interesting than Panel (a). The extreme skewness of the χ2(1) regressor +apparently affects the results quite a bit, even when G = 84. Although it under-rejects for small +values of G, using CV3 with the t(G−1) distribution over-rejects for larger values, where it is the +worst method. We note that G = 24 in Figures 2 and 3 is near where the curve for CV3 crosses +the .05 line in Figure 4. The best method is WCR-S in every case. It performs remarkably well +for G ≥ 30. However, all three WCR methods perform well for the larger values of G. The only +bootstrap method that does not perform particularly well for these values is WCU-B. By most +standards, of course, every method shown in Panel (b) of Figure 4 works very well, unless G is +21 + +Figure 4: Rejection frequencies as a function of G +12 +18 +24 +30 +36 +42 +48 +54 +60 +66 +72 +78 +84 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +..................................................................................................................................................................................................................................................................................................................................................................... +...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +.............................................................................................................................. +CV3 +......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +................................................................................................... +WCR-C +.................................................................................. +............... +WCR-B +......................................................................................................................................................................................................................................................................................................................................................... +................................................................. +WCR-S +............................................................................ +............... +WCU-B +G +Rej. Freq. +(a) N(0, 1) Regressor +12 +18 +24 +30 +36 +42 +48 +54 +60 +66 +72 +78 +84 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +..................................................................................................................................................................................................................................................................................................................................................................... +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +.............................................................................................................................. +CV3 +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +................................................................................................... +WCR-C +............................................................................... +............... +WCR-B +............................................................................................................................................................................................................................................................................................................................................ +................................................................. +WCR-S +............................................................................. +............... +WCU-B +G +Rej. Freq. +(b) χ2(1) Regressor +Notes: The vertical axes show rejection frequencies for tests of βk = 0 in (39) at the .05 level. Results are based +on 400,000 replications, with γ = 2, k = 10, ρ = 0.10, and B = 399 bootstrap samples. There are between 12 +and 84 clusters, all multiples of 6, with 400 observations per cluster on average. +less than about 30. For G = 84, CV3 is the worst method, but even it rejects only 5.49% of the +time. For comparison, CV1 rejects 9.04% of the time, and CV2 rejects 7.15%. The best method, +WCR-S, rejects 4.97% of the time. +Many applications of cluster-robust inference involve treatment at the cluster level, and +existing methods generally perform very poorly when either the number of treated clusters or +the number of control clusters is small. Using CV1 with the t(G − 1) distribution or WCU-C +leads to severe over-rejection, and using WCR-C leads to severe under-rejection (MacKinnon +and Webb, 2017, 2018). Our next set of experiments therefore focuses on the model +ygi = β1 + Zgiβ2 + βkxg + ugi, +(41) +where xg is a treatment dummy, Zgi is a row vector of other regressors, and ugi is generated by +a random-effects model with intra-cluster correlation ρ. The treatment dummy equals 1 for G1 +of the G clusters and 0 for the remaining G0 = G − G1. The clusters that are treated are chosen +at random. The Zgi consist of eight more dummy variables. For each of these variables and each +cluster, a probability πg between 0.25 and 0.75 is chosen at random for each replication. Then +each observation for that variable in that cluster equals 1 with probability πg and 0 otherwise. +Thus all the regressors are dummies, which vary at the individual level in a way that varies +across clusters. +Figure 5 shows rejection frequencies based on the t(G − 1) distribution for six cases. In the +left-hand column, there are 12 clusters and 4800 observations. In the right-hand column, there +are 24 clusters and 9600 observations. The value of γ is 0 in the top row, 2 in the middle row, +and 4 in the bottom row. The number of treated observations G1 varies between 2 and G−2 on +22 + +Figure 5: Rejection frequencies based on t(G − 1) distribution for treatment regression +2 +3 +4 +5 +6 +7 +8 +9 +10 +................. +0.40 +................. +0.30 +................. +0.20 +................. +0.15 +................. +0.10 +................. +0.05 +................. +0.03 +..................................................................................................................................................................................................................................................................................................................... +....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +...............................................................................CV1 +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +...............................................................................CV2 +.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +...............................................................................CV3 +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +................................................................CV3J +G1 +Rej. freq. +(a) G = 12, γ = 0 +2 +3 +4 +5 +6 +7 +8 +9 +10 +................. +0.40 +................. +0.30 +................. +0.20 +................. +0.15 +................. +0.10 +................. +0.05 +................. +0.03 +..................................................................................................................................................................................................................................................................................................................... +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +................................................................................................................................................................................................................................................................................................................................................................................................................................................ +G1 +Rej. freq. +(c) G = 12, γ = 2 +2 +3 +4 +5 +6 +7 +8 +9 +10 +................. +0.40 +................. +0.30 +................. +0.20 +................. +0.15 +................. +0.10 +................. +0.05 +................. +0.03 +..................................................................................................................................................................................................................................................................................................................... +...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +........................................................................................................................................................................................................................................................................................................................................................................................................................... +G1 +Rej. freq. +(e) G = 12, γ = 4 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +................. +0.40 +................. +0.30 +................. +0.20 +................. +0.15 +................. +0.10 +................. +0.05 +................. +0.03 +...................................................................................................................................................................................................................................................................................................................................................... +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +............................................................................................. CV1 +................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +............................................................................................. CV2 +.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. +............................................................................................. CV3 +...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +......................................................................... CV3J +G1 +Rej. freq. +(b) G = 24, γ = 0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +................. +0.40 +................. +0.30 +................. +0.20 +................. +0.15 +................. +0.10 +................. +0.05 +................. +0.03 +...................................................................................................................................................................................................................................................................................................................................................... +................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. +............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +G1 +Rej. freq. +(d) G = 24, γ = 2 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +................. +0.40 +................. +0.30 +................. +0.20 +................. +0.15 +................. +0.10 +................. +0.05 +................. +0.03 +...................................................................................................................................................................................................................................................................................................................................................... +.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. +............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. +................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +G1 +Rej. freq. +(f) G = 24, γ = 4 +Notes: The vertical axes, which have been subjected to a square-root transformation, show rejection frequencies +for tests of βk = 0 in (41) at the .05 level. The horizontal axes show G1, the number of treated clusters. Results +are based on 400,000 replications, with k = 10 regressors and ρ = 0.10. There are either 12 or 24 clusters, with +400 observations per cluster on average. Treated clusters are chosen at random. +the horizontal axes. It would have been impossible to set G1 = 1 or G1 = G − 1, because CV2, +CV3, and CV3J cannot be computed in those cases. For the jackknife-based estimators, this is +obvious from (15). When there is just one treated cluster, and it happens to be the omitted one, +then the coefficient of interest in ˆβ(g) is not identified. +As previous work has shown, tests that use CV1 tend to over-reject severely when either +23 + +Figure 6: Bootstrap rejection frequencies for treatment regression +2 +3 +4 +5 +6 +7 +8 +9 +10 +0.01 +0.03 +0.05 +0.07 +0.09 +..................................................................................................................................................................................................................................................................................................................... +................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +................................................................WCR-C +........................................................................................... +..........WCR-B +....................................................................... +..........WCU-B +........................................................................................................................................................................................................................................................................................................................................................................................................................................ +...............................................WCR-S +G1 +Rej. Freq. +(a) G = 12, γ = 0 +2 +3 +4 +5 +6 +7 +8 +9 +10 +0.01 +0.03 +0.05 +0.07 +0.09 +..................................................................................................................................................................................................................................................................................................................... +................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +........................................................................... +................................................................. +................................................................................................................................................................................................................................................................................................................................................................................. +G1 +Rej. Freq. +(c) G = 12, γ = 2 +2 +3 +4 +5 +6 +7 +8 +9 +10 +0.01 +0.03 +0.05 +0.07 +0.09 +..................................................................................................................................................................................................................................................................................................................... +...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +..................................................................... +................................................................ +........................................................................................................................................................................................................................................................................................................................................................ +G1 +Rej. Freq. +(e) G = 12, γ = 4 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +0.01 +0.03 +0.05 +0.07 +0.09 +...................................................................................................................................................................................................................................................................................................................................................... +.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +......................................................................... WCR-C +................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +....................................................... WCR-S +..................................................................................................................... +........... WCR-B +............................................................................................... +........... WCU-B +G1 +Rej. Freq. +(b) G = 24, γ = 0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +0.01 +0.03 +0.05 +0.07 +0.09 +...................................................................................................................................................................................................................................................................................................................................................... +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +.................................................................................................. +.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +......................................................................................... +G1 +Rej. Freq. +(d) G = 24, γ = 2 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +0.01 +0.03 +0.05 +0.07 +0.09 +...................................................................................................................................................................................................................................................................................................................................................... +................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +........................................................................................ +................................................................................................................................................................................................................................................................................................................................................................................................................................................... +..................................................................................... +G1 +Rej. Freq. +(f) G = 24, γ = 4 +Notes: The vertical axes show rejection frequencies for tests of βk = 0 in (41) at the .05 level. The horizontal +axes show G1, the number of treated clusters. Results are based on 400,000 replications, with k = 10, ρ = 0.10, +and B = 399 bootstrap samples. There are either 12 or 24 clusters, with 400 observations per cluster on average. +G0 or G1 is small. This is evident in Figure 5. The over-rejection is worst in Panel (f), where +both γ and G are largest. CV2 over-rejects less than CV1, but it still does not work very well, +except perhaps for values of G1 near G/2 when γ = 0; see Panels (a) and (b). In contrast, CV3 +and CV3J, which perform almost identically, are much less prone to over-reject than the other +two CRVEs. They actually under-reject for values of G1 fairly near G/2 when γ = 0, and they +perform very well for values of G1 near G/2 when γ = 2. Oddly, CV3 and CV3J over-reject less +seriously for extreme values of G1 when γ is large than when γ is small. +24 + +Figure 6 shows results for four bootstrap tests for the same set of experiments as in Figure 5. +When γ = 0, all three variants of the WCR bootstrap perform almost identically. However, as +γ increases, their performance starts to differ. WCR-S seems to reject least frequently, which +is a good thing for intermediate values of G1 and a bad thing for extreme values. In contrast, +WCR-B under-rejects least severely for extreme values of G1. However, for intermediate values, +it over-rejects less than WCR-C but more than WCR-S. +The most surprising results in Figure 6 are the ones for the unrestricted wild bootstraps. +We do not report results for WCU-C or WCU-S, because they would have required a much +longer vertical axis. WCU-C rejects almost 28% of the time in its worst case (G = 24, G1 = 2, +γ = 4), and WCU-S rejects over 12% of the time in its worst case (G = 24, G1 = 2, γ = 0). In +contrast, WCU-B is arguably the best method overall when G = 12, and it performs very well +for intermediate values of G1 when G = 24. In addition, it never over-rejects as severely as CV3 +for extreme values of G1. +Simulations in Djogbenou et al. (2019) suggest that many methods work poorly when one +cluster is much bigger than the others. Even when γ = 4, the largest cluster in our experiments is +never dramatically larger than all the rest, although this happens quite often in empirical work. +For instance, more than half of all incorporations in the United States occur in Delaware (Hu +and Spamann, 2020). This implies that studies of the effects of corporate governance based on +changes in state laws, where standard errors are clustered by state of incorporation, are likely to +encounter severe errors of inference. To investigate this phenomenon, we create artificial samples +with 50 clusters based on data for incorporations by year and state from Spamann and Wilkinson +(2019). There are 205,566 observations, of which 108,538, or 52.80%, are for Delaware. The +second-largest cluster is Nevada, with 17,010 or 8.27%, and the smallest is Montana, with 101 +or 0.05%. +We perform a set of experiments similar to the ones in Figures 5 and 6 using these artificial +samples. There are 10 regressors, generated in the same way as before, with one exception. +Because investigators are surely aware of whether or not the largest cluster (Delaware) is treated, +it is always treated in our experiments. The other clusters to be treated (between 1 and 47 +of them) are chosen at random. Because the largest cluster is always treated, the rejection +frequencies are no longer the same for G1 and G − G1 treated clusters. However, since this is +a pure treatment model, the results for G1 treated clusters that include Delaware must be the +same as the results for G − G1 treated clusters that exclude Delaware. +The results in Figure 7 are striking. In Panel (a), using either CV1 or CV2 leads to over- +rejection that varies between severe and extreme. +Using CV3 and CV3J also leads to over- +rejection, but it is much less severe. For between 20 and 41 treated clusters, rejection frequencies +are less than 0.07. In Panel (b), WCU-C over-rejects severely, and WCR-C can either over- +reject or under-reject, often severely. In contrast, our new bootstrap methods work remarkably +25 + +Figure 7: Rejection frequencies when a treated cluster is very large +3 +7 +11 +15 +19 +23 +27 +31 +35 +39 +43 +47 +............... +0.70 +............... +0.60 +............... +0.50 +............... +0.40 +............... +0.30 +............... +0.20 +............... +0.15 +............... +0.10 +............... +0.05 +............... +0.03 +............... +0.01 +............... +0.00 +................................................................................................................................................................................................................................................................................................................................................................................. +............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +CV1 +.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. +CV2 +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +CV3 +................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +CV3J +G1 +(a) Rejection Frequencies Based on t(49) Distribution +3 +7 +11 +15 +19 +23 +27 +31 +35 +39 +43 +47 +............... +0.70 +............... +0.60 +............... +0.50 +............... +0.40 +............... +0.30 +............... +0.20 +............... +0.15 +............... +0.10 +............... +0.05 +............... +0.03 +............... +0.01 +............... +0.00 +................................................................................................................................................................................................................................................................................................................................................................................. +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +WCR-C +....................................................................................................... +WCU-C +..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +..............................................................................................WCR-B +.................................................................................. +.............. WCU-B +................................................................................................................................................................................................................................................................................................................................................................................................................................ +...................................................................WCR-S +G1 +(b) Wild Bootstrap Rejection Frequencies +Notes: The vertical axes show rejection frequencies for tests of βk = 0 in (41) at the .05 level. Results are based +on 400,000 replications, with k = 10, ρ = 0.10, and B = 399. There are 205,566 observations and 50 clusters, +with cluster sizes proportional to incorporations in U.S. states. The largest cluster is always treated, and the +other clusters are treated at random. The number of treated clusters varies from 2 to 14 by 1, from 16 to 36 by +2, and then from 38 to 48 by 1. +well. The best of them is WCU-B, which always rejects less than 9% of the time and sometimes +rejects just about 5% of the time. WCR-S and WCR-B also perform much better than WCR-C, +except when G1 is very large, in which case they under-reject severely. +Even though it is based on real data, the distribution of cluster sizes in the experiments +reported in Figure 7 is very extreme. Although the performance of CV3 and three of our new +bootstrap methods is far from perfect, it is generally very much better than that of existing +methods. Thus it appears that jackknife-based methods are remarkably robust to heterogeneity +in cluster sizes. +6.2 +Test Power +It is natural to worry that a new test may be less powerful than existing tests, especially when +it performs much better under the null hypothesis. In this section, we therefore investigate test +power. Studying power is tricky, because it is unreasonable to compare tests that have noticeably +different rejection frequencies under the null. If, for example, an asymptotic test rejects 15% of +the time under the null and a bootstrap test based on it rejects 6% of the time, then we would +expect the asymptotic test to have substantially more power than the bootstrap test. But the +additional power may be entirely spurious, simply reflecting the finite-sample over-rejection by +the asymptotic test. +One way to compare tests with different rejection frequencies under the null is to “size-adjust” +them. But this approach has two serious conceptual difficulties. First, size-adjusted tests are +infeasible. What do we learn by comparing tests that cannot actually be performed? Second, +26 + +there are often many ways to size-adjust a given test, and they may yield quite different results. +The idea of size-adjustment is to base rejection frequencies for tests under the alternative on +critical values calculated by simulation under the null. But, in general, there exists an infinite +number of DGPs that satisfy the null hypothesis. If they all yield the same critical values, then +there is no problem. But if they yield different critical values, as will often be the case, then we +have to choose which null DGP to use. It seems natural to make the null DGP used for critical +values as close as possible to the alternative DGP. Davidson and MacKinnon (2006) suggests +a particular way of doing this, based on the Kullback-Leibler information criterion, but this +approach means using a different critical value for each set of values of the parameters under test. +To avoid the difficulties just discussed, we focus on four cases where the tests of interest +all perform quite well under the null. They are treatment experiments similar to the ones in +Figures 5 and 6, with G = 24, N = 9600, and k = 5. In Panels (a) and (b), G1 = 12, so that +precisely half the clusters are treated. In Panels (c) and (d), G1 = 6, so that the effects of having +few treated clusters are apparent but not severe. In order to avoid excessive power loss, we use +B = 999 for the bootstrap tests. We use k = 5 instead of k = 10 partly to reduce computational +cost and partly to improve test performance under the null. +Figure 8 shows rejection frequencies as a function of βk, the actual coefficient on the treatment +dummy in (41), when the null hypothesis is that βk = 0. In Panels (a) and (c), γ = 0, so that +every cluster has exactly 400 observations. In Panel (a), the perfectly balanced case, all five +power functions are visually indistinguishable. In Panel (c), where only six clusters are treated, +CV3 has noticeably more power than any of the bootstrap methods, which are all but identical. +In Panels (b) and (d), cluster sizes vary from 32 to 1513. All tests are now substantially +less powerful than in Panels (a) and (c), because, whenever there is intra-cluster correlation, the +information content of a sample declines as the cluster sizes become more variable. The most +striking result in both panels is that WCU-B has noticeably less power than any of the other +methods. This is especially true in Panel (d), where WCU-B over-rejects modestly under the +null but becomes by far the least powerful method for larger values of βk. +The pattern for CV3 is similar but much less pronounced. Under the null hypothesis, it +over-rejects slightly under the null in Panel (b) and noticeably in Panel (d), with rejection +frequencies of 0.0612 and 0.0775, respectively. But for large enough values of βk, it has less power +than WCR-C and WCR-S, especially in Panel (d). The latter two methods also have slightly +more power than WCR-B in Panel (b) and noticeably more in Panel (d) for large values of βk. +Interestingly, WCR-V, which for clarity is not shown in the figure, has somewhat less power +than either WCR-C or WCR-S in Panels (b) and (d) where cluster sizes vary a lot. In contrast, +it is almost indistinguishable from both these methods in Panels (a) and (c) where cluster sizes +are constant. +Based on these results, the procedure of choice appears to be WCR-S. For larger values of βk, +27 + +Figure 8: Power functions for several tests +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +.............................................................................................................................................................................................................................................................................................................................................. +...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +................................................................................................... +CV3 +.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. +................................................................................ +WCR-C +............................................................................................................................ +.............. +WCR-B +.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. +............................................................... +WCR-S +............................................................................................................................ +.............. +WCU-B +β +Rej. Rate +(a) Equal-Sized Clusters, G1 = 12, γ = 0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +.............................................................................................................................................................................................................................................................................................................................................. +β +Rej. Rate +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +...................................................................................................CV3 +................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ +................................................................................WCR-C +................................................................................................................... +..............WCR-B +....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +............................................................... WCR-S +................................................................................................................ +..............WCU-B +(b) Variable-Sized Clusters, G1 = 12, γ = 4 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +.............................................................................................................................................................................................................................................................................................................................................. +.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +...................................................................................................CV3 +...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +................................................................................WCR-C +....................................................................................................................... +..............WCR-B +......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +............................................................... WCR-S +....................................................................................................................... +..............WCU-B +β +Rej. Rate +(c) Equal-Sized Clusters, G1 = 6, γ = 0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +.............................................................................................................................................................................................................................................................................................................................................. +β +Rej. Rate +.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. +...................................................................................................CV3 +.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. +................................................................................WCR-C +............................................................................................................... +..............WCR-B +........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +............................................................... WCR-S +....................................................................................................... +..............WCU-B +(d) Variable-Sized Clusters, G1 = 6, γ = 4 +Notes: The vertical axes show rejection frequencies for tests at the .05 level. Results are based on 400,000 +replications, with G = 24, N = 9600, k = 5, ρ = 0.10, and B = 999. The hypothesis being tested is βk = 0 in +(41). The horizontal axes show the values of β in the DGP. +it is always one of the two most powerful tests. WCR-C has similar power, and it also works well +under the null in these experiments, but it is much more prone to over-reject than WCR-S in +Figures 3, 4, 6 and 7. Happily, WCR-S is already available in computationally efficient packages +for Stata, R, and Python; see Section 5. +6.3 +Confidence Intervals +Cluster-robust standard errors and bootstrap methods are often used to form confidence inter- +vals. Although we do not perform any Monte Carlo experiments explicitly to study the proper- +ties of confidence intervals, these can be inferred from Figure 8 and the results in Section 6.1. +28 + +Most confidence intervals are implicitly or explicitly obtained by inverting a hypothesis test. +When such a test has approximately the correct rejection frequency, the resulting confidence in- +terval must have approximately correct coverage. Similarly, when such a test has high power, +the resulting confidence interval must be relatively short. +In many of the experiments in Section 6.1, tests based on CV3 and the t(G − 1) distribution +are much less prone to over-reject than tests based on CV1. This suggests that the coverage of +confidence intervals based on CV3 standard errors will often be much better than the coverage +of ones based on CV1 standard errors. Even more reliable intervals may often (but not always) +be obtained by using the WCR-S or WCR-B bootstraps, which perform much better than the +classic WCR-C bootstrap in many cases. The WCU-B bootstrap also performs well in many +cases under the null, but the results in Panels (b) and (d) of Figure 8 suggest that, when cluster +sizes vary a lot, intervals based on it may be longer than ones based on WCR-B, which in turn +may be slightly longer than ones based on WCR-S. +The WCR-S bootstrap has excellent performance in many of the experiments of Section 6.1, +seems to have slightly better power than WCR-B in Panels (b) and (d) of Figure 8, and is easy to +compute. Therefore, we tentatively recommend that confidence intervals should be obtained by +inverting WCR-S bootstrap tests. However, inverting WCR-B bootstrap tests, or simply using +CV3 standard errors and the t(G − 1) distribution, would often lead to very similar intervals. +Of course, it is easier to obtain a confidence interval by using a standard error and the +t(G − 1) distribution than by inverting a bootstrap test, and it is easier to invert any form of +WCU bootstrap test than any form of WCR bootstrap test. However, the computational cost +of inverting WCR bootstrap tests can be remarkably small, even for very large samples; see +Roodman et al. (2019, Section 3.5) and MacKinnon (2022, Section 3.4). +7 +Empirical Examples +In this section, we consider three empirical examples. These suggest that the new bootstrap +procedures proposed in Section 5 may sometimes yield results very similar to those from the +existing WCR-C and WCU-C procedures, but they may also yield results which differ noticeably +from those and from each other. +7.1 +Minimum Wages and Hours Worked +Our first example is based on MacKinnon et al. (2022a, Section 8). It exploits differences in +the minimum wage across states and years to estimate the impact of minimum wages on hours +worked for teenagers. +The data on hours at the individual level from the American Community Survey (ACS) are +obtained from IPUMS (Ruggles et al., 2020) and cover the years 2005–2019. The minimum +wage data come from Neumark (2019) and are collapsed to state-year averages to match the +ACS frequency. We restrict attention to teenagers aged 16–19, keeping only individuals who are +29 + +Table 2: Example 1, minimum wages and hours worked +Estimate +Std. error +t-statistic +P value +HC1 +−0.15389 +0.02825 +−5.4471 +0.0000 +CV1 +−0.15389 +0.06231 +−2.4697 +0.0170 +CV3 +−0.15389 +0.06713 +−2.2925 +0.0261 +Wild cluster bootstrap P values +WCR-C +0.0362 +WCU-C +0.0207 +WCR-V +0.0352 +WCU-V +0.0186 +WCR-S +0.0374 +WCU-S +0.0227 +WCR-B +0.0371 +WCU-B +0.0203 +Notes: There are 492,827 observations, 51 clusters, and 79 coefficients, including state and year +fixed effects. The coefficient of interest is β in (42). Bootstrap P values use B = 999,999. +children of the respondent to the survey and who have never been married. We drop individuals +who had completed one year of college by age 16 and those reporting in excess of 60 hours +usually worked per week. We also restrict attention to individuals who identify as either black +or white. There are 492,827 observations in 51 clusters, which correspond to all 50 states plus +the District of Columbia. +The model we estimate is +yist = α + βmwst + Zistγ + δs + ηt + uist, +(42) +where yist is usual hours worked per week for individual i. The parameter of interest is β, which +is the coefficient on mwst, the minimum wage in state s at time t. The row vector Zist collects a +large set of individual-level controls, including race, gender, age, and education. There are also +state and year fixed effects, denoted by δs and ηt, respectively. +As MacKinnon et al. (2022a) discusses, clustering could in principle be done at several +different levels. However, the one that is most appealing and seems to be supported by the data +is clustering at the state level. The 51 clusters vary considerably in size. The smallest has 258 +observations, and the largest has 35,995. The ratio of these numbers is more than twice as large +as for γ = 4 in the experiments of Section 6.1. The mean number of observations per cluster +is 9,663, and the median is 7,082. This suggests that inference based on CV1 and the t(50) +distribution may not be reliable. Other measures of cluster heterogeneity, which are discussed +in the original paper, lead to the same conclusion. +Table 2 presents our key results. As expected, the CV3 t-statistic is somewhat smaller than +the CV1 t-statistic, and the P value based on the t(50) distribution is therefore somewhat larger. +The four WCR P values are larger than either of them, but still below 0.05, and the four +WCU P values are notably smaller than the WCR ones. Because B is so large, larger than +would normally be needed, the simulation standard errors for the WCR bootstrap P values are +30 + +about 0.0002. +Based on how similar the four WCR P values are, and on how well many of the WCR methods +perform in the experiments of Section 6.1, we tentatively conclude that the “true” P value for +the test of β = 0 is probably between 0.034 and 0.039. Thus the null hypothesis can safely be +rejected at the 0.05 level but not at the 0.01 level. +7.2 +Political Turnover and Test Scores +The second example comes from Akhtari, Moreira and Trucco (2022). This paper examines +the impact of political turnover on the quality of public services. +Specifically, it examines +several outcomes following close mayoral elections in Brazil. One of these outcomes is the test +scores of fourth-grade students. The paper uses a regression discontinuity design to identify the +treated and control municipalities, but it conducts the analysis using OLS. We replicate one +such regression, found in Table 3, Column 5 of the original paper: +scoreimt+1 = α + βI(IVMmt < 0) + γIVMmt + δI(IVMmt < 0)IVMmt + ηscoreimt + ϵimt. +(43) +The dependent variable is the test score one year after an election. IVMmt is the incumbent +vote margin in the close election which occurs in year t. Accordingly, the treatment variable +is I(IVMmt < 0), which equals 1 when the incumbent party loses the election and a turnover +occurs, and the coefficient of interest is β. This regression is estimated using a sample which is +determined by a selected bandwidth. While the paper considers several bandwidths, we focus +on the bandwidth 0.110, as this results in the largest sample. +The paper clusters the standard errors at the municipality level. Since there are 2101 munici- +palities, many of them located close to each other, it seems possible that this level of clustering is +too fine. We therefore consider state-level clustering. However, there are only 26 states in Brazil, +and they vary in size from 420 to 64,953 with partial leverages from 0.000234 to 0.179318 (Mac- +Kinnon et al., 2022b). With this much heterogeneity across clusters, relying on CV1 may be risky. +Table 3 presents our key results. As expected, the CV1 standard error for clustering by state +is smaller than the CV3 standard error. Contrary to our expectations, however, both are a bit +smaller than the CV1 standard error for clustering by municipality. The four WCR P values are +similar to each other and to the P value based on the CV1 t-statistic and the t(25) distribution. +Surprisingly, the four WCU P values are noticeably larger than the WCR ones. Nevertheless, +since every test rejects at the 0.05 level, there seems to be rather strong evidence against the +null hypothesis. +7.3 +Patronage in the British Empire +The third example is taken from Xu (2018), which explores the effect of patronage in the colonial +era of Britain on the appointment of governors to colonies. Part of the analysis examines whether +the extent to which the current secretary of state and a governor are “connected” led to more +31 + +Table 3: Example 2, political turnover and test scores +Estimate +Std. error +t-statistic +P value +HC1 +−0.06684 +0.00528 +−12.6616 +0.0000 +CV1 (munic.) +−0.06684 +0.02430 +−2.7505 +0.0060 +CV1 +−0.06684 +0.02204 +−3.0326 +0.0056 +CV3 +−0.06684 +0.02411 +−2.7722 +0.0104 +Wild cluster bootstrap P values +WCR-C +0.0047 +WCU-C +0.0193 +WCR-V +0.0057 +WCU-V +0.0235 +WCR-S +0.0046 +WCU-S +0.0212 +WCR-B +0.0056 +WCU-B +0.0236 +Notes: There are 429,979 observations, 26 clusters, and 5 coefficients. The coefficient of interest +is β in (43). Bootstrap P values use B = 999,999. +desirable colony postings. We replicate the results of one such regression, found in Table 3, +Column 3 of the original paper: +log(revenue)ist = α + β1connectedit + β2colonies servedit + γi + τt + δit + ϵist. +(44) +Here log(revenue)ist is the initial revenue for colony s when governor i was appointed in year t. +The main variable of interest is connectedit, which is a binary variable set equal to 1 when the +governor and the secretary share connections such as having attended the same elite boarding +school, or Oxford or Cambridge, or both being in the aristocracy, or having shared ancestry. +The variable colonies servedit is the number colonies in which the governor has served up to the +year of appointment. The regression also has fixed effects for governors (γi), years (τt), and the +duration of the governorship (δit). +The paper clusters the standard errors at the bilateral pair (or dyad) level between the +secretary of state and the governor. However, the dependent variable is observed at multiple +times for each colony, so it seems likely that there would be dependence across observations for +the same colony. The regression does not include colony fixed effects, which would have reduced +this dependence, because, with so many other fixed effects, it was impossible to include them. +Thus, it seems plausible that the standard errors should be clustered at the colony level instead +of the dyad level, and we investigate this approach. +Switching from dyadic clustering to clustering by colony actually reduces the CV1 standard +error. However, even though there are 70 colonies, they are quite unbalanced; the number of +observations per colony ranges from 4 to 104. Partial leverages also vary greatly, and they seem +to be roughly proportional to cluster sizes. Perhaps in consequence, the CV3 standard error is +47% larger than the CV1 one. +In view of the dramatic difference between the CV1 and CV3 t-statistics, the various wild +32 + +Table 4: Example 3, patronage in the British empire +Estimate +Std. error +t-statistic +P value +HC1 +0.17722 +0.07573 +2.3401 +0.0193 +CV1 (dyadic) +0.17722 +0.09933 +1.7842 +0.0750 +CV1 +0.17722 +0.08702 +2.0366 +0.0455 +CV3 +0.17722 +0.12810 +1.3834 +0.1710 +Wild cluster bootstrap P values +WCR-C +0.0535 +WCU-C +0.0575 +WCR-V +0.0704 +WCU-V +0.0738 +WCR-S +0.0656 +WCU-S +0.0725 +WCR-B +0.0621 +WCU-B +0.0678 +Notes: There are 3510 observations, 70 clusters, and 573 coefficients. The coefficient of interest +is β1 in (44). Bootstrap P values use B = 99,999 because, with 573 regressors, the computations +for WCR/WCU-V and WCR/WCU-B are much more expensive than for the previous examples. +bootstrap methods provide valuable information. The bootstrap P values are all somewhat larger +than the one for the CV1 t-statistic based on the t(69) distribution, but they are all much smaller +than the corresponding one for the CV3 t-statistic. The smallest bootstrap P value is the one +for the classic WCR-C method. At 0.0535, it is not much larger than the one based on the t(69) +distribution. Surprisingly, every WCU P value is larger than the corresponding WCR P value. +This example is deliberately extreme, because the number of regressors (573) is unusually +large relative to the number of observations (3510). Perhaps in consequence, the full coefficient +vectors ˆβ(g) are not identified for 61 out of the 70 clusters. However, since the ˆβ(g) +1 +coefficients +are always identified, we used a generalized inverse to compute both CV3 and the bootstrap +DGPs for the WCR/WCU-S and WCR/WCU-B bootstraps. The alternative approach of trying +to estimate a variance matrix based on just 9 out of 70 clusters seems very dubious, and it yields +an implausibly small standard error of just 0.0379. +Because k is so large in this example, we suspect that the t-test based on CV3 may be prone +to under-reject, and that both the t-test based on CV1 and the WCR-C bootstrap test may +be prone to over-reject; see Figure 3. Nevertheless, the P values for the new WCR bootstrap +methods are only modestly larger than the WCR-C P value. The fact that all the bootstrap +P values lie between 0.0535 and 0.0738 suggests that the “true” P value probably also lies +within, or at least not too far outside, this interval. We conclude that there seems to be only +weak evidence against the null hypothesis. +8 +Conclusion and Recommendations +The classic CV1 estimator given in (6) is by far the most popular CRVE for linear regression +models, but standard errors based on it are often much too small. The cluster jackknife estimator, +33 + +often called CV3, has been known for many years but is much less widely used. In Section 3, we +discuss how to compute CV3 in a computationally efficient fashion. Except when all clusters are +tiny, this is the fastest available method for computing it; see Section 4. Inference based on CV3 +and the Student’s t(G − 1) distribution seems to be much more reliable than inference based +on CV1 and that distribution; see Section 6.1. This accords with theoretical results in Hansen +(2022), which provides no simulations and cites the ones in this paper. +Although combining CV3 standard errors and the t distribution often works well, it does not +always do so. Bootstrap methods may well perform better, and they also provide a valuable +robustness check. In Section 5, we prove some simple, but by no means obvious, algebraic results +about the relationship between cluster jackknife estimates and score vectors at the cluster level. +These results allow us to obtain new and easy-to-compute variants of the wild cluster bootstrap. +These typically perform better than the classic variants, now called WCR-C and WCU-C. The +eight new and existing variants are summarized in Table 1. Of these, the ones that use CV1 +together with modified bootstrap score vectors, called WCR-S and WCU-S, are particularly easy +to compute. They have already been incorporated into packages for Stata and R. +Prior to this paper, there were already quite a few methods for inference in linear regression +models with clustered disturbances (MacKinnon et al., 2022a), and Section 5 has added six +new variants of the wild cluster bootstrap. Empiricists may reasonably ask what methods they +should use in practice. As discussed in detail in MacKinnon et al. (2022a,b), the first thing to +do is to investigate the clustering structure of the model and dataset. For instance, it is good +practice to calculate the effective number of clusters (Carter et al., 2017) as well as various +measures of leverage and influence at the cluster level (MacKinnon et al., 2022b). When these +measures indicate that clusters are well-balanced, and the (effective) number of clusters is large +(say, more than 100), CV1 and CV3 should yield very similar standard errors. In such cases, it +is probably safe to rely on CV3 standard errors together with the t(G − 1) distribution. +However, the number of clusters will often be much less than 100. Moreover, measures of +cluster-level leverage and influence may indicate that clusters are not well-balanced. This can +happen, for example, when cluster sizes vary a lot, when there are few treated clusters, or when +the distributions of key regressors vary greatly across clusters. In such cases, CV1 and CV3 +can yield quite different standard errors. Recall Table 4, where the CV3 standard error is 47% +larger than the CV1 standard error, even though there are 70 clusters. This may be because +so many singularities are encountered when computing the omit-1-cluster estimates. Whenever +CV1 and CV3 differ substantially, bootstrap P values or confidence intervals are likely to be more +reliable than conventional ones based on either of those CRVEs, and it is probably a good idea +to compute both WCR-C and WCR-S P values. +In most cases, it is advisable to compute wild cluster bootstrap P values and/or confidence +intervals using a large number of bootstrap samples, such as 9,999. This is usually not computa- +34 + +tionally difficult. However, there might be exceptions when either the number of clusters or the +number of regressors is unusually large. Of course, when the number of clusters is very large, the +bootstrap will not be needed unless the clusters are severely unbalanced, but that can happen. +If we had to recommend just one method, it would be the WCR-S bootstrap proposed in +Section 5. This method uses ordinary CV1 standard errors, which makes it easy to compute, +but the bootstrap DGP employs restricted scores that have been transformed using the cluster +jackknife. In some of our experiments, the WCR-S bootstrap works substantially better than +the classic (and popular) WCR-C bootstrap; see, in particular, Figures 2–4 and Figure 7. We +generally do not recommend using the unrestricted wild cluster bootstrap, except perhaps as a +robustness check or when it is desired to generate a large number of confidence intervals using +just one set of bootstrap samples. +References +Akhtari M, Moreira D, Trucco L. 2022. Political turnover, bureaucratic turnover, and the quality +of public services. American Economic Review 112: 442–493. +Bell RM, McCaffrey DF. 2002. Bias reduction in standard errors for linear regression with multi- +stage samples. Survey Methodology 28: 169–181. +Bester CA, Conley TG, Hansen CB. 2011. Inference with dependent data using cluster covariance +estimators. Journal of Econometrics 165: 137–151. +Brewer M, Crossley TF, Joyce R. 2018. Inference with difference-in-differences revisited. Journal +of Econometric Methods 7: 1–16. +Cameron AC, Gelbach JB, Miller DL. 2008. Bootstrap-based improvements for inference with +clustered errors. 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American Economic +Review 108: 3170–3198. +37 + diff --git a/ZdE3T4oBgHgl3EQfcgrr/content/tmp_files/load_file.txt b/ZdE3T4oBgHgl3EQfcgrr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a1ccd2153e88fb6483d1e6c4dd2eccbffae6e855 --- /dev/null +++ b/ZdE3T4oBgHgl3EQfcgrr/content/tmp_files/load_file.txt @@ -0,0 +1,42843 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf,len=42842 +page_content='Fast and Reliable Jackknife and Bootstrap Methods for Cluster-Robust Inference∗ James G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' MacKinnon† Queen’s University mackinno@queensu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='ca Morten Ørregaard Nielsen Aarhus University mon@econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='dk Matthew D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Webb Carleton University matt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='webb@carleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='ca January 12, 2023 Abstract We provide computationally attractive methods to obtain jackknife-based cluster-robust variance matrix estimators (CRVEs) for linear regression models estimated by least squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We also propose several new variants of the wild cluster bootstrap, which involve these CRVEs, jackknife-based bootstrap data-generating processes, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Extensive simula- tion experiments suggest that the new methods can provide much more reliable inferences than existing ones in cases where the latter are not trustworthy, such as when the number of clusters is small and/or cluster sizes vary substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Three empirical examples illus- trate the new methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Keywords: clustered data, grouped data, cluster-robust variance estimator, CRVE, clus- ter sizes, wild cluster bootstrap JEL Codes: C10, C12, C21, C23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' ∗We are grateful to David Drukker, Alexander Fischer, David Roodman, the Co-Editor, Francis Vella, an anonymous referee, and seminar participants at Aarhus University, Carleton University, University of Toronto, and New York Camp Econometrics 2022 for helpful comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' MacKinnon and Webb thank the Social Sciences and Humanities Research Council of Canada (SSHRC grants 435-2016-0871 and 435-2021-0396) for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Nielsen thanks the Danish National Research Foundation for financial support (DNRF Chair grant number DNRF154).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Address: Department of Economics, 94 University Avenue, Queen’s University, Kingston, Ontario K7L 3N6, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Email: mackinno@queensu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' 613-533-2293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Fax 613-533-6668.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='04527v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='EM] 11 Jan 2023 1 Introduction In applications of linear regression models to many fields of economics and other disciplines, it is common to divide the sample into disjoint clusters and employ a cluster-robust variance matrix estimator (or CRVE) for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' These estimators are based on the assumption that the disturbances of the regression model are uncorrelated across clusters, but they allow for arbitrary patterns of dependence and heteroskedasticity within each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The literature on cluster- robust inference has grown rapidly in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Cameron and Miller (2015) is a classic survey article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Conley, Gonçalves and Hansen (2018) surveys a broader class of methods for dependent data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' MacKinnon, Nielsen and Webb (2022a) provides a guide that explores the implications of key theoretical results for empirical practice, with an emphasis on bootstrap methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' There are several CRVEs for ordinary least squares (OLS) estimates of linear regression mod- els;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, mainly for computational reasons, almost all empirical work to date has made use of the simplest one, usually known as CV1, which is the default in Stata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Cluster- robust tests and confidence intervals based on CV1 may or may not yield reliable inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Whether they do so depends primarily on the number of clusters G and how homogeneous these are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' When all clusters are roughly equal in size and approximately balanced, asymptotic infer- ence based on CV1 seems to be fairly reliable whenever G is at least moderately large (say 50 or more).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, even when G is very large, cluster-robust t-tests and Wald tests are at risk of severe over-rejection, and cluster-robust confidence intervals are at risk of severe under-coverage in at least two situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The first is when one or a few clusters are much larger than the rest, and the second is when the only “treated” observations belong to just a few clusters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Djogbe- nou, MacKinnon and Nielsen (2019) discusses the first case, and MacKinnon and Webb (2017, 2018) discuss the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Alternatives to CV1 have been known since Bell and McCaffrey (2002), but computational difficulties have kept them from widespread use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The first contribution of this paper, which is discussed in Section 3, is to provide a fast method for computing jackknife-based CRVEs, of which the simplest is generally known as CV3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' By explicitly using the cluster jackknife for computation, our method makes it feasible to employ CV3 for inference even in very large samples with very large clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Because CV3 standard errors used to be hard to compute, there has been very little work comparing the finite-sample performance of t-tests based on CV3 with those of similar procedures based on CV1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' a partial exception is Niccodemi and Wansbeek (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The second contribution of this paper is to compare the finite-sample properties of these tests, and also ones based on CV2, by simulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In concurrent work that cites our simulations, Hansen (2022) provides important theoretical results which suggest that asymptotic inference based on CV3 is generally more reliable, and more conservative, than asymptotic inference based on CV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Existing bootstrap methods for cluster-robust inference are all based on CV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The best 2 known of these (and until now the best performing one) seems to be the wild cluster restricted (or WCR) bootstrap proposed in Cameron, Gelbach and Miller (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' There is also a closely related procedure called the wild cluster unrestricted (or WCU) bootstrap, which generally does not work quite as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The asymptotic validity of these procedures is proved in Djogbenou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (2019), which also analyzes their higher-order asymptotic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Until a few years ago, the WCR and WCU bootstraps were computationally expensive for large samples, but that is no longer the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Roodman, MacKinnon, Nielsen and Webb (2019) describes a remarkably ef- ficient implementation in the Stata package boottest, and MacKinnon (2022) discusses other methods for fast computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The boottest routines are now available as a Julia package which can be also be called from R, Python, and Stata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The package fwildclusterboot im- plements the boottest method natively in R (Fischer and Roodman, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The third contribution of this paper is to propose several new variants of the wild cluster bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' One modification simply replaces CV1 by CV3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The other, which requires some new results, involves modifying the bootstrap data-generating process, or DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Modern treatments of the wild cluster bootstrap, such as MacKinnon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (2022a), express the bootstrap DGP as a function of the empirical scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We show how to make the bootstrap DGP more closely resemble the (unknown) true DGP by transforming the residuals before forming the scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The transformation we propose is based on the jackknife.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Accordingly, it does not actually require any calculations that explicitly involve residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This makes it very fast when the number of clusters is small relative to the sample size, even when the latter is extremely large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The next section establishes notation and briefly reviews the literature on asymptotic cluster- robust inference for the linear regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Section 3 then provides a new computational method for CV3, which is conceptually simple and extremely fast in many cases, as we demon- strate in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Next, Section 5 discusses several ways of modifying the wild cluster boot- strap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Simulation results in Section 6 suggest that our new versions of the WCR and WCU bootstraps perform better, sometimes very much better, than the original ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This is partic- ularly true when cluster sizes vary greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' One modified version of the WCR bootstrap that uses transformed scores seems to work especially well in most settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Section 7 presents three empirical examples in which our methods are likely to be more reliable than existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Sec- tion 8 concludes with a brief discussion of the methods that we recommend in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' 2 The Linear Regression Model with Clustering Consider the linear regression model yi = x⊤ i β+ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' If we divide the data into G disjoint clusters, where the allocation of observations to clusters is assumed to be known, this can be written as yg = Xgβ + ug, g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (1) 3 The g th cluster has Ng observations, and the total sample size is N = �G g=1 Ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In (1), Xg is an Ng × k matrix of regressors, β is a k-vector of coefficients, yg is an Ng-vector of observations on the regressand, and ug is an Ng-vector of disturbances (or error terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Stacking the yg yields the N-vector y, stacking the Xg yields the N × k matrix X, and stacking the ug yields the N- vector u, so that (1) can be rewritten as y = Xβ + u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The OLS estimator of β is ˆβ = (X⊤X)−1X⊤y = β0 + (X⊤X)−1X⊤u, (2) where the second equality depends on the assumption that the data are actually generated by (1) with true value β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Thus, if sg = X⊤ g ug is the score vector for the g th cluster, ˆβ − β0 = (X⊤X)−1 G � g=1 X⊤ g ug = � G � g=1 X⊤ g Xg �−1 G � g=1 sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (3) Obtaining valid inferences evidently requires assumptions about the score vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For a correctly specified model, E(sg) = 0 for all g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We further assume that E(sgs⊤ g ) = Σg and E(sgs⊤ g′) = 0, g, g′ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , G, g′ ̸= g, (4) where Σg is the symmetric, positive semidefinite variance matrix of the scores for the g th cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The second assumption in (4) is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' It states that the scores for every cluster are uncorrelated with the scores for every other cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' From the rightmost expression in (3), we see that the distribution of ˆβ depends on the disturbance subvectors ug only through the distribution of the score vectors sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' It follows immediately that an estimator of Var( ˆβ) should be based on the usual sandwich formula, (X⊤X)−1 � G � g=1 Σg � (X⊤X)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (5) Every CRVE replaces the Σg in (5) by functions of the Xg and the residual subvectors ˆug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' There is more than one way to do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Since Σg is the expectation of sgs⊤ g , the simplest approach is just to replace it by ˆsg ˆs⊤ g , where ˆsg = X⊤ g ˆug is the empirical score vector for the g th cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' If in addition we multiply by a correction for degrees of freedom, we obtain CV1: ˆV1( ˆβ) = G(N − 1) (G − 1)(N − k)(X⊤X)−1 � G � g=1 ˆsg ˆs⊤ g � (X⊤X)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (6) This is by far the most widely-used CRVE in practice, and it is the default in Stata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The leading scalar is chosen so that, when G = N, ˆV1( ˆβ) reduces to the familiar HC1 estimator (MacKinnon and White, 1985) that is robust only to heteroskedasticity of unknown form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Inference about β is typically based on cluster-robust t-statistics and Wald statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' If βj denotes the j th element of β and β0j is its value under the null hypothesis, then the appropriate 4 t-statistic is tj = ˆβj − β0j se1(ˆβj) , (7) where ˆβj is the OLS estimate, and se1(ˆβj) is the square root of the j th diagonal element of ˆV1( ˆβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Under extremely strong assumptions (Bester, Conley and Hansen, 2011), it can be shown that tj asymptotically follows the t(G − 1) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Conventional “asymptotic” inference is based on this distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We should expect inferences based on CV1 to be reliable if the sum of the sg, suitably normalized, is well approximated by a multivariate normal distribution with mean zero, and if the sg are well approximated by the ˆsg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' But asymptotic inference can be misleading when either or both of these approximations is poor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see Djogbenou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (2019) and MacKinnon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Whether or not the first approximation is a good one depends on the model and the data, and there is not much the investigator can do about it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' But the second approximation can, in principle, be improved by using modified empirical score vectors instead of the ˆsg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Two CRVEs based on this idea, usually known as CV2 and CV3, were proposed (under different names) in Bell and McCaffrey (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' These are the cluster analogs of the hetero- skedasticity-consistent variance matrix estimators HC2 and HC3 proposed in MacKinnon and White (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' All of these estimators are designed to compensate, in different ways, for the shrinkage and intra-cluster correlation of the residuals induced by least squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The CV2 variance matrix is CV2: ˆV2( ˆβ) = (X⊤X)−1 � G � g=1 `sg `s⊤ g � (X⊤X)−1, (8) where the modified score vectors `sg are defined as `sg = X⊤ g M −1/2 gg ˆug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (9) Here Mgg = INg − Xg(X⊤X)−1X⊤ g is the g th diagonal block of the projection matrix MX, which satisfies ˆu = MXu, and M −1/2 gg is the symmetric square root of its inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The CV2 estimator has been recommended in Imbens and Kolesár (2016) and Pustejovsky and Tipton (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Following Bell and McCaffrey (2002), these papers provide methods for computing critical values based on t and F distributions with computed degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The CV3 variance matrix is very similar to CV2, but, as we explain in Section 3, it is based on the jackknife.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The usual definition is CV3: ˆV3( ˆβ) = G − 1 G (X⊤X)−1 � G � g=1 ´sg ´s⊤ g � (X⊤X)−1, (10) 5 where now the modified score vectors ´sg are defined as ´sg = X⊤ g M −1 gg ˆug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (11) The rescaling factor (G − 1)/G in (10) is the analog of the factor (N − 1)/N that occurs in jackknife variance matrix estimators at the individual level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This factor implicitly assumes that all clusters are the same size and perfectly balanced, with disturbances that are independent and homoskedastic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' an alternative is proposed in Niccodemi and Wansbeek (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Although (8) and (10) look simple enough, computing either CV2 or CV3 has until recently been extremely expensive, or even computationally infeasible, when any of the Ng are large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The problem is that, before computing (11), we apparently need to rescale the residual vector ˆug for each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This involves storing and inverting the Ng ×Ng matrix Mgg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Before computing (9), we also need to compute the symmetric square roots of the Mgg, and this requires calculating their eigenvalues and eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Of course, when all clusters are very small, this is not difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' When G = N, CV2 reduces to HC2, and CV3 reduces to HC3, both of which can be computed very quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Niccodemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (2020) has recently proposed a method that is much faster for large clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Versions of this method apply to both CV2 and CV3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Instead of rescaling the residual vectors, it calculates the score vectors `sg or ´sg directly using equations that do not involve any Ng × Ng matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' A revised version of this method, which appears to be new, works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' First, form the k × k matrices Ag = (X⊤X)−1/2X⊤ g Xg(X⊤X)−1/2, g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (12) Then, for (8), calculate the rescaled score vectors `sg = (X⊤X)1/2(Ik − Ag)−1/2(X⊤X)−1/2ˆsg, g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , G, (13) and, for (10), calculate the rescaled score vectors ´sg = (X⊤X)1/2(Ik − Ag)−1(X⊤X)−1/2ˆsg, g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (14) These rescaled score vectors are used in (8) and (10) as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Unless all the clusters are very small, computing CV2 and CV3 using (13) and (14) is much faster than computing them using (9) and (11);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In the case of CV3, however, an even faster and more intuitive method is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This jackknife-based method, which we discuss in the next section, can be extremely fast when N is large and G is much smaller than N, so that at least some clusters are large;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' 6 3 Jackknife Variance Matrix Estimators The jackknife is a simple method for reducing bias and estimating standard errors by omitting observations sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Tukey (1958) suggested using the jackknife to estimate standard errors, and Miller (1974) is a classic reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The key idea of the cluster jackknife is to compute G sets of parameter estimates, each of which omits one cluster at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In this section, we use it to compute two closely related CRVEs in an efficient fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The OLS estimates of β when each cluster is omitted in turn are ˆβ(g) = (X⊤X − X⊤ g Xg)−1(X⊤y − X⊤ g yg), g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (15) It is easy to obtain the ˆβ(g) in a computationally efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We start by calculating the cluster-level matrices and vectors X⊤ g Xg and X⊤ g yg, g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (16) Unless G is very large, this involves very little cost beyond that of computing ˆβ, because we can use the quantities in (16) to construct X⊤X and X⊤y and then use (2) to obtain ˆβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For typical values of k, it should then be reasonably inexpensive to calculate ˆβ(g) for every cluster using (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The main cost, beyond that of computing ˆβ, is that we need to calculate the inverse of a k × k matrix for each of the ˆβ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The cluster jackknife estimator of Var( ˆβ) is the cluster analog of the usual jackknife variance matrix estimator given in Efron (1981), among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' It is defined as CV3J: ˆV3J( ˆβ) = G − 1 G G � g=1 ( ˆβ(g) − ¯β)( ˆβ(g) − ¯β)⊤, (17) where ¯β = G−1 �G g=1 ˆβ(g) is the sample average of the ˆβ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Notice that (17) calculates the variance matrix around ¯β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Centering around ¯β is common in jackknife variance estimation, but it is also common to center around ˆβ, as in Bell and McCaffrey (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' There is a very close relationship between ˆV3J( ˆβ) and ˆV3( ˆβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In fact, ˆV3( ˆβ) = G − 1 G G � g=1 ( ˆβ(g) − ˆβ)( ˆβ(g) − ˆβ)⊤, (18) which is just (17) with ¯β replaced by ˆβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This follows from (10) and (11) because (X⊤X)−1´sg = (X⊤X)−1X⊤ g M −1 gg ˆug = ˆβ − ˆβ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (19) Note that the summation in (18) is unchanged if ˆβ(g) − ˆβ is replaced by ˆβ − ˆβ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Although the second equality in (19) is not new, it will turn out to be very useful in Section 5, 7 and so we now prove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The middle expression in (19) can be written as (X⊤X)−1X⊤ g M −1 gg yg − (X⊤X)−1X⊤ g M −1 gg Xg(X⊤X)−1X⊤y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (20) Using the updating formula (X⊤X − X⊤ g Xg)−1 = (X⊤X)−1 + (X⊤X)−1X⊤ g M −1 gg Xg(X⊤X)−1, (21) ˆβ(g) can be written as the sum of four terms, the first of which is just ˆβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Thus the right-hand side of (19) can be written as (X⊤X)−1X⊤ g M −1 gg Xg(X⊤X)−1X⊤ g yg + (X⊤X)−1X⊤ g yg − (X⊤X)−1X⊤ g M −1 gg Xg(X⊤X)−1X⊤y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (22) The last term in (22) is identical to the last term in (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The first two terms in (22) can be rewritten as (X⊤X)−1X⊤ g M −1 gg Pggyg + (X⊤X)−1X⊤ g yg, where Pgg = Xg(X⊤X)−1X⊤ g is the g th diagonal block of the matrix PX = I − MX, so that Pgg = I − Mgg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Inserting this straightforwardly yields the result that (X⊤X)−1X⊤ g M −1 gg Pggyg + (X⊤X)−1X⊤ g yg = (X⊤X)−1X⊤ g M −1 gg (I − Mgg)yg + (X⊤X)−1X⊤ g yg = (X⊤X)−1X⊤ g M −1 gg yg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (23) The right-hand side of (23) is the first term in (20), which proves the second equality in (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' When Ng = 1 for all g, ˆV3J( ˆβ) is numerically equal to the original HC3 estimator proposed in MacKinnon and White (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The modern version of HC3, which uses ˆβ instead of ¯β and omits the factor of N/(N − 1), is due to Davidson and MacKinnon (1993, Chapter 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Both cluster jackknife estimators may be used to compute cluster-robust t-statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Since there are G terms in the summation, it is natural to compare these with quantiles of the t(G−1) distribution, as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' These procedures should almost always be more conservative than t-tests based on CV1 (Hansen, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We expect CV3 and CV3J to be very similar in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This issue will be investigated in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='1, where we conclude that it is reasonable to focus on CV3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Both CV3 and CV3J have been available in Stata for some years by using the options “vce(jackknife,mse)” and “vce(jackknife)”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, the implementations discussed here are much more efficient when G is not very small, and are available in Stata and R packages, both named summclust;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see MacKinnon, Nielsen and Webb (2022c) and Fischer (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Both packages also calculate a number of summary statistics that may be used to assess the reliability of cluster-robust inference as described in MacKinnon, Nielsen and Webb (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' There may be cases in which the matrix X⊤X − X⊤ g Xg in (15) is singular for one or more values of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' If so, then at least some elements of ˆβ(g) cannot be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This can happen in otherwise well-specified models when there are cluster-level fixed effects, and in that case the 8 solution is simply to partial them out before running the regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In other cases where a singularity occurs, there are two possible courses of action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The first is to modify (17) and (18) so that the summation is taken only over values of g for which ˆβ(g) can be estimated, and G is replaced by the number of clusters for which that is the case (this is the approach followed in the native Stata implementations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see also Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' When there are only a few problematic clusters, this approach may be attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' But since ˆβ and ¯β would then be based on different samples, it seems likely that CV3J and CV3 may differ more than they would usually do, which suggests that it may be safer to use the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The second course of action is to replace the inverse in (15) by a generalized inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In practice, this means that coefficients that cannot be identified are replaced by zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' When the elements of ˆβ(g) that are of primary interest can always be identified, this approach may be attractive, especially when there are many problematic clusters, as in the example of Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' 4 Speed of Computation The CV3 estimator can be challenging to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Following Bell and McCaffrey (2002), it is natural to employ what we call the “residual method” based on (10) and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' To compute the modified score vector ´sg for the g th cluster, this method uses the Ng-vector of residuals ˆug and the Ng × Ng matrix M −1 gg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Unless every Ng is small, storing and inverting the Mgg matrices is computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Indeed, for even moderately large values of the Ng, this can be effectively impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' A much faster method, recently proposed in Niccodemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (2020) and revised modestly in Section 2, uses (14) to obtain the modified score vectors ´sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Since it operates directly on the score vectors ˆsg, we call it the “score method.” An even faster approach, discussed in Section 3, computes the ˆβ(g) using (15) and then calculates their variance matrix as (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For obvious reasons, we refer to this as the “jackknife method.” To compare timings for the residual, score, and jackknife methods, we generate two datasets with N = 1, 048, 576 = 220 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In one case, there are 20 regressors, and in the other case there are 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The observations are divided into G equal-sized clusters, where G varies from 16 to 512K and K denotes 1024 = 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Thus the cluster size M = N/G varies from 2 to 64K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Figure 1 shows the time in seconds, on a log2 scale, for each of the three methods and the two datasets as a function of cluster sizes M = N/G, which vary from 2 to 64K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' These times include the time required to compute the OLS estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For both the jackknife and score methods, there is considerable overlap between the computations needed for the OLS estimates and the ones needed for CV3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Thus, for large clusters, the cost of computing the OLS estimates and CV3 together using one or both of these methods was sometimes less than the cost of computing the OLS estimates alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This is probably because of cache congestion, which seems to be alleviated by forming X⊤X on a cluster-by-cluster basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For large clusters, the speed of all methods could 9 Figure 1: Timings for three ways to compute CV3 2 4 8 16 32 64 128 256 512 1K 2K 4K 8K 16K 32K 64K 1/32 1/8 1/2 2 8 32 128 512 2K 8K 32K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='. k = 40 M Time in Seconds Notes: The sample size is N = 1, 048, 576 = 1024K, where K = 1024 = 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The number of clusters varies from 16 to 512K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' All clusters have M = N/G observations, so that cluster sizes vary from 2 to 64K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The number of regressors k is either 20 or 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Times required to compute ˆβ are included;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' All computations were performed in Fortran using one core of an Intel i9-13900K processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' almost certainly be increased by using a fast BLAS implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, in the interest of programming ease, we have not done this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The jackknife and score methods are already very fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In Figure 1, the residual method works well for very small values of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' It is always the fastest method for M ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We did not perform any timings for M = 1, where CV3 reduces to HC3, because we would have needed a different program that eliminated the loops within each cluster to obtain optimal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' But the residual method is certainly the fastest one for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, its cost rises very rapidly as M increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Results for this method are only shown for M ≤ 4096, because using it for larger values would have been prohibitively costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For the largest values of M, the cost of the residual method is almost the same for k = 20 and k = 40, because it is dominated by the computations needed to form and invert the Mgg matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In contrast, both the score and jackknife methods become faster as M increases and G consequently decreases, except that, when k = 40, they are both a bit slower for M = 64K than for M = 32K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This probably occurs because of cache congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The jackknife method is always quicker than the score method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For small values of M, it seems to be faster by a factor of about 12 when k = 20 and by a factor of about 26 when k = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, the advantage of the jackknife method gradually diminishes as M increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' When M = 64K, so that there are 10 only 16 clusters, the jackknife method is only slightly faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' It is easy to see that the jackknife method will have a big advantage over the residual method whenever cluster sizes vary much, even if most of them are very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Imagine a sample with, say, 1000 equal-sized clusters and M = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For such a sample, the residual and jackknife methods will perform about the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Suppose we then merge 100 of the tiny clusters into one large cluster with 500 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Doing this will reduce the cost of the jackknife method slightly, but it will greatly increase the cost of the residual method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Indeed, when there is even a single very large cluster, the latter inevitably becomes extremely slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Based on these results, the jackknife method for computing CV3 is clearly the procedure of choice unless all clusters are tiny (say, Ng ≤ 5 for all g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For datasets with large clusters, an efficient implementation of this method (such as the one provided by the summclust package mentioned in Section 3), can compute both the OLS estimates and the CV3 variance matrix in roughly the same amount of time as a reasonably fast program for the OLS estimates alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' 5 New Versions of the Wild Cluster Bootstrap The existing WCR bootstrap is based on CV1 standard errors and the restricted empirical score vectors defined in (25) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Henceforth, we will refer to this as the classic WCR bootstrap, or WCR-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' It often works well, but not always.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We therefore propose three new versions of the WCR bootstrap, along with three corresponding versions of the WCU bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' These are based on two distinct modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' One involves replacing CV1 by CV3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The other involves modifying the scores used in the bootstrap DGP, in the hope that the modified bootstrap DGP will provide a better approximation to the unknown process that actually generated the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We first discuss the bootstrap DGPs for the classic wild cluster bootstraps, WCR-C and WCU-C, expressing them in terms of scores instead of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This approach is intuitive and computationally attractive (Roodman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' MacKinnon, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In terms of the G score vectors, a generic wild cluster bootstrap DGP is s∗b g = v∗b g ¨sg, g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , G, b = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , B, (24) where b indexes bootstrap samples, v∗b g is a random variate with mean 0 and variance 1, and the ¨sg are empirical score vectors to be discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In most cases, it seems to be best to generate the v∗b g using the Rademacher distribution, which takes the values 1 and −1 with equal probabilities (Davidson and Flachaire, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Djogbenou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, since the number of possible Rademacher bootstrap samples that are distinct from the original sample is only 2G − 1, it is better to use a distribution with more mass points, such as the six-point distribution proposed in Webb (2022), when G is less than about 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The vector ¨sg in (24) is an empirical score vector for the g th cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For the WCU-C bootstrap, it is simply the unrestricted empirical score vector ˆsg = X⊤ g ˆug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For the WCR-C 11 bootstrap, it is the restricted empirical score vector ˜sg defined as ˜sg = X⊤ g yg − X⊤ g Xg ˜β, g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , G, (25) where ˜β is the vector of OLS estimates under the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Like ˆβ, ˜sg is a k-vector, even though some elements of ˜β may equal zero or satisfy other linear restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The bootstrap DGP (24) looks very much like the one for the wild score cluster bootstrap for nonlinear models proposed in Kline and Santos (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In the context of (1), however, it is just a different way of writing the bootstrap DGP for the wild cluster bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In order to calculate a bootstrap P value or a bootstrap confidence interval, we need to compute B bootstrap test statistics indexed by b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' These depend only on the bootstrap scores in (24) and the matrix (X⊤X)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For each bootstrap sample, we use s∗b g to obtain a bootstrap estimate, not of β itself, but of the vector δ = β − ¨β, where ¨β = ˜β for the WCR-C bootstrap and ¨β = ˆβ for the WCU-C bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This estimate is simply ˆδ∗b = (X⊤X)−1 G � g=1 s∗b g = (X⊤X)−1s∗b, (26) where s∗b = �G g=1 s∗b g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' When v∗b g = 1 for all g, the bootstrap sample is the same as the original sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In this very special case, ˆδ∗b = 0 for the WCU-C bootstrap, and ˆδ∗b = ˆβ − ˜β for the WCR-C bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' If we are testing the hypothesis that βj = 0, where βj is an element of β, then we just need to multiply the j th row of (X⊤X)−1 by s∗b in order to obtain ˆδ∗b j , the j th element of δ∗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The bootstrap t-statistic is then equal to t∗b j = ˆδ∗b j se(ˆδ∗b j ) , (27) where se(·) denotes the standard error formula used to obtain tj, the original t-statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We automatically get the correct numerator, which is ˆβ∗b j for the WCR-C bootstrap, since ¨β = ˜β, and ˆβ∗b j − ˆβj for the WCU-C bootstrap, since ¨β = ˆβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' As usual, a symmetric bootstrap P value is then given by P ∗ S(tj) = 1 B � I � |t∗b j | > |tj| � , (28) where I(·) denotes the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The bootstrap P value in (28) is simply the fraction of the bootstrap samples for which |t∗b j | is more extreme than |tj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The value of B should be chosen so that α(B + 1) is an integer, where α is the level of the test (Racine and MacKinnon, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' It is common to use B = 999, but B = 9,999 and (when feasible) B = 99,999 are better choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In the classic versions of the wild cluster bootstrap, the standard error formula in (27) is se1(·), the square root of the j th diagonal element of CV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' But the results in Section 3 make it equally feasible to use standard errors based on CV3, even in large samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This gives us new versions of both the WCR and WCU bootstraps, which we will refer to as WCR-V and 12 WCU-V, because only the variance matrices have changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The bootstrap standard errors can be calculated without computing an entire variance matrix for each bootstrap sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For example, the CV3 standard error of ˆδ∗b j is just se3(ˆδ∗b j ) = � �G − 1 G G � g=1 �ˆδ∗b j(g) − ˆδ∗b j �2 � � 1/2 , (29) where ˆδ∗b j(g) is the j th element of the vector ˆδ∗b (g) = (X⊤X − X⊤ g Xg)−1(s∗b − s∗b g ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (30) Only ˆδ∗b j and the ˆδ∗b j(g) need to be computed for each bootstrap sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In (26) and (30), the first terms are invariant across bootstrap samples and only need to be computed once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We now have two versions of the WCR bootstrap, WCR-C and WCR-V, and two versions of the WCU bootstrap, WCU-C and WCR-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The two WCR bootstraps use the bootstrap DGP (24) with ¨sg = ˜sg, and the two WCU bootstraps use the bootstrap DGP (24) with ¨sg = ˆsg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The “C” and “V” versions calculate both the actual and bootstrap test statistics using se1(·) and se3(·), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' These bootstrap methods use the restricted or unrestricted empirical scores in their raw form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' But empirical scores differ from true scores, because residuals differ from disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' It therefore seems attractive to replace the empirical score vectors by modified score vectors that implicitly rescale the residuals on a cluster-by-cluster basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This is analogous to methods discussed in Davidson and Flachaire (2008) and MacKinnon (2013) for the ordinary wild bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, quite a lot more algebra is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We first consider the WCU bootstrap, since this case is slightly easier to deal with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In principle, we could simply replace the vectors ¨sg in (24) with the modified empirical score vectors ´sg defined in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, using (11) is expensive, or even computationally infeasible, for large clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' But the result (19) lets us compute ´sg very rapidly as ´sg = X⊤X � ˆβ − ˆβ(g)� , g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (31) For large clusters, using (14) to compute the ´sg is much faster than using (11), but using (31) is faster still;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This yields two new bootstrap methods, which we will refer to as WCU-S and WCU-B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The WCU-S bootstrap (S for score) employs the modified score vectors ´sg instead of ˆsg, but it uses the familiar se1(·) standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The WCU-B bootstrap (B for both) employs both the modified score vectors and the se3(·) standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Finding the analogous versions of the WCR bootstrap takes a bit more work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We need to specify a restricted wild bootstrap DGP based on modified score vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Suppose the restrictions have the usual linear form, Rβ = r, for a given matrix R and a given vector r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We can write this equivalently in terms of free parameters, φ, as β = Hφ + h for a given matrix H and a 13 given vector h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Then the modified score vectors are ˙sg = X⊤ g ˜ M −1 gg (yg − Xg ˜β), (32) which are the analogs of the ´sg from (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Here ˜ Mgg is the g th diagonal block of the projection matrix ˜ M = I− ˜ X( ˜ X⊤ ˜ X)−1 ˜ X⊤, where ˜ X = XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, evaluating (32) is computationally infeasible when the clusters are not all small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We need to replace (32) by something that is feasible for any sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The first step is to compute the restricted estimates ˜β = H ˜φ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Here ˜y = y − Xh and ˜φ = ( ˜ X⊤ ˜ X)−1 ˜ X⊤ ˜y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The corresponding estimates when each cluster is omitted in turn are ˜β(g) = H ˜φ(g) + h, where ˜φ(g) = ( ˜ X⊤ ˜ X − ˜ X⊤ g ˜ Xg)−1( ˜ X⊤ ˜y − ˜ X⊤ g ˜yg), g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (33) Then it can be shown that ˙sg = X⊤ g ˜yg − X⊤ g ˜ Xg ˜φ(g), g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (34) To see that (32) and (34) are equal, note that the right-hand side of (34) is X⊤ g � ˜yg − ˜ Xg( ˜ X⊤ ˜ X − ˜ X⊤ g ˜ Xg)−1( ˜ X⊤ ˜y − ˜ X⊤ g ˜yg) � = X⊤ g � ˜yg − ˜ Xg � ( ˜ X⊤ ˜ X)−1 + ( ˜ X⊤ ˜ X)−1 ˜ X⊤ g ˜ M −1 gg ˜ Xg( ˜ X⊤ ˜ X)−1� ( ˜ X⊤ ˜y − ˜ X⊤ g ˜yg) � , where the equality uses the updating formula (21) applied to ˜ X, ˜ Xg, and ˜ M −1 gg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Then we use the fact that ˜φ = ( ˜ X⊤ ˜ X)−1 ˜ X⊤ ˜y together with the relation ˜ Xg( ˜ X⊤ ˜ X)−1 ˜ X⊤ g = ˜Pgg = I − ˜ Mgg to rewrite the last expression as X⊤ g � ˜yg − ˜ Xg ˜φ − (I − ˜ Mgg) ˜ M −1 gg ˜ Xg ˜φ + (I − ˜ Mgg)˜yg + (I − ˜ Mgg) ˜ M −1 gg (I − ˜ Mgg)˜yg � = X⊤ g ˜ M −1 gg (˜yg − ˜ Xg ˜φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (35) Replacing ˜yg by yg − Xgh and ˜ Xg by XgH, and using the fact that H ˜φ = ˜β − h, the right- hand side of (35) equals (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' An important special case is the restriction that βk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This is obtained by setting R = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , 0, 1) and r = 0, or, equivalently, H = (Ik−1, 0)⊤ and h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In this case, we find that ˜ X = X1, which contains the first k − 1 columns of X, and ˜φ = ˜β1 = (X⊤ 1 X1)−1X⊤ 1 y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The corresponding estimates when each cluster is omitted in turn are ˜β(g) 1 = (X⊤ 1X1 − X⊤ 1gX1g)−1(X⊤ 1 y − X⊤ 1gyg), g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , G, (36) where X1g contains the first k − 1 columns of Xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Then (34) reduces to ˙sg = X⊤ g yg − X⊤ g X1g ˜β(g) 1 , g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (37) 14 Exactly the same arguments that led to (34) can be applied to the modified unrestricted empirical scores, giving us ´sg = X⊤ g yg − X⊤ g Xg ˆβ(g), g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (38) Either (31) or (38) can be used to compute the ´sg, and both are computationally attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, in situations where both ˙sg and ´sg need to be computed, (38) may offer some pro- gramming advantages relative to (31) due to its similarity to (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' It may seem puzzling that the scalar factors in (6) and (10) do not appear in the bootstrap DGPs that correspond to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The reason is that rescaling all the bootstrap scores by the same factor has no impact on the resulting bootstrap t-statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' From (26) and (30), it is easy to see that multiplying all the s∗b g by a scalar C simply makes ˆδ∗b and all the ˆδ∗b (g) larger by a factor of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' But this also makes the empirical scores for every bootstrap sample larger by the same factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Therefore, from (6), (8), and (10), the variance matrices become larger by a factor of C 2 and the standard errors by a factor of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The factors of C in the numerator and denominator of t∗b j cancel out, leaving the bootstrap t-statistics unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, if we chose not to studentize the test statistic, it would make sense to multiply the right-hand side of (24) by the square root of G(N − 1)/((G − 1)(N − k)) for methods that use CV1 and by the square root of (G − 1)/G for methods that use CV3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Doing this should improve the correspondence between the bootstrap DGP and the unknown process that actually generated the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' An unstudentized test statistic for βj = 0 is just ˆβj, and its bootstrap analog would be ˆδ∗b j , which equals ˆβ∗b j for WCR and ˆβ∗b j − ˆβj for WCU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The usual theory of higher- order refinements for the bootstrap suggests that it is generally better to studentize (Hall, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, there may be cases in which unstudentized test statistics are of interest (Canay, Santos and Shaikh, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Nevertheless, since we have eight studentized bootstrap methods to study, we do not consider unstudentized ones further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' To generate the transformed scores needed for the WCR/WCU-S and WCR/WCU-B boot- straps, (31) and (38) must be used for all G clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In the event that ˆβ(g) and ˜β(g) cannot be calculated for, say, cluster h, we have two choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The simplest is to replace the inverses in (15) and (36) by generalized inverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Alternatively, we could use ˆsh instead of ´sh and ˜sh instead of ˙sh, along with the transformed scores for the remaining clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The latter would be appro- priate if we have chosen to omit the problematic clusters when computing the cluster-jackknife variance matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see the discussion at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Table 1 provides a convenient summary of the eight wild cluster bootstrap methods that we have discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Conceptually, they differ along two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The horizontal dimension represents the way in which the standard errors for both the actual and bootstrap test statistics are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The vertical dimension represents the score vectors used in the four versions of the bootstrap DGP (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Note that the boottest and fwildclusterboot packages now provide ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='Table 1: Eight versions of the wild cluster bootstrap ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='Standard errors based on ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='Scores in bootstrap DGP (24) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='CV1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='CV3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='Null hypothesis imposed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='˜sg defined in (25) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='WCR-C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='WCR-V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='˙sg defined in (37) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='WCR-S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='WCR-B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='Null hypothesis not imposed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='ˆsg = X⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='g ˆug ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='WCU-C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='WCU-V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='´sg defined in (31) or (38) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='WCU-S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='WCU-B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='Notes: WCR-C and WCU-C are the classic versions of the wild cluster restricted and wild ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='cluster unrestricted bootstraps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' WCR-S and WCU-S employ transformed scores with the usual CV1 variance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' WCR-V and WCU-V employ the usual scores with the CV3 variance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' WCR-B and WCU-B employ both transformed scores and CV3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' fast implementations of the WCR/WCU-S bootstraps as well as the classic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This is possible because, in contrast to the WCR/WCU-V and WCR/WCU-B bootstraps, the former do not involve any jackknife calculations for the bootstrap samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Once the transformed scores have been computed, the fast bootstrap algorithm proposed in Roodman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (2019) applies directly to the WCR/WCU-S bootstraps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' It seems highly likely that all the methods discussed in this section are asymptotically valid, in the sense that, under suitable regularity conditions, the rejection frequencies for any test converge to the nominal level of the test as G → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Formal proofs could be obtained by modifying the arguments in Djogbenou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For the WCU bootstrap methods, the key fact is that the modified empirical score vectors ´sg computed using (31) or (38) are asymptotically equal to the ordinary empirical score vectors ˆsg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For the WCR bootstrap methods, the key fact is that the modified restricted empirical score vectors ˙sg defined in (34) are asymptotically equal to the ordinary restricted empirical score vectors ˜sg in (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' 6 Monte Carlo Simulations Previous simulation results in MacKinnon and Webb (2017, 2018), Brewer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (2018), Djog- benou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (2019), MacKinnon (2022), and several other papers have shown that the reliability of both bootstrap and asymptotic methods for cluster-robust inference depends heavily on the number of clusters, the extent to which cluster sizes vary, and (in the case of treatment effects) both the number of treated clusters and their sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Many of our experiments therefore focus on these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The model we consider is ygi = β1 + k � j=2 βjXjgi + ugi, g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , G, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , Ng, (39) 16 where the ugi are generated by a normal random-effects model with intra-cluster correlation ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The way in which the k −1 non-constant regressors are generated varies across the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The hypothesis to be tested is that βk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In most of our experiments, there are N = 400G observations, which are divided among the G clusters using the formula Ng = � N exp(γg/G) �G j=1 exp(γj/G) � , g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' , G − 1, (40) where [x] means the integer part of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The value of NG is then set to N − �G−1 g=1 Ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The key parameter here is γ, which determines how uneven the cluster sizes are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' When γ = 0 and N/G is an integer, (40) implies that Ng = N/G for all g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For γ > 0, cluster sizes vary more and more as γ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The largest value of γ that we use is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In that case, when G = 24 and N = 9600, the largest cluster (1513 observations) is about 47 times as large as the smallest cluster (32 observations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In contrast, when γ = 2, the largest cluster (899 observations) is just under seven times as large as the smallest (130 observations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The sample sizes that we employ are unusually large for experiments of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Since cluster-robust inference is often used with samples that have hundreds of thousands or even millions of observations, we want our results to apply to such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In preliminary experiments, we found that the results tended to change slightly, but systematically, as small values of N/G were increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Results for N/G > 400 are very similar to ones for N/G = 400, so we use 400 in all the experiments based on (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Because the bootstrap samples are generated using scores, the cost of the experiments increases much less than proportionally with N/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' All experiments use 400,000 replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This number is so large that experimental ran- domness is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The most important determinant of computational cost is k, the number of regressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' As can be seen from (24) and (34) or (38), generating each bootstrap sample in- volves O(k2G) operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' So does calculating the test statistics using either CV1 or CV3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Thus the experiments can be somewhat costly when k is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Nevertheless, many of our experiments involve k ≥ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We do this because results in MacKinnon (2022) suggest that the performance of many methods of inference deteriorates as k increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Previous Monte Carlo experiments, which often use k ≤ 3, may therefore have tended to give too optimistic a picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' It might seem that substantial savings could be achieved by partialing out all regressors except the one(s) of interest prior to performing the bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, this trick only works in certain special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For methods based on the jackknife, it is easy to see the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' If we were to partial out some of the regressors prior to computing the delete-one-cluster estimates in (15), then the computed ˆβ(g) would depend on the values of the partialed-out regressors for the full sample, including those in the g th cluster which was supposed to be deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Consequently, the values of the delete-one-cluster estimates would be incorrect if we partialed out any regressor 17 that affects more than one cluster (such as industry-level fixed effects with firm-level clustering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' An important exception is when the regressors that are partialed out are cluster fixed effects or fixed effects at a finer level (such as firm-level fixed effects with industry-level clustering), because each of them affects only some or all of the observations within a single cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In fact, it is essential to partial out fixed effects of this type, because the coefficient on any regressor that is non-zero only for the g th cluster cannot be identified from a sample which omits that cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='1 Test Size The experiments in this subsection deal with rejection frequencies under the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We consider both asymptotic tests based on the t(G−1) distribution and the wild cluster bootstrap tests listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Figure 2 focuses on variation in cluster sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In these experiments, there are always 9600 observations, 24 clusters, and 10 regressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Cluster sizes vary according to (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Regressors 2 through k − 1 in (39) are normally distributed according to a random-effects model that yields intra-cluster correlations of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The test regressor either follows the same normal distribution as the others (in the three panels on the left), or a χ2(1) distribution (in the three panels on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In the latter case, it is obtained by squaring a normally distributed random variable that is generated by the same random-effects model as the other regressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The disturbances are also generated by such a model, but with ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We focus on rejection frequencies for a test that βk = 0 at the 5% level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The results for asymptotic tests, based on the t(23) distribution and shown in Panels (a) and (b), are striking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Note that a square-root transformation has been applied to the vertical axis to prevent these panels from being too tall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Tests based on CV1 over-reject substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The extent of the over-rejection increases with γ, and, except for γ = 4, it is more severe in Panel (b) than in Panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' A regressor that follows the χ2(1) distribution necessarily has some extreme values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' These become points of high leverage, which makes inference more difficult in Panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Although tests based on CV2 always reject considerably less often than ones based on CV1, they also over-reject significantly and to an extent that increases with γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In contrast, tests based on CV3 and CV3J either under-reject slightly all the time, in Panel (a), or under-reject very slightly for larger values of γ, in Panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The results for CV3 and CV3J are extremely similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The latter always rejects more often than the former, because the difference between (17) and (18) is the positive semi-definite matrix ((G − 1)/G)( ˆβ − ¯β)( ˆβ − ¯β)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Since CV3 tends to under-reject slightly in Figure 2, it might seem that CV3J is to be preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, as we shall see, there are also many cases in which CV3 over-rejects, and CV3J therefore over-rejects slightly more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In practice, it would be perfectly reasonable to report either CV3 or CV3J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We never encountered a case in which it made any real difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The results for the WCR bootstrap tests, shown in Panels (c) and (d), are surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In 18 Figure 2: Rejection frequencies as a function of γ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='5 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' WCU-S γ Rej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (f) χ2(1) Regressor, WCU Bootstraps Notes: The vertical axes show rejection frequencies for tests of βk = 0 in (39) at the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='05 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Results are based on 400,000 replications, with B = 399 bootstrap samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' There are 24 clusters, 9600 observations, and 10 regressors, with ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The extent to which cluster sizes vary increases with γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' the past, WCR-C has been the only variant of the WCR bootstrap, and numerous Monte Carlo experiments have suggested that it is the procedure of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' But WCR-B performs notably better than WCR-C for every value of γ, and both WCR-V and WCR-S perform better still.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Note that, although these two procedures perform almost the same here, this is not true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Oddly, all the WCR procedures perform better in Panel (d), where the test regressor is highly skewed, than they do in Panel (c), where it is Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The rather mediocre performance of WCR-C must be due, at least in part, to the fact that k = 10, which is a larger number than 19 Figure 3: Rejection frequencies as a function of k 2 4 6 8 10 12 14 16 18 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' WCU-S k Rej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (f) χ2(1) Regressor, WCU Bootstraps Notes: The vertical axes show rejection frequencies for tests of βk = 0 in (39) at the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='05 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Results are based on 400,000 replications, with γ = 2, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='10, and B = 399 bootstrap samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' There are 24 clusters, 9600 observations, and k regressors, where k varies from 2 to 20 by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' has been used in most previous experiments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see Figure 3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Some of the results for the WCU bootstrap tests, shown in Panels (e) and (f), are also sur- prising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' It is not a surprise that WCU-C rejects more often than WCR-C or that its perfor- mance is much worse in Panel (f) than in Panel (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, the fact that the other three WCU procedures over-reject much less often than WCU-C may well be surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In both pan- els, WCU-B is clearly the procedure of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' WCU-V and WCU-S perform much better than 20 WCU-C, but worse than WCU-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In Panels (c) and (d), the differences between WCU-V and WCU-S are small, but larger than the differences between WCR-V and WCR-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Figure 3 is similar to Figure 2, but the number of regressors k is now on the horizontal axis, and γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In Panels (a) and (b), CV1 over-rejects to an increasing extent as k increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' So does CV2, although it always over-rejects considerably less than CV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In contrast, CV3 and CV3J over-reject modestly for small values of k and under-reject modestly for large ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Panels (c) and (d) look a lot like the same panels in Figure 2, even though what is on the horizontal axis is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' WCR-C performs quite well for very small values of k, but it over- rejects more and more severely as k increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' WCR-B performs much better than WCR-C, but WCR-V and WCR-S perform even better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In Panel (d), where the test regressor is highly skewed, they both perform extremely well for all values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Panels (e) and (f) also look a lot like the same panels in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' WCU-C performs quite poorly, over-rejecting more and more severely as k increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In contrast, WCU-B performs quite well in Panel (e) and fairly well in Panel (f), and there is no tendency for its performance to deteriorate as k increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' As before, the two other bootstrap methods generally perform much better than WCU-C but slightly worse than WCU-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In the next set of experiments, we focus on what happens as G increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Figure 4 shows rejection frequencies as functions of G, which varies from 12 to 84 by 6, and implicitly also N, since N = 400G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In these experiments, γ = 2 and k = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We report results for only five methods, instead of twelve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We omit CV1 and CV2, because they never perform very well, and CV3J because it is almost identical to CV3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Among the restricted bootstrap methods, we report WCR-C, because it was until now the procedure of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We also report WCR-S and WCR-B, but we do not report WCR-V, because it yields results nearly identical to those of WCR-S and is harder to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Among the unrestricted bootstrap methods, we report only WCU-B, because it always seems to outperform the other WCU methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In Panel (a), using CV3 with the t(G−1) distribution under-rejects quite noticeably for very small values of G, but it performs extremely well for G ≥ 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The bootstrap methods always over-reject, with WCR-C always the worst of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For G ≥ 42, however, all the bootstrap methods perform very well, with WCR-S the winner by a tiny margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Panel (b) is more interesting than Panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The extreme skewness of the χ2(1) regressor apparently affects the results quite a bit, even when G = 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Although it under-rejects for small values of G, using CV3 with the t(G−1) distribution over-rejects for larger values, where it is the worst method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We note that G = 24 in Figures 2 and 3 is near where the curve for CV3 crosses the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='05 line in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The best method is WCR-S in every case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' It performs remarkably well for G ≥ 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, all three WCR methods perform well for the larger values of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The only bootstrap method that does not perform particularly well for these values is WCU-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' By most standards, of course, every method shown in Panel (b) of Figure 4 works very well, unless G is 21 Figure 4: Rejection frequencies as a function of G 12 18 24 30 36 42 48 54 60 66 72 78 84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='09 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' WCR-S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' WCU-B G Rej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (b) χ2(1) Regressor Notes: The vertical axes show rejection frequencies for tests of βk = 0 in (39) at the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='05 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Results are based on 400,000 replications, with γ = 2, k = 10, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='10, and B = 399 bootstrap samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' There are between 12 and 84 clusters, all multiples of 6, with 400 observations per cluster on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' less than about 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For G = 84, CV3 is the worst method, but even it rejects only 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='49% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For comparison, CV1 rejects 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='04% of the time, and CV2 rejects 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The best method, WCR-S, rejects 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='97% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Many applications of cluster-robust inference involve treatment at the cluster level, and existing methods generally perform very poorly when either the number of treated clusters or the number of control clusters is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Using CV1 with the t(G − 1) distribution or WCU-C leads to severe over-rejection, and using WCR-C leads to severe under-rejection (MacKinnon and Webb, 2017, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Our next set of experiments therefore focuses on the model ygi = β1 + Zgiβ2 + βkxg + ugi, (41) where xg is a treatment dummy, Zgi is a row vector of other regressors, and ugi is generated by a random-effects model with intra-cluster correlation ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The treatment dummy equals 1 for G1 of the G clusters and 0 for the remaining G0 = G − G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The clusters that are treated are chosen at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The Zgi consist of eight more dummy variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For each of these variables and each cluster, a probability πg between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='25 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='75 is chosen at random for each replication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Then each observation for that variable in that cluster equals 1 with probability πg and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Thus all the regressors are dummies, which vary at the individual level in a way that varies across clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Figure 5 shows rejection frequencies based on the t(G − 1) distribution for six cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In the left-hand column, there are 12 clusters and 4800 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In the right-hand column, there are 24 clusters and 9600 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The value of γ is 0 in the top row, 2 in the middle row, and 4 in the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The number of treated observations G1 varies between 2 and G−2 on 22 Figure 5: Rejection frequencies based on t(G − 1) distribution for treatment regression 2 3 4 5 6 7 8 9 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='. G1 Rej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (f) G = 24, γ = 4 Notes: The vertical axes, which have been subjected to a square-root transformation, show rejection frequencies for tests of βk = 0 in (41) at the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='05 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The horizontal axes show G1, the number of treated clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Results are based on 400,000 replications, with k = 10 regressors and ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' There are either 12 or 24 clusters, with 400 observations per cluster on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Treated clusters are chosen at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' the horizontal axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' It would have been impossible to set G1 = 1 or G1 = G − 1, because CV2, CV3, and CV3J cannot be computed in those cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For the jackknife-based estimators, this is obvious from (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' When there is just one treated cluster, and it happens to be the omitted one, then the coefficient of interest in ˆβ(g) is not identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' As previous work has shown, tests that use CV1 tend to over-reject severely when either 23 Figure 6: Bootstrap rejection frequencies for treatment regression 2 3 4 5 6 7 8 9 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='01 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' G1 Rej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (f) G = 24, γ = 4 Notes: The vertical axes show rejection frequencies for tests of βk = 0 in (41) at the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='05 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The horizontal axes show G1, the number of treated clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Results are based on 400,000 replications, with k = 10, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='10, and B = 399 bootstrap samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' There are either 12 or 24 clusters, with 400 observations per cluster on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' G0 or G1 is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This is evident in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The over-rejection is worst in Panel (f), where both γ and G are largest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' CV2 over-rejects less than CV1, but it still does not work very well, except perhaps for values of G1 near G/2 when γ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see Panels (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In contrast, CV3 and CV3J, which perform almost identically, are much less prone to over-reject than the other two CRVEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' They actually under-reject for values of G1 fairly near G/2 when γ = 0, and they perform very well for values of G1 near G/2 when γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Oddly, CV3 and CV3J over-reject less seriously for extreme values of G1 when γ is large than when γ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' 24 Figure 6 shows results for four bootstrap tests for the same set of experiments as in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' When γ = 0, all three variants of the WCR bootstrap perform almost identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, as γ increases, their performance starts to differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' WCR-S seems to reject least frequently, which is a good thing for intermediate values of G1 and a bad thing for extreme values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In contrast, WCR-B under-rejects least severely for extreme values of G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, for intermediate values, it over-rejects less than WCR-C but more than WCR-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The most surprising results in Figure 6 are the ones for the unrestricted wild bootstraps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We do not report results for WCU-C or WCU-S, because they would have required a much longer vertical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' WCU-C rejects almost 28% of the time in its worst case (G = 24, G1 = 2, γ = 4), and WCU-S rejects over 12% of the time in its worst case (G = 24, G1 = 2, γ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In contrast, WCU-B is arguably the best method overall when G = 12, and it performs very well for intermediate values of G1 when G = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In addition, it never over-rejects as severely as CV3 for extreme values of G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Simulations in Djogbenou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (2019) suggest that many methods work poorly when one cluster is much bigger than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Even when γ = 4, the largest cluster in our experiments is never dramatically larger than all the rest, although this happens quite often in empirical work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For instance, more than half of all incorporations in the United States occur in Delaware (Hu and Spamann, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This implies that studies of the effects of corporate governance based on changes in state laws, where standard errors are clustered by state of incorporation, are likely to encounter severe errors of inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' To investigate this phenomenon, we create artificial samples with 50 clusters based on data for incorporations by year and state from Spamann and Wilkinson (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' There are 205,566 observations, of which 108,538, or 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='80%, are for Delaware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The second-largest cluster is Nevada, with 17,010 or 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='27%, and the smallest is Montana, with 101 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='05%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We perform a set of experiments similar to the ones in Figures 5 and 6 using these artificial samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' There are 10 regressors, generated in the same way as before, with one exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Because investigators are surely aware of whether or not the largest cluster (Delaware) is treated, it is always treated in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The other clusters to be treated (between 1 and 47 of them) are chosen at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Because the largest cluster is always treated, the rejection frequencies are no longer the same for G1 and G − G1 treated clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, since this is a pure treatment model, the results for G1 treated clusters that include Delaware must be the same as the results for G − G1 treated clusters that exclude Delaware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The results in Figure 7 are striking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In Panel (a), using either CV1 or CV2 leads to over- rejection that varies between severe and extreme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Using CV3 and CV3J also leads to over- rejection, but it is much less severe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For between 20 and 41 treated clusters, rejection frequencies are less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In Panel (b), WCU-C over-rejects severely, and WCR-C can either over- reject or under-reject, often severely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In contrast, our new bootstrap methods work remarkably 25 Figure 7: Rejection frequencies when a treated cluster is very large 3 7 11 15 19 23 27 31 35 39 43 47 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='WCR-S G1 (b) Wild Bootstrap Rejection Frequencies Notes: The vertical axes show rejection frequencies for tests of βk = 0 in (41) at the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='05 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Results are based on 400,000 replications, with k = 10, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='10, and B = 399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' There are 205,566 observations and 50 clusters, with cluster sizes proportional to incorporations in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The largest cluster is always treated, and the other clusters are treated at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The number of treated clusters varies from 2 to 14 by 1, from 16 to 36 by 2, and then from 38 to 48 by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The best of them is WCU-B, which always rejects less than 9% of the time and sometimes rejects just about 5% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' WCR-S and WCR-B also perform much better than WCR-C, except when G1 is very large, in which case they under-reject severely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Even though it is based on real data, the distribution of cluster sizes in the experiments reported in Figure 7 is very extreme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Although the performance of CV3 and three of our new bootstrap methods is far from perfect, it is generally very much better than that of existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Thus it appears that jackknife-based methods are remarkably robust to heterogeneity in cluster sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='2 Test Power It is natural to worry that a new test may be less powerful than existing tests, especially when it performs much better under the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In this section, we therefore investigate test power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Studying power is tricky, because it is unreasonable to compare tests that have noticeably different rejection frequencies under the null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' If, for example, an asymptotic test rejects 15% of the time under the null and a bootstrap test based on it rejects 6% of the time, then we would expect the asymptotic test to have substantially more power than the bootstrap test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' But the additional power may be entirely spurious, simply reflecting the finite-sample over-rejection by the asymptotic test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' One way to compare tests with different rejection frequencies under the null is to “size-adjust” them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' But this approach has two serious conceptual difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' First, size-adjusted tests are infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' What do we learn by comparing tests that cannot actually be performed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Second, 26 there are often many ways to size-adjust a given test, and they may yield quite different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The idea of size-adjustment is to base rejection frequencies for tests under the alternative on critical values calculated by simulation under the null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' But, in general, there exists an infinite number of DGPs that satisfy the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' If they all yield the same critical values, then there is no problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' But if they yield different critical values, as will often be the case, then we have to choose which null DGP to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' It seems natural to make the null DGP used for critical values as close as possible to the alternative DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Davidson and MacKinnon (2006) suggests a particular way of doing this, based on the Kullback-Leibler information criterion, but this approach means using a different critical value for each set of values of the parameters under test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' To avoid the difficulties just discussed, we focus on four cases where the tests of interest all perform quite well under the null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' They are treatment experiments similar to the ones in Figures 5 and 6, with G = 24, N = 9600, and k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In Panels (a) and (b), G1 = 12, so that precisely half the clusters are treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In Panels (c) and (d), G1 = 6, so that the effects of having few treated clusters are apparent but not severe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In order to avoid excessive power loss, we use B = 999 for the bootstrap tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We use k = 5 instead of k = 10 partly to reduce computational cost and partly to improve test performance under the null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Figure 8 shows rejection frequencies as a function of βk, the actual coefficient on the treatment dummy in (41), when the null hypothesis is that βk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In Panels (a) and (c), γ = 0, so that every cluster has exactly 400 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In Panel (a), the perfectly balanced case, all five power functions are visually indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In Panel (c), where only six clusters are treated, CV3 has noticeably more power than any of the bootstrap methods, which are all but identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In Panels (b) and (d), cluster sizes vary from 32 to 1513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' All tests are now substantially less powerful than in Panels (a) and (c), because, whenever there is intra-cluster correlation, the information content of a sample declines as the cluster sizes become more variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The most striking result in both panels is that WCU-B has noticeably less power than any of the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This is especially true in Panel (d), where WCU-B over-rejects modestly under the null but becomes by far the least powerful method for larger values of βk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The pattern for CV3 is similar but much less pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Under the null hypothesis, it over-rejects slightly under the null in Panel (b) and noticeably in Panel (d), with rejection frequencies of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0612 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0775, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' But for large enough values of βk, it has less power than WCR-C and WCR-S, especially in Panel (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The latter two methods also have slightly more power than WCR-B in Panel (b) and noticeably more in Panel (d) for large values of βk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Interestingly, WCR-V, which for clarity is not shown in the figure, has somewhat less power than either WCR-C or WCR-S in Panels (b) and (d) where cluster sizes vary a lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In contrast, it is almost indistinguishable from both these methods in Panels (a) and (c) where cluster sizes are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Based on these results, the procedure of choice appears to be WCR-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For larger values of βk, 27 Figure 8: Power functions for several tests 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' WCR-S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='.WCU-B (d) Variable-Sized Clusters, G1 = 6, γ = 4 Notes: The vertical axes show rejection frequencies for tests at the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='05 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Results are based on 400,000 replications, with G = 24, N = 9600, k = 5, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='10, and B = 999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The hypothesis being tested is βk = 0 in (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The horizontal axes show the values of β in the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' it is always one of the two most powerful tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' WCR-C has similar power, and it also works well under the null in these experiments, but it is much more prone to over-reject than WCR-S in Figures 3, 4, 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Happily, WCR-S is already available in computationally efficient packages for Stata, R, and Python;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='3 Confidence Intervals Cluster-robust standard errors and bootstrap methods are often used to form confidence inter- vals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Although we do not perform any Monte Carlo experiments explicitly to study the proper- ties of confidence intervals, these can be inferred from Figure 8 and the results in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' 28 Most confidence intervals are implicitly or explicitly obtained by inverting a hypothesis test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' When such a test has approximately the correct rejection frequency, the resulting confidence in- terval must have approximately correct coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Similarly, when such a test has high power, the resulting confidence interval must be relatively short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In many of the experiments in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='1, tests based on CV3 and the t(G − 1) distribution are much less prone to over-reject than tests based on CV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This suggests that the coverage of confidence intervals based on CV3 standard errors will often be much better than the coverage of ones based on CV1 standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Even more reliable intervals may often (but not always) be obtained by using the WCR-S or WCR-B bootstraps, which perform much better than the classic WCR-C bootstrap in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The WCU-B bootstrap also performs well in many cases under the null, but the results in Panels (b) and (d) of Figure 8 suggest that, when cluster sizes vary a lot, intervals based on it may be longer than ones based on WCR-B, which in turn may be slightly longer than ones based on WCR-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The WCR-S bootstrap has excellent performance in many of the experiments of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='1, seems to have slightly better power than WCR-B in Panels (b) and (d) of Figure 8, and is easy to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Therefore, we tentatively recommend that confidence intervals should be obtained by inverting WCR-S bootstrap tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, inverting WCR-B bootstrap tests, or simply using CV3 standard errors and the t(G − 1) distribution, would often lead to very similar intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Of course, it is easier to obtain a confidence interval by using a standard error and the t(G − 1) distribution than by inverting a bootstrap test, and it is easier to invert any form of WCU bootstrap test than any form of WCR bootstrap test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, the computational cost of inverting WCR bootstrap tests can be remarkably small, even for very large samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see Roodman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (2019, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='5) and MacKinnon (2022, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' 7 Empirical Examples In this section, we consider three empirical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' These suggest that the new bootstrap procedures proposed in Section 5 may sometimes yield results very similar to those from the existing WCR-C and WCU-C procedures, but they may also yield results which differ noticeably from those and from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='1 Minimum Wages and Hours Worked Our first example is based on MacKinnon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (2022a, Section 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' It exploits differences in the minimum wage across states and years to estimate the impact of minimum wages on hours worked for teenagers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The data on hours at the individual level from the American Community Survey (ACS) are obtained from IPUMS (Ruggles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=', 2020) and cover the years 2005–2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The minimum wage data come from Neumark (2019) and are collapsed to state-year averages to match the ACS frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We restrict attention to teenagers aged 16–19, keeping only individuals who are 29 Table 2: Example 1, minimum wages and hours worked Estimate Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' error t-statistic P value HC1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='15389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='02825 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='4471 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0000 CV1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='15389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='06231 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='4697 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0170 CV3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='15389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='06713 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='2925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0261 Wild cluster bootstrap P values WCR-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0362 WCU-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0207 WCR-V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0352 WCU-V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0186 WCR-S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0374 WCU-S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0227 WCR-B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0371 WCU-B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0203 Notes: There are 492,827 observations, 51 clusters, and 79 coefficients, including state and year fixed effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The coefficient of interest is β in (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Bootstrap P values use B = 999,999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' children of the respondent to the survey and who have never been married.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We drop individuals who had completed one year of college by age 16 and those reporting in excess of 60 hours usually worked per week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We also restrict attention to individuals who identify as either black or white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' There are 492,827 observations in 51 clusters, which correspond to all 50 states plus the District of Columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The model we estimate is yist = α + βmwst + Zistγ + δs + ηt + uist, (42) where yist is usual hours worked per week for individual i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The parameter of interest is β, which is the coefficient on mwst, the minimum wage in state s at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The row vector Zist collects a large set of individual-level controls, including race, gender, age, and education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' There are also state and year fixed effects, denoted by δs and ηt, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' As MacKinnon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (2022a) discusses, clustering could in principle be done at several different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, the one that is most appealing and seems to be supported by the data is clustering at the state level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The 51 clusters vary considerably in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The smallest has 258 observations, and the largest has 35,995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The ratio of these numbers is more than twice as large as for γ = 4 in the experiments of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The mean number of observations per cluster is 9,663, and the median is 7,082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This suggests that inference based on CV1 and the t(50) distribution may not be reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Other measures of cluster heterogeneity, which are discussed in the original paper, lead to the same conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Table 2 presents our key results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' As expected, the CV3 t-statistic is somewhat smaller than the CV1 t-statistic, and the P value based on the t(50) distribution is therefore somewhat larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The four WCR P values are larger than either of them, but still below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='05, and the four WCU P values are notably smaller than the WCR ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Because B is so large, larger than would normally be needed, the simulation standard errors for the WCR bootstrap P values are 30 about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Based on how similar the four WCR P values are, and on how well many of the WCR methods perform in the experiments of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='1, we tentatively conclude that the “true” P value for the test of β = 0 is probably between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='034 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Thus the null hypothesis can safely be rejected at the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='05 level but not at the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='01 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='2 Political Turnover and Test Scores The second example comes from Akhtari, Moreira and Trucco (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This paper examines the impact of political turnover on the quality of public services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Specifically, it examines several outcomes following close mayoral elections in Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' One of these outcomes is the test scores of fourth-grade students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The paper uses a regression discontinuity design to identify the treated and control municipalities, but it conducts the analysis using OLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We replicate one such regression, found in Table 3, Column 5 of the original paper: scoreimt+1 = α + βI(IVMmt < 0) + γIVMmt + δI(IVMmt < 0)IVMmt + ηscoreimt + ϵimt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (43) The dependent variable is the test score one year after an election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' IVMmt is the incumbent vote margin in the close election which occurs in year t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Accordingly, the treatment variable is I(IVMmt < 0), which equals 1 when the incumbent party loses the election and a turnover occurs, and the coefficient of interest is β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This regression is estimated using a sample which is determined by a selected bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' While the paper considers several bandwidths, we focus on the bandwidth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='110, as this results in the largest sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The paper clusters the standard errors at the municipality level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Since there are 2101 munici- palities, many of them located close to each other, it seems possible that this level of clustering is too fine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We therefore consider state-level clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, there are only 26 states in Brazil, and they vary in size from 420 to 64,953 with partial leverages from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='000234 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='179318 (Mac- Kinnon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' With this much heterogeneity across clusters, relying on CV1 may be risky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Table 3 presents our key results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' As expected, the CV1 standard error for clustering by state is smaller than the CV3 standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Contrary to our expectations, however, both are a bit smaller than the CV1 standard error for clustering by municipality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The four WCR P values are similar to each other and to the P value based on the CV1 t-statistic and the t(25) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Surprisingly, the four WCU P values are noticeably larger than the WCR ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Nevertheless, since every test rejects at the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='05 level, there seems to be rather strong evidence against the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='3 Patronage in the British Empire The third example is taken from Xu (2018), which explores the effect of patronage in the colonial era of Britain on the appointment of governors to colonies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Part of the analysis examines whether the extent to which the current secretary of state and a governor are “connected” led to more 31 Table 3: Example 2, political turnover and test scores Estimate Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' error t-statistic P value HC1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='06684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='00528 −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='6616 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0000 CV1 (munic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=') −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='06684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='02430 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='7505 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0060 CV1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='06684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='02204 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0326 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0056 CV3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='06684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='02411 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='7722 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0104 Wild cluster bootstrap P values WCR-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0047 WCU-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0193 WCR-V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0057 WCU-V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0235 WCR-S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0046 WCU-S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0212 WCR-B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0056 WCU-B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0236 Notes: There are 429,979 observations, 26 clusters, and 5 coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The coefficient of interest is β in (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Bootstrap P values use B = 999,999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' desirable colony postings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We replicate the results of one such regression, found in Table 3, Column 3 of the original paper: log(revenue)ist = α + β1connectedit + β2colonies servedit + γi + τt + δit + ϵist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (44) Here log(revenue)ist is the initial revenue for colony s when governor i was appointed in year t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The main variable of interest is connectedit, which is a binary variable set equal to 1 when the governor and the secretary share connections such as having attended the same elite boarding school, or Oxford or Cambridge, or both being in the aristocracy, or having shared ancestry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The variable colonies servedit is the number colonies in which the governor has served up to the year of appointment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The regression also has fixed effects for governors (γi), years (τt), and the duration of the governorship (δit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The paper clusters the standard errors at the bilateral pair (or dyad) level between the secretary of state and the governor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, the dependent variable is observed at multiple times for each colony, so it seems likely that there would be dependence across observations for the same colony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The regression does not include colony fixed effects, which would have reduced this dependence, because, with so many other fixed effects, it was impossible to include them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Thus, it seems plausible that the standard errors should be clustered at the colony level instead of the dyad level, and we investigate this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Switching from dyadic clustering to clustering by colony actually reduces the CV1 standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, even though there are 70 colonies, they are quite unbalanced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' the number of observations per colony ranges from 4 to 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Partial leverages also vary greatly, and they seem to be roughly proportional to cluster sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Perhaps in consequence, the CV3 standard error is 47% larger than the CV1 one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In view of the dramatic difference between the CV1 and CV3 t-statistics, the various wild 32 Table 4: Example 3, patronage in the British empire Estimate Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' error t-statistic P value HC1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='17722 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='07573 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='3401 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0193 CV1 (dyadic) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='17722 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='09933 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='7842 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0750 CV1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='17722 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='08702 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0366 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0455 CV3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='17722 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='12810 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='3834 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='1710 Wild cluster bootstrap P values WCR-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0535 WCU-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0575 WCR-V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0704 WCU-V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0738 WCR-S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0656 WCU-S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0725 WCR-B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0621 WCU-B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0678 Notes: There are 3510 observations, 70 clusters, and 573 coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The coefficient of interest is β1 in (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Bootstrap P values use B = 99,999 because, with 573 regressors, the computations for WCR/WCU-V and WCR/WCU-B are much more expensive than for the previous examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' bootstrap methods provide valuable information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The bootstrap P values are all somewhat larger than the one for the CV1 t-statistic based on the t(69) distribution, but they are all much smaller than the corresponding one for the CV3 t-statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The smallest bootstrap P value is the one for the classic WCR-C method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' At 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0535, it is not much larger than the one based on the t(69) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Surprisingly, every WCU P value is larger than the corresponding WCR P value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This example is deliberately extreme, because the number of regressors (573) is unusually large relative to the number of observations (3510).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Perhaps in consequence, the full coefficient vectors ˆβ(g) are not identified for 61 out of the 70 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, since the ˆβ(g) 1 coefficients are always identified, we used a generalized inverse to compute both CV3 and the bootstrap DGPs for the WCR/WCU-S and WCR/WCU-B bootstraps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The alternative approach of trying to estimate a variance matrix based on just 9 out of 70 clusters seems very dubious, and it yields an implausibly small standard error of just 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Because k is so large in this example, we suspect that the t-test based on CV3 may be prone to under-reject, and that both the t-test based on CV1 and the WCR-C bootstrap test may be prone to over-reject;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Nevertheless, the P values for the new WCR bootstrap methods are only modestly larger than the WCR-C P value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The fact that all the bootstrap P values lie between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0535 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='0738 suggests that the “true” P value probably also lies within, or at least not too far outside, this interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We conclude that there seems to be only weak evidence against the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' 8 Conclusion and Recommendations The classic CV1 estimator given in (6) is by far the most popular CRVE for linear regression models, but standard errors based on it are often much too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The cluster jackknife estimator, 33 often called CV3, has been known for many years but is much less widely used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In Section 3, we discuss how to compute CV3 in a computationally efficient fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Except when all clusters are tiny, this is the fastest available method for computing it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Inference based on CV3 and the Student’s t(G − 1) distribution seems to be much more reliable than inference based on CV1 and that distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This accords with theoretical results in Hansen (2022), which provides no simulations and cites the ones in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Although combining CV3 standard errors and the t distribution often works well, it does not always do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Bootstrap methods may well perform better, and they also provide a valuable robustness check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In Section 5, we prove some simple, but by no means obvious, algebraic results about the relationship between cluster jackknife estimates and score vectors at the cluster level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' These results allow us to obtain new and easy-to-compute variants of the wild cluster bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' These typically perform better than the classic variants, now called WCR-C and WCU-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' The eight new and existing variants are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Of these, the ones that use CV1 together with modified bootstrap score vectors, called WCR-S and WCU-S, are particularly easy to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' They have already been incorporated into packages for Stata and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Prior to this paper, there were already quite a few methods for inference in linear regression models with clustered disturbances (MacKinnon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=', 2022a), and Section 5 has added six new variants of the wild cluster bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Empiricists may reasonably ask what methods they should use in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' As discussed in detail in MacKinnon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' (2022a,b), the first thing to do is to investigate the clustering structure of the model and dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' For instance, it is good practice to calculate the effective number of clusters (Carter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=', 2017) as well as various measures of leverage and influence at the cluster level (MacKinnon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' When these measures indicate that clusters are well-balanced, and the (effective) number of clusters is large (say, more than 100), CV1 and CV3 should yield very similar standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In such cases, it is probably safe to rely on CV3 standard errors together with the t(G − 1) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, the number of clusters will often be much less than 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Moreover, measures of cluster-level leverage and influence may indicate that clusters are not well-balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This can happen, for example, when cluster sizes vary a lot, when there are few treated clusters, or when the distributions of key regressors vary greatly across clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In such cases, CV1 and CV3 can yield quite different standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Recall Table 4, where the CV3 standard error is 47% larger than the CV1 standard error, even though there are 70 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This may be because so many singularities are encountered when computing the omit-1-cluster estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Whenever CV1 and CV3 differ substantially, bootstrap P values or confidence intervals are likely to be more reliable than conventional ones based on either of those CRVEs, and it is probably a good idea to compute both WCR-C and WCR-S P values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In most cases, it is advisable to compute wild cluster bootstrap P values and/or confidence intervals using a large number of bootstrap samples, such as 9,999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This is usually not computa- 34 tionally difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' However, there might be exceptions when either the number of clusters or the number of regressors is unusually large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Of course, when the number of clusters is very large, the bootstrap will not be needed unless the clusters are severely unbalanced, but that can happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' If we had to recommend just one method, it would be the WCR-S bootstrap proposed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' This method uses ordinary CV1 standard errors, which makes it easy to compute, but the bootstrap DGP employs restricted scores that have been transformed using the cluster jackknife.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' In some of our experiments, the WCR-S bootstrap works substantially better than the classic (and popular) WCR-C bootstrap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' see, in particular, Figures 2–4 and Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' We generally do not recommend using the unrestricted wild cluster bootstrap, except perhaps as a robustness check or when it is desired to generate a large number of confidence intervals using just one set of bootstrap samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' References Akhtari M, Moreira D, Trucco L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Political turnover, bureaucratic turnover, and the quality of public services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' American Economic Review 112: 442–493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE3T4oBgHgl3EQfcgrr/content/2301.04527v1.pdf'} +page_content=' Bell RM, McCaffrey DF.' metadata={'source': 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+++ b/_NA0T4oBgHgl3EQfPf82/content/tmp_files/2301.02175v1.pdf.txt @@ -0,0 +1,725 @@ +Jet Energy Scale and Resolution Measurements in CMS +Garvita Agarwal𝑎,∗ +𝑎University at Buffalo - State University of New York, +210 Talbert Hall, Buffalo, NY 14260, United States +E-mail: garvitaa@buffalo.edu +Measurements of jet energy scale (JES) and resolution (JER) are presented, based on the legacy reconstruction +of 13 TeV proton-proton collision data collected by the CMS experiment during the LHC Run 2 period from +2016-2018. Precision measurement of JES is of the utmost importance for the vast majority of physics +measurements and searches at CMS. The high pileup, a harsh radiation environment, and time-dependent +variations in detector response and calibration, all make precision JES measurement a challenging task. We +present in-situ derivations of JES and JER based on CMS Run 2 data, as well as on simulated samples using +various advanced techniques. +41st International Conference on High Energy physics - ICHEP2022 +6-13 July, 2022 +Bologna, Italy +∗on behalf of the CMS collaboration +© Copyright owned by the author(s) under the terms of the Creative Commons +Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). +https://pos.sissa.it/ +arXiv:2301.02175v1 [hep-ex] 5 Jan 2023 + +Jet Energy Scale and Resolution Measurements in CMS +Garvita Agarwal +1. +Introduction +Quarks and gluons are produced abundantly in high-energy proton-proton collisions at the LHC. Color +confinement causes the quarks and gluons to fragment and hadronize into a spray of stable particles (𝑐𝜏 > +1 cm) called jets. Proper calibration of jets, i.e. ensuring that the energy and momentum of the reconstructed +jet matches that of the quark/gluon-initiated jet, is extremely crucial for Standard Model (SM) measurements +and Beyond Standard Model (BSM) searches. Furthermore, the achieved calibration precision defines the +accuracy of many measurements and the sensitivity of searches in CMS [1] such as in the very precise +measurement of the top quark mass [2]. +Figure 1: Pileup distribution in data for proton-proton collisions observed during Run 2. [3] +Jet calibration is a challenging task due to time-dependent changes in both the detector response and +calibration and high pileup (PU), which are additional particles originating from secondary proton-proton +interactions in the same and neighboring bunch crossings. During Run 2, on average 29 PU interactions per +bunch-crossing were observed (Figure 1). Several techniques, both at event-level and jet-level, can be used +to limit the impact of PU on jet energy scale and resolution. An overview of jet reconstruction procedure, +PU mitigation methods, and the jet calibration sequence is presented in the following. +2. +Jet Reconstruction +2.1 Event Reconstruction +Particles produced in proton-proton collisions pass through the CMS detector leaving hits in the tracking +system and depositing energies in the electromagnetic and hadronic calorimeters (ECAL and HCAL respec- +tively). The hits in the tracker are seeded, built using pattern recognition, and fitted to recover the trajectory +of the charged particles. In the calorimeters, the energy deposits are reconstructed as pulses, where the +amplitude of the reconstructed pulse corresponds to the measured energy of the particle. However, due to the +finite decay time of the signal in the calorimeters, the total signal contains contributions from the previous +and next bunches (Figure 2). Simultaneous pulse shape fitting is performed for both the ECAL and HCAL +separately to resolve the signal corresponding to the current in-time pulse and to remove contributions coming +from out-of-time pulses. The information from the ECAL and HCAL is combined using the Particle Flow +(PF) [4] algorithm to form clusters. A precise calibration is then performed on these calorimeter clusters to +correctly reconstruct neutral particles with the right energy scale. The reconstructed tracks are linked to PF +clusters to form charged electromagnetic and hadronic candidates. PF clusters without linked tracks form +neutral hadronic and electromagnetic candidates. Muons being minimum ionising particles pass through +2 + +(13 TeV) +Recorded luminosity [fb-'] +CMS +oPP(13 TeV) = 69.2 mb +2016-2018:= +29 +2018:<μ>= 32 +2017:<μ>= 32 +2016:= 23 +T +10 +20 +30 +40 +50 +60 +70 +80 +06 +100 +Mean number of interactions per crossingJet Energy Scale and Resolution Measurements in CMS +Garvita Agarwal +the entire detector and are reconstructed from hits in the inner and outer tracking systems. At this stage, +by combining information from various sub-detectors, a global event description is provided where all final +state particles are identified as a charged hadron, neutral hadron, electron, photon or muon candidates. +Figure 2: Single channel reconstruction in ECAL [5] and HCAL [6]. The dots are the digitized data samples, red +distribution is the fitted in-time pulse, and light blue distributions are fitted out-of-time pulses. +2.2 Event-level PU Mitigation +PU particles produce additional tracks and deposits in the calorimeters which can overlap with that of the +jets. A majority of PU is from charged particles which can be reduced using the charged hadron subtraction +(CHS) method [4] , which removes charged particles originating from PU vertices. This technique, however, +only works within the tracker covered region, and it does not remove neutral PU contribution. Another +complementary technique is pileup per particle identification, or PUPPI [3], where on an event-by-event +basis a probability is calculated for each particle describing the degree to which they are pileup-like. These +weights are then used to re-scale the four-momenta of the particles. As a result, physics objects such as +jets and missing energy, and jet substructure variables such as soft-drop mass [7] and N-subjettiness [8] are +expected to be less susceptible to PU when PUPPI is used. +2.3 Jet Clustering +At CMS, PF candidates are clustered into jets using the anti-k𝑇 [9] algorithm which is infrared and +collinear safe. The default PU mitigation methods for Run 2 were to use CHS for narrow jets and PUPPI for +large-area jets. These large-area jets are used in boosted topologies where jet substructure plays an important +role. The default for Run 3 is to use PUPPI for both narrow and large area jets. +3. +Jet Calibrations +CMS follows a factorised approach to calibrating jets which is explained below. Run 2 legacy recon- +struction results shown in Figures 3 to 6 are of PF+CHS jets clustered using anti-k𝑇 with R = 0.4. +3.1 PU Offset Corrections +The first step in jet calibration is to estimate and subtract the offset energy coming from PU and noise. +In simulation, this is done by taking the average difference in transverse momentum (p𝑇 ) between matched +jets, with and without PU overlay, in QCD multi-jet samples and evaluating it as a function of p𝑝𝑡𝑐𝑙 +𝑇 +, |𝜂|, and +mean number of pileup interactions per crossing (⟨𝜇⟩). Residual offset corrections for data are derived using +the random cone method which takes the average of PF candidate momenta in a randomly placed cone in +3 + +CMs Simulation +(13 TeV) +Energy (GeV) +8 +- Total +Endcap + - In-time +7 +Out-of-time +6 + Observed +5 + = 20 +4 +3 +2 +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Time sampleCMSPreliminary2018 +13TeV +Charge [fC] +Data +Run315645,LS331,Event329952782 +50 +Total fit +IEta-6,IPhi 37,Depth 1 +In-timepulse +40 +Previouspulse +Nextpulse +30 +Baseline +20 +10 +0 +0 +2 +4 +6 +Time sliceJet Energy Scale and Resolution Measurements in CMS +Garvita Agarwal +zero-bias data and simulated samples. The offset contributions from different PF candidates as a function of 𝜂 +for data and simulation are shown in Figure 3 (left). The light red fraction corresponds to the charged hadrons +associated to pileup vertices that are removed by the CHS algorithm. The data-to-simulation scale-factors +for the offset residual corrections are shown in Figure 3 (right). +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +> (GeV) +µ +> / < +T + 100 GeV +jet +T +p +2016 early +2016 late +2017 +2018 +Syst. unc. +Figure 6: JER versus p𝑇 for varying levels of PU, 𝜇 (left). JER data-to-simulation scale-factors vs. |𝜂 𝑗𝑒𝑡 | with the +statistical uncertainty shown at each point, while the total systematic uncertainty (gray band) shown around 1. [10] +References +[1] “The CMS experiment at the CERN LHC. The Compact Muon Solenoid experiment,” JINST, +vol. 3, p. S08004. 361 p, 2008, also published by CERN Geneva in 2010. [Online]. Available: +https://cds.cern.ch/record/1129810 +[2] “A profile likelihood approach to measure the top quark mass in the lepton+jets channel at √𝑠 = 13 TeV,” +CERN, Geneva, Tech. Rep., 2022. [Online]. Available: https://cds.cern.ch/record/2806509 +[3] “Pileup mitigation at CMS in 13 TeV data,” CERN, Geneva, Tech. Rep., 2019. [Online]. Available: +https://cds.cern.ch/record/2683784 +[4] M. Dordevic, “The CMS Particle Flow Algorithm,” EPJ Web Conf., vol. 191, p. 02016. 7 p, 2018. +[Online]. Available: https://cds.cern.ch/record/2678077 +[5] “Reconstruction of signal amplitudes in the CMS electromagnetic calorimeter in the presence of +overlapping proton-proton interactions,” JINST, vol. 15, p. P10002. 44 p, Oct 2020. [Online]. +Available: https://cds.cern.ch/record/2721995 +[6] “HCAL Out Of Time Pileup Subtraction and Energy Reconstruction,” 2018. [Online]. Available: +https://cds.cern.ch/record/2320408 +[7] A. J. Larkoski, S. Marzani, G. Soyez, and J. Thaler, “Soft drop,” Journal of High Energy Physics, vol. +2014, no. 5, may 2014. [Online]. Available: https://doi.org/10.1007%2Fjhep05%282014%29146 +[8] J. +Thaler +and +K. +V. +Tilburg, +“Identifying +boosted +objects +with +n-subjettiness,” +Journal +of +High +Energy +Physics, +vol. +2011, +no. +3, +mar +2011. +[Online]. +Available: +https: +//doi.org/10.1007%2Fjhep03%282011%29015 +[9] M. Cacciari, G. P. Salam, and G. Soyez, “The anti-k𝑡 jet clustering algorithm,” Journal +of High Energy Physics, +vol. 2008, +no. 04, +pp. 063–063, +apr 2008. [Online]. Available: +https://doi.org/10.1088%2F1126-6708%2F2008%2F04%2F063 +[10] “Jet energy scale and resolution measurement with Run 2 Legacy Data Collected by CMS at 13 TeV,” +2021. [Online]. Available: http://cds.cern.ch/record/2792322 +[11] “Jet energy scale and resolution performance with 13 TeV data collected by CMS in 2016-2018,” +2020. [Online]. Available: https://cds.cern.ch/record/2715872 +6 + diff --git a/_NA0T4oBgHgl3EQfPf82/content/tmp_files/load_file.txt b/_NA0T4oBgHgl3EQfPf82/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f96e0fc2cac083e3f3a4b39ac1c97be19e51d2fa --- /dev/null +++ b/_NA0T4oBgHgl3EQfPf82/content/tmp_files/load_file.txt @@ -0,0 +1,283 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf,len=282 +page_content='Jet Energy Scale and Resolution Measurements in CMS Garvita Agarwal𝑎,∗ 𝑎University at Buffalo - State University of New York, 210 Talbert Hall, Buffalo, NY 14260, United States E-mail: garvitaa@buffalo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='edu Measurements of jet energy scale (JES) and resolution (JER) are presented, based on the legacy reconstruction of 13 TeV proton-proton collision data collected by the CMS experiment during the LHC Run 2 period from 2016-2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' Precision measurement of JES is of the utmost importance for the vast majority of physics measurements and searches at CMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' The high pileup, a harsh radiation environment, and time-dependent variations in detector response and calibration, all make precision JES measurement a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' We present in-situ derivations of JES and JER based on CMS Run 2 data, as well as on simulated samples using various advanced techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' 41st International Conference on High Energy physics - ICHEP2022 6-13 July, 2022 Bologna, Italy ∗on behalf of the CMS collaboration © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='it/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='02175v1 [hep-ex] 5 Jan 2023 Jet Energy Scale and Resolution Measurements in CMS Garvita Agarwal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' Introduction Quarks and gluons are produced abundantly in high-energy proton-proton collisions at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' Color confinement causes the quarks and gluons to fragment and hadronize into a spray of stable particles (𝑐𝜏 > 1 cm) called jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' Proper calibration of jets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' ensuring that the energy and momentum of the reconstructed jet matches that of the quark/gluon-initiated jet, is extremely crucial for Standard Model (SM) measurements and Beyond Standard Model (BSM) searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' Furthermore, the achieved calibration precision defines the accuracy of many measurements and the sensitivity of searches in CMS [1] such as in the very precise measurement of the top quark mass [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' Figure 1: Pileup distribution in data for proton-proton collisions observed during Run 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' [3] Jet calibration is a challenging task due to time-dependent changes in both the detector response and calibration and high pileup (PU), which are additional particles originating from secondary proton-proton interactions in the same and neighboring bunch crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' During Run 2, on average 29 PU interactions per bunch-crossing were observed (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' Several techniques, both at event-level and jet-level, can be used to limit the impact of PU on jet energy scale and resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' An overview of jet reconstruction procedure, PU mitigation methods, and the jet calibration sequence is presented in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' Jet Reconstruction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='1 Event Reconstruction Particles produced in proton-proton collisions pass through the CMS detector leaving hits in the tracking system and depositing energies in the electromagnetic and hadronic calorimeters (ECAL and HCAL respec- tively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' The hits in the tracker are seeded, built using pattern recognition, and fitted to recover the trajectory of the charged particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' In the calorimeters, the energy deposits are reconstructed as pulses, where the amplitude of the reconstructed pulse corresponds to the measured energy of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' However, due to the finite decay time of the signal in the calorimeters, the total signal contains contributions from the previous and next bunches (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' Simultaneous pulse shape fitting is performed for both the ECAL and HCAL separately to resolve the signal corresponding to the current in-time pulse and to remove contributions coming from out-of-time pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' The information from the ECAL and HCAL is combined using the Particle Flow (PF) [4] algorithm to form clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' A precise calibration is then performed on these calorimeter clusters to correctly reconstruct neutral particles with the right energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' The reconstructed tracks are linked to PF clusters to form charged electromagnetic and hadronic candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' PF clusters without linked tracks form neutral hadronic and electromagnetic candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=" Muons being minimum ionising particles pass through 2 (13 TeV) Recorded luminosity [fb-'] CMS oPP(13 TeV) = 69." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='2 mb 2016-2018:= 29 2018:<μ>= 32 2017:<μ>= 32 2016:= 23 T 10 20 30 40 50 60 70 80 06 100 Mean number of interactions per crossingJet Energy Scale and Resolution Measurements in CMS Garvita Agarwal the entire detector and are reconstructed from hits in the inner and outer tracking systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' At this stage, by combining information from various sub-detectors, a global event description is provided where all final state particles are identified as a charged hadron, neutral hadron, electron, photon or muon candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' Figure 2: Single channel reconstruction in ECAL [5] and HCAL [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' The dots are the digitized data samples, red distribution is the fitted in-time pulse, and light blue distributions are fitted out-of-time pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='2 Event-level PU Mitigation PU particles produce additional tracks and deposits in the calorimeters which can overlap with that of the jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' A majority of PU is from charged particles which can be reduced using the charged hadron subtraction (CHS) method [4] , which removes charged particles originating from PU vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' This technique, however, only works within the tracker covered region, and it does not remove neutral PU contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' Another complementary technique is pileup per particle identification, or PUPPI [3], where on an event-by-event basis a probability is calculated for each particle describing the degree to which they are pileup-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' These weights are then used to re-scale the four-momenta of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' As a result, physics objects such as jets and missing energy, and jet substructure variables such as soft-drop mass [7] and N-subjettiness [8] are expected to be less susceptible to PU when PUPPI is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='3 Jet Clustering At CMS, PF candidates are clustered into jets using the anti-k𝑇 [9] algorithm which is infrared and collinear safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' The default PU mitigation methods for Run 2 were to use CHS for narrow jets and PUPPI for large-area jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' These large-area jets are used in boosted topologies where jet substructure plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' The default for Run 3 is to use PUPPI for both narrow and large area jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' Jet Calibrations CMS follows a factorised approach to calibrating jets which is explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' Run 2 legacy recon- struction results shown in Figures 3 to 6 are of PF+CHS jets clustered using anti-k𝑇 with R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='1 PU Offset Corrections The first step in jet calibration is to estimate and subtract the offset energy coming from PU and noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' In simulation, this is done by taking the average difference in transverse momentum (p𝑇 ) between matched jets, with and without PU overlay, in QCD multi-jet samples and evaluating it as a function of p𝑝𝑡𝑐𝑙 𝑇 , |𝜂|, and mean number of pileup interactions per crossing (⟨𝜇⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' Residual offset corrections for data are derived using the random cone method which takes the average of PF candidate momenta in a randomly placed cone in 3 CMs Simulation (13 TeV) Energy (GeV) 8 Total Endcap In-time 7 Out-of-time 6 Observed 5 = 20 4 3 2 0 0 1 2 3 4 5 6 7 8 9 Time sampleCMSPreliminary2018 13TeV Charge [fC] Data Run315645,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='LS331,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='Event329952782 50 Total fit IEta-6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='IPhi 37,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='Depth 1 In-timepulse 40 Previouspulse Nextpulse 30 Baseline 20 10 0 0 2 4 6 Time sliceJet Energy Scale and Resolution Measurements in CMS Garvita Agarwal zero-bias data and simulated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' The offset contributions from different PF candidates as a function of 𝜂 for data and simulation are shown in Figure 3 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' The light red fraction corresponds to the charged hadrons associated to pileup vertices that are removed by the CHS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' The data-to-simulation scale-factors for the offset residual corrections are shown in Figure 3 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NA0T4oBgHgl3EQfPf82/content/2301.02175v1.pdf'} +page_content='8 > (GeV) µ > / < T 1 hour to compress a 100M parameter model. Thus, this is unsuitable for 100B- +parameter models. Even the fastest known accurate post-training method, AdaPrune [22], requires a few minutes to +prune a 100M model. Assuming best-case linear runtime scaling, this extrapolates to several hundreds of hours (a few +weeks) of computation for a GPT3-sized Transformer. +Despite extremely large models being a highly active research area for the past several years, to the best of our knowledge, +so far no model with 10+ billion parameters has been accurately pruned to nontrivial amounts of sparsity. This suggests +that scaling up existing methods might actually lead to significantly higher runtime costs than optimistically estimated +in the previous paragraph2 and/or bring other major unexpected challenges like overfitting to calibration data. In this +work, we introduce the first method that is fast enough to run on 100+ billion parameter models in a few hours on a +single GPU and accurate enough to prune them to sparsity levels of 50% to 60% without significant performance drop. +3 +The SparseGPT Algorithm +3.1 +Fast Approximate Reconstruction +Motivation. +As outlined in Section 2, for a fixed pruning mask M, the optimal values of all weights in the mask +can be calculated by solving the sparse reconstruction problem corresponding to each row. However, doing so exactly +requires inverting the Hessian matrix corresponding to the values preserved by the pruning mask Mi for row i, i.e. +computing (HMi)−1, for all rows 1 ≤ i ≤ drow. One such inversion takes O(d3 +col) time, for a total computational +complexity of O(drow · d3 +col) over drow rows. In practical terms, for a Transformer model, this means that the overall +runtime scales with the 4th power of the hidden dimension dhidden and will thus clearly be infeasible to run on the largest +GPT variants. To arrive at a practical algorithm, we need to improve the overall runtime by at least a full factor of +dhidden, corresponding to a > 10000× compute reduction for models with more than 100 billion parameters. We will +achieve this via a series of careful approximations. +2In the context of quantization, there is evidence [47] that optimization steps also have to be scaled with increasing model size for +AdaPrune-like methods, which would lead to quadratic rather than linear runtime scaling. +3 + +Massive Language Models Can Be Accurately Pruned in One-Shot +PREPRINT +Different Row-Hessian Challenge. +The high computational complexity of optimally reconstrucing the unpruned +weights following Equation 3 mainly stems from the fact that solving each row requires the individual inversion of a +O(dcol × dcol) matrix. This is because the row masks Mi are generally different and (HMi)−1 ̸= (H−1)Mi, i.e., the +inverse of a masked Hessian does not equal the masked version of the full inverse. This is illustrated also in Figure 2. If +all row-masks were the same, then we would only need to compute a single shared inverse, as H = 2XX⊤ depends +just on the layer inputs which are the same for all rows. +select & invert +reconstruct +Figure 2: +Illustration of the Row- +Hessian challenge. Rows are pruned +independently, pruned weights are in +white. Hessian information is used for +weight reconstruction. +Since masks +may be different per row, the inverse +computation must be performed inde- +pendently for each row. +Such a constraint could be enforced in the mask selection, but this would +have a major impact on the final model accuracy, as sparsifying weights in +big structures, like entire columns, is known to be much more difficult than +pruning them individually3. The key towards designing an approximation +algorithm that is both accurate and efficient lies in enabling the reuse of +Hessians between rows with distinct pruning masks. We now propose an +algorithm that achieves this in a principled manner. +Equivalent Iterative Perspective. +To motivate our algorithm, we first have +to look at the row-wise weight reconstruction from a different iterative per- +spective, using the classic OBS update [19, 43, 10]. Assuming a quadratic +approximation of the loss, for which the current weights w are optimal, the +OBS update δm provides the optimal adjustment of the remaining weights +to compensate for the removal of the weight at index m: +δm = − +wm +[H−1]mm +· H−1 +:,m incurring error εm = +w2 +m +2[H−1]mm +. +(4) +Since the loss function corresponding to the layer-wise pruning of one row +of W is a quadratic, the OBS formula is exact in this case. Hence, w + δm +is the optimal weight reconstruction corresponding to mask {m}C. Further, +given an optimal sparse reconstruction w(M) corresponding to mask M, we +can apply OBS again to find the optimal reconstruction for mask M′ = +M − {m}. Consequently, this means that instead of solving for a full mask +M = {m1, . . . , mp}C directly, we could iteratively apply OBS to individually +prune the weights m1 up until mp in order, one-at-a-time, reducing an initially +complete mask to M, and will ultimately arrive at the same optimal solution +as applying the standard closed-form linear regression reconstruction with the +full M directly. +Optimal Partial Updates. Applying the OBS update δm potentially adjusts the values of all available parameters (in +the current mask M) in order to compensate for the removal of wm as much as possible. However, what if we wanted +to update only the weights in a subset U ⊆ M, of all remaining unpruned weights? Thus, we could still benefit from +significant error compensation using only weights in U while reducing the cost of applying OBS. +Such a partial update can indeed be accomplished by simply computing the OBS update using HU, the Hessian +corresponding to U, rather than HM, and updating only wU. Importantly, the loss of our particular layer-wise problem +remains quadratic also for U and the OBS updates are still optimal: the restriction to U does not incur any extra +approximation error by itself, only the error compensation might not be as effective, as less weights are available for +adjustment. At the same time, if |U| < |M|, then inverting HU will be a lot faster than inverting HM. We will now +utilize this mechanism to accomplish our goal of synchronizing the masked Hessians across all rows of W. +Hessian Synchronization. In the following, assume a fixed ordering of the input features j = 1, . . . , dcol. Since those +are typically arranged randomly, we will just preserve the given order for simplicity, but any permutation could in +principle be chosen. Next, we define a sequence of dcol index subsets Uj recursively as +Uj+1 = Uj − {j} with U1 = {1, . . . , dcol}. +(5) +In words, starting with U1 being the set of all indices, each subset Uj+1 is created by removing the smallest index from +the previous subset Uj. These subsets also impose a sequence of inverse Hessians (HUj)−1 = ((2XX⊤)Uj)−1 which +we are going to share across all rows of W. Crucially, following [11], the updated inverse (HUj+1)−1 can be calculated +efficiently by removing the first row and column, corresponding to j in the original H, from the inverse of (HUj)−1 in +3For example, structured (column-wise) pruning ResNet50 to 50% structured sparsity without accuracy loss is challenging, even +with extensive retraining [30], while unstructured pruning to 90% sparsity is easily achievable with state-of-the-art methods [6, 39]. +4 + +Massive Language Models Can Be Accurately Pruned in One-Shot +PREPRINT +O(d2 +col) time via one step of Gaussian elimination: +(HUj+1)−1 = +� +(HUj)−1 − +1 +[(HUj)−1]11 +· (HUj)−1 +:,1 (HUj)−1 +1,: +� +1:,1: with (HU1)−1 = H−1. +(6) +Hence, the entire sequence of dcol inverse Hessians can be calculated recursively in O(d3 +col) time, i.e. at similar cost to a +single extra matrix inversion on top of the initial one for H−1. +Once some weight wk has been pruned, it should not be updated anymore. Further, when we prune wk, we want to +update as many unpruned weights as possible for maximum error compensation. This leads to the following strategy: +iterate through the Uj and their corresponding inverse Hessians (HUj)−1 in order and prune wj if j ∈ Mi, for all rows +i. Importantly, each inverse Hessian (HUj)−1 is computed only once and reused to remove weight j in all rows where +it is part of the pruning mask. A visualization of the algorithm can be found in Figure 3. +elimination +update +prune +Figure 3: Visualization of the SparseGPT reconstruction algorithm. Given a fixed pruning mask M, we incrementally +prune weights in each column of the weight matrix W, using a sequence of Hessian inverses (HUj)−1, and updating +the remainder of the weights in those rows, located to the “right” of the column being processed. Specifically, the +weights to the “right” of a pruned weight (dark blue) will be updated to compensate for the pruning error, whereas the +unpruned weights do not generate updates (light blue). +Computational Complexity. The overall cost of the approximate reconstruction process thus consists of three parts: +(a) the computation of the initial Hessian, which takes time Θ(n · d2 +col) where n is the number of input samples used4, +(b) iterating through the inverse Hessian sequence in time O(d3 +col) and (c) the reconstruction/pruning itself. The latter +can be upper bounded by the time it takes to apply (4) to all drow rows of W for all dcol columns in turn, which is +O(dcol · drow · dcol). In total, this sums up to O(d3 +col + drow · d2 +col). For Transformer models, this is simply O(d3 +hidden), +and is thus a full dhidden-factor more efficient than exact reconstruction. This means that we have reached our initial +goal, as this complexity will be sufficient to make our scheme practical, even for extremely large models. +Weight Freezing Interpretation. While we have motivated the SparseGPT algorithm as an approximation to the exact +reconstruction using optimal partial updates, there is also another interesting view of this scheme. Specifically, consider +an exact greedy framework which compresses a weight matrix column by column, always optimally updating all not +yet compressed weights in each step [11, 9]. At first glance, SparseGPT does not seem to fit into this framework as we +only compress some of the weights in each column and also only update a subset of the uncompressed weights. Yet, +mechanically, “compressing” a weight ultimately means fixing it to some specific value and ensuring that it is never +“decompressed” again via some future update, i.e. that it is frozen. Hence, by defining column-wise compression as: +compress(wj)i = 0 if i ̸∈ Mi and wj +i otherwise, +(7) +i.e. zeroing weights not in the mask and fixing the rest to their current value, our algorithm can be interpreted as an +exact column-wise greedy scheme. As we will show later, this perspective allows us to cleanly merge sparsification and +quantization into a single compression pass, as well as inherit some other algorithmic enhancements from post-training +quantization [9]. +3.2 +Adaptive Mask Selection +So far, we have only focused on the reconstruction aspect, i.e. assuming a fixed pruning mask M. How should this +mask be decided? One simple option would be to follow AdaPrune [22] and choose the mask for the whole layer in +4Taking the number of samples n to be a small multiple of dcol is sufficient for good results, even on very large models. +5 + +Massive Language Models Can Be Accurately Pruned in One-Shot +PREPRINT +advance using e.g. the standard magnitude criterion [50] or including also second-order information [10]. However, +recent work [8, 11] has shown that the updates applied during the pruning process change weights significantly due to +correlations, and that taking this into account for the mask selection yields significantly more accurate results. This +insight can be integrated into SparseGPT by adaptively choosing the mask while running the reconstruction pass. +frozen +not yet pruned +p% sparse +Figure 4: Mask selection. +One obvious way of doing so would be to pick the p% easiest weights to prune in +each column i just when it is compressed, which will lead to p% overall sparsity. +At the same time, this approach has one big disadvantage: the sparsity cannot be +distributed non-uniformly across columns. This is a significant restriction, which +will generally make unstructured pruning more difficult. This is particularly +problematic for massive language models, as they are known to have a small +number of highly sensitive outlier features [3, 46]. Further, [11] observe that +some OPT models appear to have a large number of dead RELUs in the earlier +layers, leading to many columns that are trivial to prune. +We propose to alleviate this disadvantage while still exploiting the significant +accuracy gains of adaptive weight selection during pruning via iterative blocking. +More precisely, we always select the pruning mask for Bs = 128 columns at- +a-time based on the OBS reconstruction error ε from Equation (4), using the +diagonal values in our Hessian sequence. We then perform the next Bs updates as +discussed in the previous section, before selecting the mask for the next block, and +so on. This procedure allows non-uniform selection per column, in particular also +using the corresponding Hessian information5, while at the same time considering +also previous weight updates in the selection process. +3.3 +Extension to Semi-Structured Sparsity +While so far we have only discussed unstructured pruning, SparseGPT is also easily adapted to semi-structured patterns +such as the popular n:m sparsity format [49, 22] which delivers speedups in its 2:4 implementation on Ampere NVIDIA +GPUs. Specifically, every consecutive m weights should contain exactly n zeros. Hence, we can simply choose +blocksize Bs = m and then enforce the zeros-constraint in the mask selection for each row by picking the n weights +which incur the lowest error as per Equation (4). A similar strategy could also be applied for other semi-structured +pruning patterns. Finally, we note that a larger Bs would not be useful in this semi-structured scenario since zeros +cannot be distributed non-uniformly between different column-sets of size m. +3.4 +Full Algorithm Pseudocode +Algorithm 1 The SparseGPT algorithm. We prune the layer matrix W to p% unstructured sparsity given inverse +Hessian H−1 = (2XX⊤ + λI)−1, lazy batch-update blocksize B and adaptive mask selection blocksize Bs; each Bs +consecutive columns will be p% sparse. +M ← 1drow×dcol +// 0/1 pruning mask +E ← 0drow×B +// block quantization errors +H−1 ← Cholesky(H−1) +// Hessian inverse information +for i = 0, B, 2B, . . . do +for j = i, . . . , i + B − 1 do +if j mod Bs = 0 then +M:,j:(j+Bs) ← mask of (1 − p)% weights wc ∈ W:,j:(j+Bs) with largest w2 +c/[H−1]cc +end if +E:,j−i ← W2 +:,j / [H−1]jj +// pruning error +E:,j−i ← (1 − M:,j) · E:,j−i +// freeze weights that are not pruned +W:,j:(i+B) ← E:,j−i · H−1 +j,j:(i+B) +// update weights in block +end for +W:,(i+B): ← E · H−1 +i:(i+B),(i+B): +// update all remaining weights +end for +W ← W · M +// set pruned weights to 0 +With the weight freezing interpretation discussed at the end of Section 3.1, the SparseGPT reconstruction can be cast in +the column-wise greedy framework of the recent quantization algorithm GPTQ [9]. This means we can also inherit +5For a single column j, the OBS selection criterion would degrade to just the magnitude, as [H−1]jj is constant across rows. +6 + +Massive Language Models Can Be Accurately Pruned in One-Shot +PREPRINT +several algorithmic enhancements from GPTQ, specifically: precomputing all the relevant inverse Hessian sequence +information via a Cholesky decomposition to achieve numerical robustness and applying lazy batched weight matrix +updates to improve the compute-to-memory ratio of the algorithm. Our adaptive mask selection, as well as its extensions +to semi-structured pruning, are compatible with all of those extra techniques as well. +The pseudocode in Algorithm 1 presents the the unstructured sparsity version of the SparseGPT algorithm in its fully +developed form, integrating also all relevant techniques from GPTQ discussed above. +3.5 +Joint Sparsification & Quantization +Algorithm 1 operates in the column-wise greedy framework of GPTQ, thus sharing the computationally heavy steps +of computing the Cholesky decomposition of H−1 and continuously updating W. This makes it possible to merge +both algorithms into a single joint procedure. Specifically, all weights that are frozen by SparseGPT are additionally +quantized, leading to the following generalized errors to be compensated in the following update step: +E:,j−i ← (W:,j − M:,j · quant(W:,j))2 / [H−1]jj, +(8) +where quant(w) rounds each weight in w to the nearest value on the quantization grid. Crucially, in this scheme, +sparsification and pruning are performed jointly in a single pass at essentially no extra cost over just running SparseGPT. +We emphasize that doing quantization and pruning jointly means that later pruning decisions are influenced by earlier +quantization rounding, and vice-versa. This is in contrast to prior techniques, such as OBC [11], which first sparsify a +layer and then quantize the remaining weights, where quantization consequently has no influence on pruning outcomes. +4 +Experiments +Setup. +We implement SparseGPT in PyTorch [38] and use the HuggingFace Transformers library [45] for handling +models and datasets. All experiments are conducted on a single NVIDIA A100 GPU with 80GB of memory. In this +setup, SparseGPT can fully sparsify the 175-billion-parameter models in approximately 4 hours. Similar to [47, 9], we +sparsify Transformer layers sequentially in order. This significantly reduces memory requirements, and also noticeably +improves accuracy over handling all layers in parallel. All our compression experiments are performed in one-shot, +without any finetuning, following a similar setup to that of recent work on post-training quantization of GPT-scale +models [9, 47, 3]. +For calibration data, following [9], we use 128 2048-token segments, randomly chosen from the C4 [40] dataset. This +represents generic text data crawled from internet and makes sure that our experiments remain actually zero-shot since +no task-specific data is seen during pruning. +Models, Datasets & Evaluation. +We primarily work with the OPT model family as it provides a suite of models +ranging from 125 million to 175 billion parameters, allowing us to study the scaling behavior of pruning relative to +model size. Additionally, we also consider the 176 billion parameter variant of BLOOM [42]. In general, our focus +lies on the very largest variants but we also show some results on smaller models to provide a broader picture, and in +particular ablations with respect to model size. +In terms of datasets and evaluation, we mainly focus on perplexity on the raw WikiText2 test set [31], a popular +benchmark in LLM compression literature [47, 36, 9, 46]. In the Appendix, we also show results on a set of text +segments sampled from the C4 validation set. In general, perplexity is known be a challenging and stable metric that +is well suited for evaluating the accuracy of compression methods [47, 11, 4]. Yet, for additional interpretability, we +also provide some ZeroShot accuracy results for LAMBADA [35], ARC (Easy and Challenge) [2], PIQA [44] and +StoryCloze [33]. +We emphasize that the main focus of our evaluation lies on accuracy of the sparse models, relative to the dense baseline +rather than on absolute numbers. We calculate perplexity in easily-reproducible fashion following the procedure +described by HuggingFace6: we concatenate all samples of the raw dataset with “\n\n” separators, encode and split the +entire sequence into non-overlapping segments of 2048 tokens (the maximum window size of both OPT and BLOOM), +on which the standard average causal language modelling loss is computed. The final perplexity is the exponentiated +version of this result. Different preprocessing may influence absolute numbers, but does not affect our claims, as we +mainly focus on performance relative to the dense model. Our ZeroShot evaluations are performed using GPTQ’s [9] +implementation, which is in turn based on the popular [46, 4] EleutherAI-eval-harness7. We emphasize that all dense +and sparse results were computed with exactly the same code to ensure a fair comparison. +6https://huggingface.co/docs/transformers/perplexity +7https://github.com/EleutherAI/lm-evaluation-harness +7 + +Massive Language Models Can Be Accurately Pruned in One-Shot +PREPRINT +Baselines. +We believe to be the first academic work to perform post-training pruning of massive models. (As discussed +previously, prior post-training pruning techniques have only been applied to models 1000x smaller in size, and do not +scale to GPT model sizes.) As such, there are no standard benchmarks, and we therefore primarily evaluate what levels +of sparsity are achievable with SparseGPT while maintaining close to dense model accuracy. +Nevertheless, we also compare with the highly popular magnitude pruning criterion [50] which drops the weights +of smallest absolute value, applied layerwise. This technique easily scales to extremely large models in terms of +computational efficiency, but, as we show, does not perform well in terms of accuracy. +4.1 +Results +Pruning Difficulty Scaling with Model Size. +In our first set of experiments, we study how the difficulty of sparsifying +LLMs changes with their size. For this, we consider the entire OPT model family and uniformly prune all linear +layers, excluding the embeddings and the head as is standard [41, 25], to 50% unstructured sparsity, full 4:8 or full 2:4 +semi-structured sparsity. (All three correspond to 50% overall sparsity, but the 2:4 pattern is the most stringent, followed +by 4:8 and unstructured sparsity.) The raw-WikiText2 performance numbers are given in Table 1 and visualized in +Figure 1 (right). +OPT +Sparsity +125M +350M +1.3B +2.7B +6.7B +13B +30B +66B +175B +dense +0% +27.66 +22.01 +14.63 +12.46 +10.86 +10.12 +9.56 +9.33 +8.34 +Magnitude +50% +193. +97.80 +1.7e4 +265. +969. +1.2e5 +168. +4.2e4 +4.3e4 +SparseGPT +50% +36.89 +31.60 +17.45 +13.44 +11.56 +11.12 +9.77 +9.33 +8.21 +SparseGPT +4:8 +44.75 +38.82 +20.02 +14.97 +12.54 +11.72 +10.28 +9.66 +8.45 +SparseGPT +2:4 +59.17 +50.21 +24.04 +17.14 +14.16 +12.91 +10.88 +10.10 +8.73 +Table 1: OPT perplexity results on raw-WikiText2. +One immediate finding is that the accuracy of magnitude-pruned models collapses across all scales, with larger variants +generally dropping worse, relative to smaller ones. This is in stark contrast to smaller vision models which can usually +be pruned via simple magnitude to 50% or more at very little loss of accuracy [43, 11]. This highlights the importance +of more accurate pruners in the context of extremely large generative language models, but also the fact that perplexity +is a very sensitive metric. +For SparseGPT, the trend is very different: already at 2.7B parameters, the perplexity loss is < 1 point, at 66B, there is +zero loss and at the very largest scale there is even a slight accuracy improvement over the dense baseline. (As can be +seen in Figure 1, sparse OPT models of ≤ 50% sparsity all have lower perplexity than the dense baseline, although the +differences are small.) +In general, there is a clear trend of larger models being significantly easier to sparsify, which we speculate may be due +to them being more overparametrized and also more noise resistant in general. We think a more detailed investigation +of this phenomenon would be a great topic for future work. For 4:8 and 2:4 sparsity, the behavior is very similar, +but accuracy drops are typically a bit higher due to the sparsity pattern being significantly more constrained [22]. +Nevertheless, at the largest scale, the perlexity increases are only 0.11 and 0.39 for 4:8 and 2:4 sparsity, respectively. +We emphasize that these sparsity pattern can actually achieve 2× speedup in practice [22, 32], with commercially +available NVIDIA Ampere GPUs already including support for 2:4 sparsity. +Sparsity Scaling for 100+ Billion Parameter Models. +Next, we take a closer look at the largest publicly-available +dense models, OPT-175B and BLOOM-176B, and investigate how their performance scales with the degree of sparsity +induced by either SparseGPT or magnitude pruning. The results are visualized in Figures 1 and 5 (both left panels). +For the OPT-175B model, for which the results are presented in Figure 1 (left), magnitude pruning can achieve at +most 10% sparsity before significant accuracy loss occurs; meanwhile, SparseGPT enables up to 60% sparsity at a +comparable perplexity increase. BLOOM-176B, for which the results are provided in Figure 5 (left), appears to be +more favorable for magnitude pruning, admitting up 30% sparsity without major loss; still, SparseGPT can deliver 50% +sparsity, a 1.66× improvement, at a similar level of perplexity degradation. Even at 80% sparsity, models compressed by +SparseGPT still score reasonable perplexities, while magnitude pruning leads to a complete collapse (> 100 perplexity) +already at 40% sparsity for OPT and 60% sparsity for BLOOM, respectively. Remarkably, SparseGPT is able to +remove around 100 billion weights from these models, with limited impact on model accuracy. +8 + +Massive Language Models Can Be Accurately Pruned in One-Shot +PREPRINT +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Sparsity +8 +10 +12 +14 +16 +18 +20 +22 +24 +Perplexity on raw-WikiText2 +BLOOM-176B Uniform Unstructured Sparsity +Magnitude +SparseGPT +Dense +10 +1 +100 +101 +102 +#Params in Billions +10 +20 +30 +40 +50 +Perplexity on raw-WikiText2 +OPT Model Family - Sparse + Quantized +3bit GPTQ +50% + 4bit +Dense +Figure 5: Left: Uniformly compressing BLOOM-176B to various sparsity levels with SparseGPT and magnitude +pruning, respectively. Right: 50% sparsity + 4-bit quantization joint compression vs. 3-bit on the OPT family. +ZeroShot Experiments. +To complement our perplexity evaluations, we now also provide results for various sparsified +variants of OPT-175B on several ZeroShot tasks. ZeroShot evaluations are known to be relatively noisy [3] but at the +same time more interpretable. All numbers are summarized in Table 2. +Method +Sparsity +Lambada +PIQA +ARC-easy +ARC-ch +StoryCloze +Average +Dense +0% +75.59 +81.07 +71.04 +43.94 +79.82 +70.29 +Magnitude +50% +00.02 +54.73 +28.03 +25.60 +47.10 +31.10 +SparseGPT +50% +76.51 +80.03 +69.65 +41.30 +78.87 +69.27 +SparseGPT +4:8 +78.77 +79.16 +68.35 +39.85 +77.02 +68.63 +SparseGPT +2:4 +79.47 +79.16 +67.08 +38.99 +76.19 +68.18 +Table 2: ZeroShot results on several datasets for sparsified variants of OPT-175B. +Overall, a similar trend to the perplexity results seems to hold with magnitude pruned models collapsing to close to +random performance on several datasets while SparseGPT models stay close to the original accuracy. However, as +expected, these numbers are significantly more noisy: for instance, 2:4 pruning appears to achieve noticeably higher +accuracy than the dense model on Lambada despite being the most constrained sparsity pattern investigated here. +Notice also that these effects ultimately average out when considering many different tasks, which is consistent to the +literature [47, 3, 4]. +Joint Sparsification & Quantization. +Finally, another interesting research direction is the combination of sparsity +and quantization, which would allow combining computational speedups from sparsity [26, 5] with memory savings +from quantization [9, 3, 4]. Specifically, if we compress a model to 50% sparse + 4-bit weights, store only the non-zero +weights and use a bitmask to indicate their positions, then this has the same overall memory consumption as 3-bit +quantization. Hence, in Figure 5 (right) we compare SparseGPT 50% + 4-bit with state-of-the-art GPTQ [9] 3-bit +numbers. While there seem to be a few outliers, 50% + 4-bit models are more accurate than their respective 3-bit +versions for several models sizes, including 175B with 8.55 vs. 8.68 3-bit. We also tested 2:4 and 4:8 in combination +with 4-bit on OPT-175B yielding encouraging 9.20 and 8.86 perplexities, which can likely be improved further using +additional quantization tricks such as blocking [9, 4]. +5 +Related Work +Model compression aims to produce more efficient models, using approaches such as pruning, quantization, and +knowledge distillation—we refer the reader to the respective surveys [21, 14, 15] for an in-depth discussion. Since the +focus of our work is on compressing massive models, with 10-100s of billions of parameters, we will mainly focus on +work in this specific area. +9 + +Massive Language Models Can Be Accurately Pruned in One-Shot +PREPRINT +Pruning Methods. To our knowledge, we are the first academic work to investigate pruning of massive GPT-scale +models, e.g. with more than 10 billion parameters. This gap in the literature may seem surprising, given both the +widespread popularity of these models, and the significant amount of existing work on pruning, e.g. [17, 7, 12, 6, +43, 41, 39, 10, 13, 25]. One justification for this gap is the fact that most existing pruning methods, such as gradual +magnitude pruning [16, 17, 12, 24], require extensive retraining following the pruning step in order to recover accuracy, +while GPT-scale models usually require massive amounts of computation and parameter tuning both for training or +finetuning [48], which renders retraining-based approaches difficult to apply. Thus, we are not aware of any work +applying such gradual pruning methods at GPT scale. +SparseGPT is a post-training method for GPT-scale models, as it does not perform any finetuning. So far, post-training +pruning methods have only been investigated at the scale of classic CNN or BERT-type models [22, 11, 27], which have +100-1000x fewer weights than our models of interest. We discussed the challenges of scaling existing post-training +methods to GPT models, and the technical relationship between SparseGPT and these methods, in Section 2. +Post-Training Quantization. By contrast, there has been a significant amount of emerging work on post-training meth- +ods for quantizing GPT-scale models, closely following the first open releases of such models [48, 42]. Specifically, the +ZeroQuant [47], LLM.int8() [3] and nuQmm [36] methods investigated the feasibility of round-to-nearest quantization +for billion-parameter models, showing that 8-bit quantization for weights is feasible via this approach, but that activation +quantization can be difficult due to the existence of outlier features. GPTQ [9] leverages approximate second-order +information for accurate quantization of weights down to 2–4 bits, for the very largest models, and shows that this +can bring inference speedups of 2-5x when coupled with efficient GPU kernels. Follow-up work by Xiao et al. [46] +investigated joint activation and weight quantization down to 8 bits per component, proposing a smoothing-based +scheme which reduces the difficulty of activation quantization and is complemented by efficient GPU kernels for fast +inference. Concurrent work by Park et al. [37] tackles the hardness of quantizing activation outliers via quadapters, a +set of learnable parameters whose goal is to scale activations channel-wise, while keeping the other model parameters +unchanged. Very recent work by Dettmers and Zettlemoyer [4] investigate scaling relationships between model size, +quantization bits, and different notions of accuracy for massive LLMs, observing a high degree of correlation between +perplexity scores and aggregated zero-shot accuracy across tasks, as well as saturation behavior. +Since it focuses on sparsification rather than quantization, SparseGPT is complementary to quantization approaches. +Specifically, as we have shown in Section 3.5, the SparseGPT algorithm can be applied in conjunction with GPTQ, +the current state-of-the-art algorithm for weight quantization, and should be compatible with activation quantization +approaches [3, 46, 37]. Thus, it would be very interesting to investigate in depth how compression errors compound +when quantization and pruning are applied in conjunction. +6 +Discussion +We have provided a new post-training pruning method called SparseGPT, specifically tailored to massive language +models from the GPT family. Our results show for the first time that large-scale generative pretrained Transformer- +family models can be compressed to high sparsity via weight pruning in one-shot, without any retraining, at low loss +of accuracy, when measured both in terms of perplexity and zero-shot performance. Specifically, we have shown the +largest open-source GPT-family models (e.g. OPT-175B and BLOOM-176B) can reach 50-60% sparsity with low +accuracy fluctuations. Surprisingly, this means that more than 100 billion weights from these models can be ignored at +inference time. Central to our approach is a new large-scale approximate sparse regression algorithm, which generalizes +to semi-structured (2:4 and 4:8) patterns, and is also compatible with existing weight quantization approaches. +Interestingly, our method is local: after each pruning step, it performs weight updates, designed to preserve the +input-output relationship for each layer. These updates are computed without any global gradient information. Thus, it +appears that the high degree of parametrization of massive GPT models allows our method to directly identify sparse +accurate models in the “close neighborhood” of the dense pretrained model. Remarkably, since our main accuracy +measure (perplexity) is extremely sensitive, it appears that the output of the generated sparse model correlates extremely +closely with that of the dense model. Our second main finding is that larger models are easier to sparsify: at a fixed +sparsity level, the relative accuracy drop for the sparse model, relative to the dense one, narrows as we increase the +model size, to the point where inducing 50% sparsity results in practically no accuracy decrease on the largest models. +This finding should be seen as very encouraging for future work on compressing such massive models. +One natural avenue for future work would be to investigate fine-tuning mechanisms for such large-scale models, which +would allow further accuracy recovery. We conjecture that this should be possible, and that probably at least 80-90% +sparsity can be achieved with progressive pruning and fine-tuning. Another extension which we plan to investigate is the +applicability of our approaches during training, to reduce the computational cost of pre-training these massive models. +10 + +Massive Language Models Can Be Accurately Pruned in One-Shot +PREPRINT +Acknowledgments +The authors gratefully acknowledge funding from the European Research Council (ERC) under the European Union’s +Horizon 2020 programme (grant agreement No. 805223 ScaleML), as well as experimental support from Eldar Kurtic, +and from the IST Austria IT department, in particular Stefano Elefante, Andrei Hornoiu, and Alois Schloegl. +References +[1] Thomas Blumensath and Mike E Davies. Iterative thresholding for sparse approximations. Journal of Fourier +Analysis and Applications, 14(5-6):629–654, 2008. +[2] Michael Boratko, Harshit Padigela, Divyendra Mikkilineni, Pritish Yuvraj, Rajarshi Das, Andrew McCallum, +Maria Chang, Achille Fokoue-Nkoutche, Pavan Kapanipathi, Nicholas Mattei, et al. 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OPT: Open pre-trained transformer language models. arXiv preprint +arXiv:2205.01068, 2022. +[49] Aojun Zhou, Yukun Ma, Junnan Zhu, Jianbo Liu, Zhijie Zhang, Kun Yuan, Wenxiu Sun, and Hongsheng Li. +Learning N:M fine-grained structured sparse neural networks from scratch. In International Conference on +Learning Representations (ICLR), 2021. +[50] Michael Zhu and Suyog Gupta. To prune, or not to prune: exploring the efficacy of pruning for model compression. +arXiv preprint arXiv:1710.01878, 2017. +13 + +Massive Language Models Can Be Accurately Pruned in One-Shot +PREPRINT +7 +Appendix +7.1 +Additional Results +In addition to our raw-WikiText2 numbers in the main paper, we now also present some OPT-175B and BLOOM-175B +results on a subsample from the C4 dataset consisting of randomly crawled website text. Specifically, we take the first +validation shard and randomly sample 256 2048-token segments (each contained in a single large enough document). +This is exactly the same sampling procedure that is also used for the calibration dataset, the latter is just sampled from +the first training shard. The results are shown in Figure 6. +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Sparsity +8 +10 +12 +14 +16 +18 +20 +22 +24 +Perplexity on C4 (sub) +Uniform Unstructured Sparsity +OPT-175B +BLOOM-176B +Figure 6: C4 perplexity for OPT-175B and BLOOM-176B at various SparseGPT sparsity levels. +In summary, these results confirm our raw-WikiText2 findings from the main paper that SparseGPT is able to achieve +50-60% sparsity with an only very minor perplexity increase. In this direct comparison between models one can also +see how BLOOM’s accuracy starts to degrade slightly more quickly, especially at the higher sparsity levels. +14 + diff --git a/_dAyT4oBgHgl3EQf3vl6/content/tmp_files/load_file.txt b/_dAyT4oBgHgl3EQf3vl6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cde4d39e6f52a4dd5c9262ef17c2f877818453a0 --- /dev/null +++ b/_dAyT4oBgHgl3EQf3vl6/content/tmp_files/load_file.txt @@ -0,0 +1,648 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf,len=647 +page_content='SPARSEGPT: MASSIVE LANGUAGE MODELS CAN BE ACCURATELY PRUNED IN ONE-SHOT PREPRINT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' VERSION JANUARY 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Elias Frantar IST Austria elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='frantar@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='at Dan Alistarh IST Austria & Neural Magic dan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='alistarh@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='at ABSTRACT We show for the first time that large-scale generative pretrained transformer (GPT) family models can be pruned to at least 50% sparsity in one-shot, without any retraining, at minimal loss of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This is achieved via a new pruning method called SparseGPT, specifically designed to work efficiently and accurately on massive GPT-family models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' When executing SparseGPT on the largest available open-source models, OPT-175B and BLOOM-176B, we can reach 60% sparsity with negligible increase in perplexity: remarkably, more than 100 billion weights from these models can be ignored at inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' SparseGPT generalizes to semi-structured (2:4 and 4:8) patterns, and is compatible with weight quantization approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 1 Introduction Large Language Models (LLMs) from the Generative Pretrained Transformer (GPT) family have shown remarkable performance on a wide range of tasks, but are difficult to deploy because of their massive size and computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' For instance, the top-performing GPT-175B model has 175 billion parameters, which total at least 320GB (counting multiples of 1024) of storage in half-precision (FP16) format, leading it to require at least five A100 GPUs with 80GB of memory each for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' It is therefore natural that there has been significant interest in reducing these costs via model compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' To date, virtually all existing GPT compression approaches have focused on quantization [3, 47, 46, 9], that is, reducing the precision of the numerical representation of individual weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' A complementary approach for model compression is pruning, which removes network elements, from individual weights (unstructured pruning) to higher-granularity components such as entire rows/columns of the weight matrices (structured pruning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This approach has a long history [28, 19], and has been applied successfully in the case of vision and smaller-scale language models and tasks [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Yet, the best-performing pruning methods require extensive retraining of the model to recover from the accuracy loss due to removed elements, which is extremely expensive in the case of GPT-scale models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' While some one-shot pruning methods also exist [22, 11], which compress the model without retraining, they are unfortunately too computationally-expensive to be applied to models with billions of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Thus, to date, there is virtually no work on accurate pruning of GPT3-scale models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' In this paper, we propose SparseGPT, the first accurate one-shot pruning method which works efficiently at the scale of models with 10-100 billion parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' SparseGPT works by reducing the pruning problem to an extremely large-scale instance of sparse regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' It is based on a new approximate sparse regression solver, used to solve a layer-wise compression problem, which is efficient enough to execute in a few hours on the largest openly-available GPT models (175B parameters), using a single GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' At the same time, SparseGPT is accurate enough to drop negligible accuracy post-pruning, without any fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' For example, when executed on the largest publicly-available generative language models (OPT-175B and BLOOM-176B), SparseGPT induces 50-60% sparsity in one-shot, with minor accuracy loss, measured either in terms of perplexity or zero-shot accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Our experimental results, for which we provide a snapshot in Figure 1, illustrate the following two key points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' First, as shown in Figure 1 (left), SparseGPT can induce uniform layer-wise sparsity of up to 60% in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' the 175-billion- parameter variant of the OPT family [48], with minor accuracy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' By contrast, the only known one-shot baseline which works at this scale, Magnitude Pruning [16, 18], preserves accuracy only until 10% sparsity, and completely arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='00774v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='LG] 2 Jan 2023 Massive Language Models Can Be Accurately Pruned in One-Shot PREPRINT collapses beyond 30% sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Second, as shown in Figure 1 (right), SparseGPT can also accurately impose sparsity in the more stringent, but hardware-friendly, 2:4 and 4:8 semi-structured sparsity patterns [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Although these patterns tend to lose additional accuracy relative to the dense baseline, especially for the smaller models, these sparsity patterns can be directly exploited to obtain computational speedups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Moreover, as we show later in the paper, the sparsity induced by our technique compounds well with additional compression obtained through quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='8 Sparsity 8 10 12 14 16 Perplexity on raw-WikiText2 OPT-175B Uniform Unstructured Sparsity Magnitude SparseGPT Dense 10 1 100 101 102 #Params in Billions 10 20 30 40 50 60 Perplexity on raw-WikiText2 OPT Model Family - SparseGPT 2:4 4:8 50% Unstructured Dense Figure 1: Left: Comparison of SparseGPT against magnitude pruning on OPT-175B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Right: Compressing the entire OPT model family (135M, 350M, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=', 66B, 175B) to different sparsity patterns using SparseGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' One interesting fact is that our method is entirely local, in the sense that it relies solely on weight updates designed to preserve the input-output relationship for each layer, which are computed without any global gradient information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' As such, it is remarkable that one can directly identify such sparse models in the “neighborhood” of dense pretrained models, whose output correlates extremely closely with that of the dense model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Another general finding, illustrated in Figure 1 (right), is that larger models are easier to sparsify: specifically, we found that, for a fixed sparsity level, the relative accuracy gap between the dense and sparse model variant narrows as we increase the model size, to the point where inducing 50% sparsity results in practically no accuracy decrease on the largest models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This observation, illustrated in full detail in the experimental section, should be seen as very encouraging for future work on compressing such massive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 2 Background Post-Training Pruning is a practical scenario where we are given a well-optimized model θ⋆, together with some calibration data, and must obtain a compressed version of θ⋆ which satisfies some compression predicate C, specifying a set of weight quantization or sparsity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Post-training compression has traditionally been investigated in the context of quantization [23, 34, 29] to reduce the computational cost of quantization-aware training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' More recently, it has been shown that it is possible to also perform accurate post-training pruning [22, 11, 27], although existing work focuses on classic CNN and Transformer models, which have less than 100 million parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Layer-Wise Pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Post-training compression usually works by splitting the full-model compression problem into layer-wise subproblems, whose solution quality is measured in terms of the ℓ2-error between the output of the uncompressed layer, and that of the compressed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Specifically, for each layer Wℓ with calibration input Xℓ, the objective is to solve the following constrained optimization problem: argmin� Wℓ ||WℓXℓ − � WℓXℓ||2 2, (1) where � Wℓ is a set of weights satisfying the compression constraint C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Specifically for pruning, Hubara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' [22] posed this problem as that of finding, for each layer ℓ, a sparsity mask1 M satisfying the constraint, and weights � Wℓ such that argminmask M,� Wℓ ||WℓXℓ − (M ⊙ � Wℓ)Xℓ||2 2, (2) where � Wℓ is a possibly-updated version of the original dense weights Wℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Once each one of these layer-wise subproblems is solved, the model can then be “stitched back together” by re-composing the compressed layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 1Throughout the paper, by sparsity mask for a given tensor we mean a binary tensor of the same dimensions, with 0 at the indices of the sparsified entries, and 1 at the other indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 2 Massive Language Models Can Be Accurately Pruned in One-Shot PREPRINT Intuitively, if the layer-wise errors are small enough, the resulting model should preserve the accuracy of the original dense model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Mask Selection & Weight Reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' A key aspect of the the layer-wise pruning problem in (2) is that both the mask M as well as the remaining weights � Wℓ are optimized jointly, which makes this problem NP-hard [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Thus, exactly solving it for larger layers is unrealistic, leading all existing approaches to resort to approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' A particularly popular approach is to separate the problem into mask selection and weight reconstruction [20, 27, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Concretely, this means to first choose a pruning mask M according to some saliency criterion, like the weight magnitude [50], and then optimize the remaining unpruned weights while keeping the mask unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Importantly, once the mask is fixed, (2) turns into a linear squared error problem, which is convex and thus easily optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' It can even be solved in closed form by applying the standard linear regression formula to each matrix row wi: wi Mi = (XMiX⊤ Mi)−1XMi(wMiXMi)⊤, (3) where XMi denotes only the subset of input features whose corresponding weights have not been pruned in row i, and wMi represents the respective weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' It is also worth noting that XMiX⊤ Mi is the problem’s Hessian matrix, which needs to be inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' The AdaPrune approach [22] has shown good results for this problem in the context of post-training pruning via magnitude-based weight selection, followed by applying SGD steps to reconstruct the remaining weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Follow-up works demonstrate that pruning accuracy can be further improved by removing the strict separation between mask selection and weight reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Iterative AdaPrune [8] performs pruning in gradual steps with reoptimization in between and OBC [11] introduces a greedy solver which removes weights one-at-a-time, fully reconstructing the remaining weights after each iteration, via efficient closed-form equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Yet, these improvements also come with increased runtime, which, as we will discuss next, is particularly problematic in the context of extremely large models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Difficulty of Scaling to 100+ Billion Parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Prior post-training techniques have been successfully applied to models up to a few hundred million parameters [22, 11, 27], on which they are able to produce good results within a few hours of computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' However, our goal in this paper is to accurately sparsify models up to 1000× larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' The currently most accurate post-training method, OBC [11], exhibits runtime scaling to the 4th power of a transformer’s hidden dimension, while taking > 1 hour to compress a 100M parameter model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Thus, this is unsuitable for 100B- parameter models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Even the fastest known accurate post-training method, AdaPrune [22], requires a few minutes to prune a 100M model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Assuming best-case linear runtime scaling, this extrapolates to several hundreds of hours (a few weeks) of computation for a GPT3-sized Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Despite extremely large models being a highly active research area for the past several years, to the best of our knowledge, so far no model with 10+ billion parameters has been accurately pruned to nontrivial amounts of sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This suggests that scaling up existing methods might actually lead to significantly higher runtime costs than optimistically estimated in the previous paragraph2 and/or bring other major unexpected challenges like overfitting to calibration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' In this work, we introduce the first method that is fast enough to run on 100+ billion parameter models in a few hours on a single GPU and accurate enough to prune them to sparsity levels of 50% to 60% without significant performance drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 3 The SparseGPT Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='1 Fast Approximate Reconstruction Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' As outlined in Section 2, for a fixed pruning mask M, the optimal values of all weights in the mask can be calculated by solving the sparse reconstruction problem corresponding to each row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' However, doing so exactly requires inverting the Hessian matrix corresponding to the values preserved by the pruning mask Mi for row i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' computing (HMi)−1, for all rows 1 ≤ i ≤ drow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' One such inversion takes O(d3 col) time, for a total computational complexity of O(drow · d3 col) over drow rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' In practical terms, for a Transformer model, this means that the overall runtime scales with the 4th power of the hidden dimension dhidden and will thus clearly be infeasible to run on the largest GPT variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' To arrive at a practical algorithm, we need to improve the overall runtime by at least a full factor of dhidden, corresponding to a > 10000× compute reduction for models with more than 100 billion parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' We will achieve this via a series of careful approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 2In the context of quantization, there is evidence [47] that optimization steps also have to be scaled with increasing model size for AdaPrune-like methods, which would lead to quadratic rather than linear runtime scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 3 Massive Language Models Can Be Accurately Pruned in One-Shot PREPRINT Different Row-Hessian Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' The high computational complexity of optimally reconstrucing the unpruned weights following Equation 3 mainly stems from the fact that solving each row requires the individual inversion of a O(dcol × dcol) matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This is because the row masks Mi are generally different and (HMi)−1 ̸= (H−1)Mi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=', the inverse of a masked Hessian does not equal the masked version of the full inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This is illustrated also in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' If all row-masks were the same, then we would only need to compute a single shared inverse, as H = 2XX⊤ depends just on the layer inputs which are the same for all rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' select & invert reconstruct Figure 2: Illustration of the Row- Hessian challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Rows are pruned independently, pruned weights are in white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Hessian information is used for weight reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Since masks may be different per row, the inverse computation must be performed inde- pendently for each row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Such a constraint could be enforced in the mask selection, but this would have a major impact on the final model accuracy, as sparsifying weights in big structures, like entire columns, is known to be much more difficult than pruning them individually3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' The key towards designing an approximation algorithm that is both accurate and efficient lies in enabling the reuse of Hessians between rows with distinct pruning masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' We now propose an algorithm that achieves this in a principled manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Equivalent Iterative Perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' To motivate our algorithm, we first have to look at the row-wise weight reconstruction from a different iterative per- spective, using the classic OBS update [19, 43, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Assuming a quadratic approximation of the loss, for which the current weights w are optimal, the OBS update δm provides the optimal adjustment of the remaining weights to compensate for the removal of the weight at index m: δm = − wm [H−1]mm H−1 :,m incurring error εm = w2 m 2[H−1]mm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' (4) Since the loss function corresponding to the layer-wise pruning of one row of W is a quadratic, the OBS formula is exact in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Hence, w + δm is the optimal weight reconstruction corresponding to mask {m}C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Further, given an optimal sparse reconstruction w(M) corresponding to mask M, we can apply OBS again to find the optimal reconstruction for mask M′ = M − {m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Consequently, this means that instead of solving for a full mask M = {m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' , mp}C directly, we could iteratively apply OBS to individually prune the weights m1 up until mp in order, one-at-a-time, reducing an initially complete mask to M, and will ultimately arrive at the same optimal solution as applying the standard closed-form linear regression reconstruction with the full M directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Optimal Partial Updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Applying the OBS update δm potentially adjusts the values of all available parameters (in the current mask M) in order to compensate for the removal of wm as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' However, what if we wanted to update only the weights in a subset U ⊆ M, of all remaining unpruned weights?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Thus, we could still benefit from significant error compensation using only weights in U while reducing the cost of applying OBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Such a partial update can indeed be accomplished by simply computing the OBS update using HU, the Hessian corresponding to U, rather than HM, and updating only wU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Importantly, the loss of our particular layer-wise problem remains quadratic also for U and the OBS updates are still optimal: the restriction to U does not incur any extra approximation error by itself, only the error compensation might not be as effective, as less weights are available for adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' At the same time, if |U| < |M|, then inverting HU will be a lot faster than inverting HM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' We will now utilize this mechanism to accomplish our goal of synchronizing the masked Hessians across all rows of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Hessian Synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' In the following, assume a fixed ordering of the input features j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' , dcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Since those are typically arranged randomly, we will just preserve the given order for simplicity, but any permutation could in principle be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Next, we define a sequence of dcol index subsets Uj recursively as Uj+1 = Uj − {j} with U1 = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' , dcol}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' (5) In words, starting with U1 being the set of all indices, each subset Uj+1 is created by removing the smallest index from the previous subset Uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' These subsets also impose a sequence of inverse Hessians (HUj)−1 = ((2XX⊤)Uj)−1 which we are going to share across all rows of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Crucially, following [11], the updated inverse (HUj+1)−1 can be calculated efficiently by removing the first row and column, corresponding to j in the original H, from the inverse of (HUj)−1 in 3For example, structured (column-wise) pruning ResNet50 to 50% structured sparsity without accuracy loss is challenging, even with extensive retraining [30], while unstructured pruning to 90% sparsity is easily achievable with state-of-the-art methods [6, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 4 Massive Language Models Can Be Accurately Pruned in One-Shot PREPRINT O(d2 col) time via one step of Gaussian elimination: (HUj+1)−1 = � (HUj)−1 − 1 [(HUj)−1]11 (HUj)−1 :,1 (HUj)−1 1,: � 1:,1: with (HU1)−1 = H−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' (6) Hence, the entire sequence of dcol inverse Hessians can be calculated recursively in O(d3 col) time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' at similar cost to a single extra matrix inversion on top of the initial one for H−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Once some weight wk has been pruned, it should not be updated anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Further, when we prune wk, we want to update as many unpruned weights as possible for maximum error compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This leads to the following strategy: iterate through the Uj and their corresponding inverse Hessians (HUj)−1 in order and prune wj if j ∈ Mi, for all rows i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Importantly, each inverse Hessian (HUj)−1 is computed only once and reused to remove weight j in all rows where it is part of the pruning mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' A visualization of the algorithm can be found in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' elimination update prune Figure 3: Visualization of the SparseGPT reconstruction algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Given a fixed pruning mask M, we incrementally prune weights in each column of the weight matrix W, using a sequence of Hessian inverses (HUj)−1, and updating the remainder of the weights in those rows, located to the “right” of the column being processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Specifically, the weights to the “right” of a pruned weight (dark blue) will be updated to compensate for the pruning error, whereas the unpruned weights do not generate updates (light blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Computational Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' The overall cost of the approximate reconstruction process thus consists of three parts: (a) the computation of the initial Hessian, which takes time Θ(n · d2 col) where n is the number of input samples used4, (b) iterating through the inverse Hessian sequence in time O(d3 col) and (c) the reconstruction/pruning itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' The latter can be upper bounded by the time it takes to apply (4) to all drow rows of W for all dcol columns in turn, which is O(dcol · drow · dcol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' In total, this sums up to O(d3 col + drow · d2 col).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' For Transformer models, this is simply O(d3 hidden), and is thus a full dhidden-factor more efficient than exact reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This means that we have reached our initial goal, as this complexity will be sufficient to make our scheme practical, even for extremely large models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Weight Freezing Interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' While we have motivated the SparseGPT algorithm as an approximation to the exact reconstruction using optimal partial updates, there is also another interesting view of this scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Specifically, consider an exact greedy framework which compresses a weight matrix column by column, always optimally updating all not yet compressed weights in each step [11, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' At first glance, SparseGPT does not seem to fit into this framework as we only compress some of the weights in each column and also only update a subset of the uncompressed weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Yet, mechanically, “compressing” a weight ultimately means fixing it to some specific value and ensuring that it is never “decompressed” again via some future update, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' that it is frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Hence, by defining column-wise compression as: compress(wj)i = 0 if i ̸∈ Mi and wj i otherwise, (7) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' zeroing weights not in the mask and fixing the rest to their current value, our algorithm can be interpreted as an exact column-wise greedy scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' As we will show later, this perspective allows us to cleanly merge sparsification and quantization into a single compression pass, as well as inherit some other algorithmic enhancements from post-training quantization [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='2 Adaptive Mask Selection So far, we have only focused on the reconstruction aspect, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' assuming a fixed pruning mask M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' How should this mask be decided?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' One simple option would be to follow AdaPrune [22] and choose the mask for the whole layer in 4Taking the number of samples n to be a small multiple of dcol is sufficient for good results, even on very large models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 5 Massive Language Models Can Be Accurately Pruned in One-Shot PREPRINT advance using e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' the standard magnitude criterion [50] or including also second-order information [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' However, recent work [8, 11] has shown that the updates applied during the pruning process change weights significantly due to correlations, and that taking this into account for the mask selection yields significantly more accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This insight can be integrated into SparseGPT by adaptively choosing the mask while running the reconstruction pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' frozen not yet pruned p% sparse Figure 4: Mask selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' One obvious way of doing so would be to pick the p% easiest weights to prune in each column i just when it is compressed, which will lead to p% overall sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' At the same time, this approach has one big disadvantage: the sparsity cannot be distributed non-uniformly across columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This is a significant restriction, which will generally make unstructured pruning more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This is particularly problematic for massive language models, as they are known to have a small number of highly sensitive outlier features [3, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Further, [11] observe that some OPT models appear to have a large number of dead RELUs in the earlier layers, leading to many columns that are trivial to prune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' We propose to alleviate this disadvantage while still exploiting the significant accuracy gains of adaptive weight selection during pruning via iterative blocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' More precisely, we always select the pruning mask for Bs = 128 columns at- a-time based on the OBS reconstruction error ε from Equation (4), using the diagonal values in our Hessian sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' We then perform the next Bs updates as discussed in the previous section, before selecting the mask for the next block, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This procedure allows non-uniform selection per column, in particular also using the corresponding Hessian information5, while at the same time considering also previous weight updates in the selection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='3 Extension to Semi-Structured Sparsity While so far we have only discussed unstructured pruning, SparseGPT is also easily adapted to semi-structured patterns such as the popular n:m sparsity format [49, 22] which delivers speedups in its 2:4 implementation on Ampere NVIDIA GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Specifically, every consecutive m weights should contain exactly n zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Hence, we can simply choose blocksize Bs = m and then enforce the zeros-constraint in the mask selection for each row by picking the n weights which incur the lowest error as per Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' A similar strategy could also be applied for other semi-structured pruning patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Finally, we note that a larger Bs would not be useful in this semi-structured scenario since zeros cannot be distributed non-uniformly between different column-sets of size m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='4 Full Algorithm Pseudocode Algorithm 1 The SparseGPT algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' We prune the layer matrix W to p% unstructured sparsity given inverse Hessian H−1 = (2XX⊤ + λI)−1, lazy batch-update blocksize B and adaptive mask selection blocksize Bs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' each Bs consecutive columns will be p% sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' M ← 1drow×dcol // 0/1 pruning mask E ← 0drow×B // block quantization errors H−1 ← Cholesky(H−1) // Hessian inverse information for i = 0, B, 2B, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' do for j = i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' i + B − 1 do if j mod Bs = 0 then M:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='j:(j+Bs) ← mask of (1 − p)% weights wc ∈ W:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='j:(j+Bs) with largest w2 c/[H−1]cc end if E:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='j−i ← W2 :,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='j / [H−1]jj // pruning error E:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='j−i ← (1 − M:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='j) · E:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='j−i // freeze weights that are not pruned W:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='j:(i+B) ← E:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='j−i · H−1 j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='j:(i+B) // update weights in block end for W:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='(i+B): ← E · H−1 i:(i+B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='(i+B): // update all remaining weights end for W ← W · M // set pruned weights to 0 With the weight freezing interpretation discussed at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='1, the SparseGPT reconstruction can be cast in the column-wise greedy framework of the recent quantization algorithm GPTQ [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This means we can also inherit 5For a single column j, the OBS selection criterion would degrade to just the magnitude, as [H−1]jj is constant across rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 6 Massive Language Models Can Be Accurately Pruned in One-Shot PREPRINT several algorithmic enhancements from GPTQ, specifically: precomputing all the relevant inverse Hessian sequence information via a Cholesky decomposition to achieve numerical robustness and applying lazy batched weight matrix updates to improve the compute-to-memory ratio of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Our adaptive mask selection, as well as its extensions to semi-structured pruning, are compatible with all of those extra techniques as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' The pseudocode in Algorithm 1 presents the the unstructured sparsity version of the SparseGPT algorithm in its fully developed form, integrating also all relevant techniques from GPTQ discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='5 Joint Sparsification & Quantization Algorithm 1 operates in the column-wise greedy framework of GPTQ, thus sharing the computationally heavy steps of computing the Cholesky decomposition of H−1 and continuously updating W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This makes it possible to merge both algorithms into a single joint procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Specifically, all weights that are frozen by SparseGPT are additionally quantized, leading to the following generalized errors to be compensated in the following update step: E:,j−i ← (W:,j − M:,j · quant(W:,j))2 / [H−1]jj, (8) where quant(w) rounds each weight in w to the nearest value on the quantization grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Crucially, in this scheme, sparsification and pruning are performed jointly in a single pass at essentially no extra cost over just running SparseGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' We emphasize that doing quantization and pruning jointly means that later pruning decisions are influenced by earlier quantization rounding, and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This is in contrast to prior techniques, such as OBC [11], which first sparsify a layer and then quantize the remaining weights, where quantization consequently has no influence on pruning outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 4 Experiments Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' We implement SparseGPT in PyTorch [38] and use the HuggingFace Transformers library [45] for handling models and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' All experiments are conducted on a single NVIDIA A100 GPU with 80GB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' In this setup, SparseGPT can fully sparsify the 175-billion-parameter models in approximately 4 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Similar to [47, 9], we sparsify Transformer layers sequentially in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This significantly reduces memory requirements, and also noticeably improves accuracy over handling all layers in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' All our compression experiments are performed in one-shot, without any finetuning, following a similar setup to that of recent work on post-training quantization of GPT-scale models [9, 47, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' For calibration data, following [9], we use 128 2048-token segments, randomly chosen from the C4 [40] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This represents generic text data crawled from internet and makes sure that our experiments remain actually zero-shot since no task-specific data is seen during pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Models, Datasets & Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' We primarily work with the OPT model family as it provides a suite of models ranging from 125 million to 175 billion parameters, allowing us to study the scaling behavior of pruning relative to model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Additionally, we also consider the 176 billion parameter variant of BLOOM [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' In general, our focus lies on the very largest variants but we also show some results on smaller models to provide a broader picture, and in particular ablations with respect to model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' In terms of datasets and evaluation, we mainly focus on perplexity on the raw WikiText2 test set [31], a popular benchmark in LLM compression literature [47, 36, 9, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' In the Appendix, we also show results on a set of text segments sampled from the C4 validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' In general, perplexity is known be a challenging and stable metric that is well suited for evaluating the accuracy of compression methods [47, 11, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Yet, for additional interpretability, we also provide some ZeroShot accuracy results for LAMBADA [35], ARC (Easy and Challenge) [2], PIQA [44] and StoryCloze [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' We emphasize that the main focus of our evaluation lies on accuracy of the sparse models, relative to the dense baseline rather than on absolute numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' We calculate perplexity in easily-reproducible fashion following the procedure described by HuggingFace6: we concatenate all samples of the raw dataset with “\\n\\n” separators, encode and split the entire sequence into non-overlapping segments of 2048 tokens (the maximum window size of both OPT and BLOOM), on which the standard average causal language modelling loss is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' The final perplexity is the exponentiated version of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Different preprocessing may influence absolute numbers, but does not affect our claims, as we mainly focus on performance relative to the dense model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Our ZeroShot evaluations are performed using GPTQ’s [9] implementation, which is in turn based on the popular [46, 4] EleutherAI-eval-harness7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' We emphasize that all dense and sparse results were computed with exactly the same code to ensure a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 6https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='co/docs/transformers/perplexity 7https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='com/EleutherAI/lm-evaluation-harness 7 Massive Language Models Can Be Accurately Pruned in One-Shot PREPRINT Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' We believe to be the first academic work to perform post-training pruning of massive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' (As discussed previously, prior post-training pruning techniques have only been applied to models 1000x smaller in size, and do not scale to GPT model sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=') As such, there are no standard benchmarks, and we therefore primarily evaluate what levels of sparsity are achievable with SparseGPT while maintaining close to dense model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Nevertheless, we also compare with the highly popular magnitude pruning criterion [50] which drops the weights of smallest absolute value, applied layerwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This technique easily scales to extremely large models in terms of computational efficiency, but, as we show, does not perform well in terms of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='1 Results Pruning Difficulty Scaling with Model Size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' In our first set of experiments, we study how the difficulty of sparsifying LLMs changes with their size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' For this, we consider the entire OPT model family and uniformly prune all linear layers, excluding the embeddings and the head as is standard [41, 25], to 50% unstructured sparsity, full 4:8 or full 2:4 semi-structured sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' (All three correspond to 50% overall sparsity, but the 2:4 pattern is the most stringent, followed by 4:8 and unstructured sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=') The raw-WikiText2 performance numbers are given in Table 1 and visualized in Figure 1 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' OPT Sparsity 125M 350M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='3B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='7B 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='7B 13B 30B 66B 175B dense 0% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='66 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='01 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='63 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='46 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='86 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='12 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='56 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='33 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='34 Magnitude 50% 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='7e4 265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='2e5 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='2e4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='3e4 SparseGPT 50% 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='89 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='60 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='45 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='44 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='56 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='12 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='77 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='33 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='21 SparseGPT 4:8 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='75 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='82 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='02 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='97 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='54 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='72 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='28 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='66 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='45 SparseGPT 2:4 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='17 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='21 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='04 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='14 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='16 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='91 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='88 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='73 Table 1: OPT perplexity results on raw-WikiText2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' One immediate finding is that the accuracy of magnitude-pruned models collapses across all scales, with larger variants generally dropping worse, relative to smaller ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This is in stark contrast to smaller vision models which can usually be pruned via simple magnitude to 50% or more at very little loss of accuracy [43, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This highlights the importance of more accurate pruners in the context of extremely large generative language models, but also the fact that perplexity is a very sensitive metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' For SparseGPT, the trend is very different: already at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='7B parameters, the perplexity loss is < 1 point, at 66B, there is zero loss and at the very largest scale there is even a slight accuracy improvement over the dense baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' (As can be seen in Figure 1, sparse OPT models of ≤ 50% sparsity all have lower perplexity than the dense baseline, although the differences are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=') In general, there is a clear trend of larger models being significantly easier to sparsify, which we speculate may be due to them being more overparametrized and also more noise resistant in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' We think a more detailed investigation of this phenomenon would be a great topic for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' For 4:8 and 2:4 sparsity, the behavior is very similar, but accuracy drops are typically a bit higher due to the sparsity pattern being significantly more constrained [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Nevertheless, at the largest scale, the perlexity increases are only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='11 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='39 for 4:8 and 2:4 sparsity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' We emphasize that these sparsity pattern can actually achieve 2× speedup in practice [22, 32], with commercially available NVIDIA Ampere GPUs already including support for 2:4 sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Sparsity Scaling for 100+ Billion Parameter Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Next, we take a closer look at the largest publicly-available dense models, OPT-175B and BLOOM-176B, and investigate how their performance scales with the degree of sparsity induced by either SparseGPT or magnitude pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' The results are visualized in Figures 1 and 5 (both left panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' For the OPT-175B model, for which the results are presented in Figure 1 (left), magnitude pruning can achieve at most 10% sparsity before significant accuracy loss occurs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' meanwhile, SparseGPT enables up to 60% sparsity at a comparable perplexity increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' BLOOM-176B, for which the results are provided in Figure 5 (left), appears to be more favorable for magnitude pruning, admitting up 30% sparsity without major loss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' still, SparseGPT can deliver 50% sparsity, a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='66× improvement, at a similar level of perplexity degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Even at 80% sparsity, models compressed by SparseGPT still score reasonable perplexities, while magnitude pruning leads to a complete collapse (> 100 perplexity) already at 40% sparsity for OPT and 60% sparsity for BLOOM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Remarkably, SparseGPT is able to remove around 100 billion weights from these models, with limited impact on model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 8 Massive Language Models Can Be Accurately Pruned in One-Shot PREPRINT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='8 Sparsity 8 10 12 14 16 18 20 22 24 Perplexity on raw-WikiText2 BLOOM-176B Uniform Unstructured Sparsity Magnitude SparseGPT Dense 10 1 100 101 102 #Params in Billions 10 20 30 40 50 Perplexity on raw-WikiText2 OPT Model Family - Sparse + Quantized 3bit GPTQ 50% + 4bit Dense Figure 5: Left: Uniformly compressing BLOOM-176B to various sparsity levels with SparseGPT and magnitude pruning, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Right: 50% sparsity + 4-bit quantization joint compression vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 3-bit on the OPT family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' ZeroShot Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' To complement our perplexity evaluations, we now also provide results for various sparsified variants of OPT-175B on several ZeroShot tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' ZeroShot evaluations are known to be relatively noisy [3] but at the same time more interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' All numbers are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Method Sparsity Lambada PIQA ARC-easy ARC-ch StoryCloze Average Dense 0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='59 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='07 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='04 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='94 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='82 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='29 Magnitude 50% 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='02 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='73 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='03 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='60 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='10 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='10 SparseGPT 50% 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='51 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='03 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='65 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='30 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='87 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='27 SparseGPT 4:8 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='77 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='16 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='35 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='85 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='02 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='63 SparseGPT 2:4 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='47 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='16 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='08 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='99 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='19 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='18 Table 2: ZeroShot results on several datasets for sparsified variants of OPT-175B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Overall, a similar trend to the perplexity results seems to hold with magnitude pruned models collapsing to close to random performance on several datasets while SparseGPT models stay close to the original accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' However, as expected, these numbers are significantly more noisy: for instance, 2:4 pruning appears to achieve noticeably higher accuracy than the dense model on Lambada despite being the most constrained sparsity pattern investigated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Notice also that these effects ultimately average out when considering many different tasks, which is consistent to the literature [47, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Joint Sparsification & Quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Finally, another interesting research direction is the combination of sparsity and quantization, which would allow combining computational speedups from sparsity [26, 5] with memory savings from quantization [9, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Specifically, if we compress a model to 50% sparse + 4-bit weights, store only the non-zero weights and use a bitmask to indicate their positions, then this has the same overall memory consumption as 3-bit quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Hence, in Figure 5 (right) we compare SparseGPT 50% + 4-bit with state-of-the-art GPTQ [9] 3-bit numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' While there seem to be a few outliers, 50% + 4-bit models are more accurate than their respective 3-bit versions for several models sizes, including 175B with 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='55 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='68 3-bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' We also tested 2:4 and 4:8 in combination with 4-bit on OPT-175B yielding encouraging 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='20 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='86 perplexities, which can likely be improved further using additional quantization tricks such as blocking [9, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 5 Related Work Model compression aims to produce more efficient models, using approaches such as pruning, quantization, and knowledge distillation—we refer the reader to the respective surveys [21, 14, 15] for an in-depth discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Since the focus of our work is on compressing massive models, with 10-100s of billions of parameters, we will mainly focus on work in this specific area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 9 Massive Language Models Can Be Accurately Pruned in One-Shot PREPRINT Pruning Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' To our knowledge, we are the first academic work to investigate pruning of massive GPT-scale models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' with more than 10 billion parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This gap in the literature may seem surprising, given both the widespread popularity of these models, and the significant amount of existing work on pruning, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' [17, 7, 12, 6, 43, 41, 39, 10, 13, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' One justification for this gap is the fact that most existing pruning methods, such as gradual magnitude pruning [16, 17, 12, 24], require extensive retraining following the pruning step in order to recover accuracy, while GPT-scale models usually require massive amounts of computation and parameter tuning both for training or finetuning [48], which renders retraining-based approaches difficult to apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Thus, we are not aware of any work applying such gradual pruning methods at GPT scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' SparseGPT is a post-training method for GPT-scale models, as it does not perform any finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' So far, post-training pruning methods have only been investigated at the scale of classic CNN or BERT-type models [22, 11, 27], which have 100-1000x fewer weights than our models of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' We discussed the challenges of scaling existing post-training methods to GPT models, and the technical relationship between SparseGPT and these methods, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Post-Training Quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' By contrast, there has been a significant amount of emerging work on post-training meth- ods for quantizing GPT-scale models, closely following the first open releases of such models [48, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Specifically, the ZeroQuant [47], LLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='int8() [3] and nuQmm [36] methods investigated the feasibility of round-to-nearest quantization for billion-parameter models, showing that 8-bit quantization for weights is feasible via this approach, but that activation quantization can be difficult due to the existence of outlier features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' GPTQ [9] leverages approximate second-order information for accurate quantization of weights down to 2–4 bits, for the very largest models, and shows that this can bring inference speedups of 2-5x when coupled with efficient GPU kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Follow-up work by Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' [46] investigated joint activation and weight quantization down to 8 bits per component, proposing a smoothing-based scheme which reduces the difficulty of activation quantization and is complemented by efficient GPU kernels for fast inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Concurrent work by Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' [37] tackles the hardness of quantizing activation outliers via quadapters, a set of learnable parameters whose goal is to scale activations channel-wise, while keeping the other model parameters unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Very recent work by Dettmers and Zettlemoyer [4] investigate scaling relationships between model size, quantization bits, and different notions of accuracy for massive LLMs, observing a high degree of correlation between perplexity scores and aggregated zero-shot accuracy across tasks, as well as saturation behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Since it focuses on sparsification rather than quantization, SparseGPT is complementary to quantization approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Specifically, as we have shown in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='5, the SparseGPT algorithm can be applied in conjunction with GPTQ, the current state-of-the-art algorithm for weight quantization, and should be compatible with activation quantization approaches [3, 46, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Thus, it would be very interesting to investigate in depth how compression errors compound when quantization and pruning are applied in conjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 6 Discussion We have provided a new post-training pruning method called SparseGPT, specifically tailored to massive language models from the GPT family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Our results show for the first time that large-scale generative pretrained Transformer- family models can be compressed to high sparsity via weight pruning in one-shot, without any retraining, at low loss of accuracy, when measured both in terms of perplexity and zero-shot performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Specifically, we have shown the largest open-source GPT-family models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' OPT-175B and BLOOM-176B) can reach 50-60% sparsity with low accuracy fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Surprisingly, this means that more than 100 billion weights from these models can be ignored at inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Central to our approach is a new large-scale approximate sparse regression algorithm, which generalizes to semi-structured (2:4 and 4:8) patterns, and is also compatible with existing weight quantization approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Interestingly, our method is local: after each pruning step, it performs weight updates, designed to preserve the input-output relationship for each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' These updates are computed without any global gradient information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Thus, it appears that the high degree of parametrization of massive GPT models allows our method to directly identify sparse accurate models in the “close neighborhood” of the dense pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Remarkably, since our main accuracy measure (perplexity) is extremely sensitive, it appears that the output of the generated sparse model correlates extremely closely with that of the dense model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Our second main finding is that larger models are easier to sparsify: at a fixed sparsity level, the relative accuracy drop for the sparse model, relative to the dense one, narrows as we increase the model size, to the point where inducing 50% sparsity results in practically no accuracy decrease on the largest models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This finding should be seen as very encouraging for future work on compressing such massive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' One natural avenue for future work would be to investigate fine-tuning mechanisms for such large-scale models, which would allow further accuracy recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' We conjecture that this should be possible, and that probably at least 80-90% sparsity can be achieved with progressive pruning and fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Another extension which we plan to investigate is the applicability of our approaches during training, to reduce the computational cost of pre-training these massive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 10 Massive Language Models Can Be Accurately Pruned in One-Shot PREPRINT Acknowledgments The authors gratefully acknowledge funding from the European Research Council (ERC) under the European Union’s Horizon 2020 programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 805223 ScaleML), as well as experimental support from Eldar Kurtic, and from the IST Austria IT department, in particular Stefano Elefante, Andrei Hornoiu, and Alois Schloegl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' References [1] Thomas Blumensath and Mike E Davies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Iterative thresholding for sparse approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Journal of Fourier Analysis and Applications, 14(5-6):629–654, 2008.' metadata={'source': 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Younes Belkada, and Luke Zettlemoyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' LLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='int8(): 8-bit matrix multiplication for transformers at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' arXiv preprint arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='07339, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' [4] Tim Dettmers and Luke Zettlemoyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' The case for 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subsample from the C4 dataset consisting of randomly crawled website text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' Specifically, we take the first validation shard and randomly sample 256 2048-token segments (each contained in a single large enough document).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' This is exactly the same sampling procedure that is also used for the calibration dataset, the latter is just sampled from the first training shard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' The results are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content='8 Sparsity 8 10 12 14 16 18 20 22 24 Perplexity on C4 (sub) Uniform Unstructured Sparsity OPT-175B BLOOM-176B Figure 6: C4 perplexity for OPT-175B and BLOOM-176B at various SparseGPT sparsity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' In summary, these results confirm our raw-WikiText2 findings from the main paper that SparseGPT is able to achieve 50-60% sparsity with an only very minor perplexity increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' In this direct comparison between models one can also see how BLOOM’s accuracy starts to degrade slightly more quickly, especially at the higher sparsity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} +page_content=' 14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQf3vl6/content/2301.00774v1.pdf'} diff --git a/a9A0T4oBgHgl3EQfGP_A/vector_store/index.faiss b/a9A0T4oBgHgl3EQfGP_A/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..1565fae613c37fbdd53c68cd91be1b3b809d2d5a --- /dev/null +++ b/a9A0T4oBgHgl3EQfGP_A/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0a20426253e282b285cd22b7ce45d2a662fc850d8a5c119bfcf970d6943cf19c +size 2555949 diff --git a/adAzT4oBgHgl3EQfnP01/content/tmp_files/2301.01576v1.pdf.txt b/adAzT4oBgHgl3EQfnP01/content/tmp_files/2301.01576v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f3f938fd0bbff82ffe0ef1b1ea37f0b895d0ee27 --- /dev/null +++ b/adAzT4oBgHgl3EQfnP01/content/tmp_files/2301.01576v1.pdf.txt @@ -0,0 +1,430 @@ +ROBOFRIEND: AN ADPATIVE STORYTELLING ROBOTIC TEDDY +BEAR – TECHNICAL REPORT +A PREPRINT +Ido Glanz∗ +Technion Autonomous Systems Program +Technion – Israel Institute of Technology +Matan Weksler∗ +Technion Autonomous Systems Program +Technion – Israel Institute of Technology +Erez Karpas +Faculty of Data and Decision Sciences +Technion – Israel Institute of Technology +karpase@technion.ac.il +Tzipi Horowitz-Kraus +Faculty of Education in Science and Technology +Technion – Israel Institute of Technology +tzipi.kraus@ed.technion.ac.il +ABSTRACT +In this paper we describe Robofriend, a robotic teddy bear for telling stories to young children. +Robofriend adapts its behavior to keep the childrens’ attention using reinforcement learning. +Keywords Robotics · Education · Reinforcement Learning +Introduction +Language exposure at an early stage of development is critical for the facilitation of brain networks associated with +language Kuhl [2004], Cardillo and Kuhl [2009], Moon et al. [2013]. Storytelling is one form of language exposure, +which was found to be associated with a greater engagement not only in language processing but also in visualization +and cognitive abilities in children Hutton et al. [2015]. Interestingly, it was suggested that it is not the storytelling itself +that is related to these improvements, but it is the interaction during the stories that amplify these abilities in children +Twait et al. [2019]. A recent study demonstrated how a group of 4–6-year-old children attending storytelling sessions +interactively vs. a group attending non-interactively (storytelling sessions on the screen), shared greater cognitive and +language abilities Twait et al. [2019]. Hence, a question was raised regarding this positive effect during interactive +(dialogic) storytelling – is the positive effect due to the human interaction? or due to the interactive nature of the +storytelling? in other words, will an interactive robot during storytelling result in similar results as the human-based +interactive condition? +To study this question, we designed Robofriend (shown in Figure 1) – a robotic teddy bear that reads young children +stories. Robofriend is constructed by taking a regular teddy bear and inserting a tablet in its belly, as well as a +rudimentary skeleton, motors and sensors that allow it to move its head and arms. Robofriend can read a story to a +small group of children, with the robot’s main objective being to engage the children, keeping their attention on the +story. Thus, although our main motivation for designing Robofriend is the scientific study described above, Robofriend +can also serve as a tool that a teacher in a daycare class can use. Robofriend can read a story to one group of children +while the teacher engages with the other children in the class. +Each story that Robofriend can tell is divided into prerecorded video segments. Typically, each segment will correspond +to showing a still image of one page in the printed book, with a human reading the text on the page. Note that Robofriend +does not perform any text to speech, the segments are all prerecorded. At the end of each segment, Robofriend chooses +which action to perform out of several actions it has available. Possible actions include asking a simple question +about the story (there is a set of prerecorded questions for each segment of the story, Robofriend chooses one of these +∗ denotes equal contribution +arXiv:2301.01576v1 [cs.RO] 4 Jan 2023 + +Robofriend TR +A PREPRINT +Item +Use +Quantity +Price +Bear doll +- +1 +US$30.00 +Display +Play video +1 +US$200.00 +Camera +Monitor kids +1 +US$79.00 +Arduino + wiring kit +Control Servos +1 +US$37.00 +Servo +Move head +4 +US$24.00 +Speakers +Playing Sound +1 +US$20.00 +Total: +US$462.00 +Table 1: Robofriend Bill of Materials +randomly), giving positive feedback (e.g., “very good children, I see you are paying attention”), or negative feedback +(e.g., “children, are you listening to the story?”). +As previously stated, the objective of Robofriend is to keep the childrens’ attention. The first step to optimizing +something is to measure it, or at least some proxy of it. Robofriend uses a camera to measure some things, which +can serve as a proxy for engagement. First, Robofriend uses computer vision to detect the faces of the children, and +the direction of their gazes. We remark that these faces are anonymous – Robofriend does not try to associate faces +to identities in any way. From these face detection, we extract several measurements: how many faces are looking +at the robot, how focused are they on the story (using their relative gaze) as well as how “jumpy” the faces are (an +"excitement" metric). Aside from the visual attributes, we also monitor the noise level as and its momentary change +(its derivative) to serve as supportive metrics capturing the children’s state. These are aggregated into a reward signal +for each camera video frame, and aggregated throughout each story segment to produce a state and reward for each +segment. +Having defined the rewards, we can now try to optimize our objective – the total sum of rewards. Of course, we do +not know in advance what is the right action to take after each story segment, nor do we have a model for how each +action will affect the children’s engagement. Therefore, we chose to use reinforcement learning to control Robofriend’s +actions. However, because young children are involved, we do not want to allow the robot to explore sequences of +actions we know are not beneficial for the children (for example, always using the negative feedback action). Therefore, +we adopt the approach of using LTL “restraining bolts” Giacomo et al. [2020], and manually encode what are the +allowed trajectories for Robofriend. +In the remainder of this paper, we describe the design of Robofriend in more detail. We also describe our preliminary +evaluation of the robot at a local daycare center. Finally, we conclude with some lessons learned and a discussion of the +ethical considerations that arose in this project. +Robofriend Design +We now describe the design of Robofriend in more detail, starting with the mechanical build. +Robofriend Mechanical Build +As previously stated Robofriend is constructed by taking a large, 1m tall, teddy bear, and instrumenting it to be able to +move its head and arms, play videos and sound, and look at the children it is reading the story to. First, we inserted an +aluminum skeleton into the robot to support the other devices. To do so, we removed the majority of the stuffing and +decoupled the head momentarily to mount the camera in the bear’s nose and create a fixture for the servo motors to +connect to. We then mounted a 12.3 inch display, which the robot displays the story segments on, as well as 4 servo +motors which are used to move the head and arms (one for each arm and 2 for the the head pan and tilt). As mentioned +above, a camera was placed into the teddy bear’s nose, to monitor the children and measure the reward signal, finally, +speakers were connected to play sound, see figure 2 for a schematic diagram. +This hardware was controlled from a PC, which was connected directly to the camera, display, and speakers. The servo +motors are controlled by an Arduino Uno (see Figure 3), which was connected to the PC as well. The controller for the +servo motors runs in a separate process on the Arduino, following instructions from the PC. +The Bill of Materials (BOM) for Robofriend is shown in Table 1, while a detailed BOM with links to each item +is available online at: shorturl.at/efBZ2. Overall, the total cost to construct Robofriend was less than US$500, +making it fairly accessible. +2 + +Robofriend TR +A PREPRINT +Figure 1: Robofriend in Home Testing +3 + +Robofriend TR +A PREPRINT +Figure 2: Hardware schematic block diagram +Robofriend Code Architecture +To operate and coordinate the different algorithms, hardware and user interactions, a proprietary python software +stack was developed and will be briefly described below. The code is available at https://github.com/IdoMatan/ +RoboFriend. +The code architecture is based on RabbitMQ, which is a ROS-like publisher-subscriber framework that implements +asynchronous parallel process control. RabbitMQ allows us to simultaneously control the robot’s servos, camera, screen +and any other needed peripherals, as well as to run the algorithms we will describe later. +Figure 5 shows the schematic structure of our software, showing the processes and the messages that are passed between +them. The main process is the StoryTeller, which coordinates the flow among the other processes and displays the video. +This process runs a loop which plays the next story segment, then calls the algorithm service to get the next action. This +loop repeats until the story ends. Throughout this loop, the robot moves its head, aiming to center its viewing angle so +to center all faces in the frame. This flow is illustrated in Figure 4. +The other services are either timer-based (e.g. send a frame every N milliseconds) or event-driven, e.g., a page-ended +message would trigger a next action calculation in the algorithm-service. All metrics, actions and useful metadata (not +4 + +Teddy bear inputs +Camera +Mic +Monitoring +Teddy bear outputs +Dashboard +Speakers +PC / Laptop +12in TFT screen +Database +(local) +Arduino Uno Micro-controller +Servo +Servo +Servo +Servo +left hand +Right hand +Head tilt +Head pan +Teddy bear control +miroRobofriend TR +A PREPRINT +Figure 3: Arduino sub-system +video footage) were logged in real-time in a local Postgres database, allowing both post-analysis of the trial as well as +live monitoring using a Grafana dashboard. +Figure 6 shows the simple GUI implemented where a user can run the app, choose one of the supported stories, an +operation mode (which will be discussed later) and start and stop the story. +Robofriend Algorithm +Having described the mechanical construction of Robofriend and the code architecture, we can now discuss the +algorithm which is used to control it. As previously mentioned, our high-level control algorithm involved using +reinforcement learning with “restraining bolts” Giacomo et al. [2020], to avoid the robot following trajectories which +we know will not be good. We begin by describing our sensing and reward function, then we describe the actions which +are available to the robot, and finally, we describe the constraints which were used as the “restraining bolts” in our +preliminary evaluation. +5 + +9 +DIGITAL +(PWM +8 +IUNO +XL +RX +ARDUINO +POWER +ANALOG +VRobofriend TR +A PREPRINT +Figure 4: Software Operational Flow +Robofriend Sensing and Reward Function +As previously mentioned, the reward is based on using computer vision to detect the children’s faces and gaze direction, +and on measuring the noise level. Specifically, we used an MTCNN neural-network-based face detector to detect the +faces of the children within the frame Zhang et al. [2016], followed by a gaze estimation step for each using GazeNet +Zemblys et al. [2018] to generate a gaze vector relative to the camera lens. This results in the following metrics: +Number of Faces The number of detected faces by the MTCNN face detection algorithm. A change in the number of +faces would likely indicate a child walking away or not looking at the camera. +Average relative gaze (attention) For each detected face (denoted by index i), a gaze vector θi, φi is predicted by +the GazeNet algorithm, where θi is the lateral (left/right) angle and φi is the vertical (up/down angle) – both +angles are relative to the center of the frame (the camera lens center). Based on these measurements, we define +the gaze component of the reward as: +rgaze := 1 +n +n +� +i=1 +cos(θi) +Roughly speaking, our attention metric, ranging from 0-1, corresponds to how focused the children are on the +robot as opposed to looking around the room. +Excitement Using consecutive frames we are able to calculate a per-face jumpiness metric corresponding to how still +the children are. This is another proxy to their attention and engagement. Formally, let us denote the positions +of (centers of) faces detected in the first image by xi, yi (for i = 1 . . . n), and the positions of the faces detected +in the next image by x′ +j, y′ +j (for j = 1 . . . m). As the faces do not have identities associated with them, we +must first align the faces in the first image to the faces in the next image. We do so greedily by finding, for +each face position xi, yi in the first image, the closest face position x′ +j, y′ +j (using Euclidean distance) in the +next image (which is also under a feasible max possible distance). We then define the jumpiness for face i by +6 + +JsonConfigfile +(storypages, +possible actions) +Start playing video - Page 1 +Choose Action +Algorithm - Policy Maker +GUI App +Grafana Dashboard +Init RabbitMQ +Start playing video -Page 2 +Init services +ChooseAuto/Manual +Choose Action +(checkbox) +2. +Choose story (start playing) +3. +Get Action if in manual mode +Start playing video - Page N +Train cycleRobofriend TR +A PREPRINT +Figure 5: Software Architecture Diagram +this Euclidean distance, and the total jumpiness is the sum of jumpiness for each face, that is: +rjump := +n +� +i=1 +min +j +� +(xi − x′ +j)2 + (yi − y′ +j)2 +Noise Level The average noise level over a 1 second period as measured by the microphone inside the teddy bear. This +is a proxy for how much the children are talking to each other instead of listening to the story. We denote this +by rnoise. +Deviation of Noise Level The derivative of the noise level, again averaged every second, denoted by rnd. The +motivation behind this metric is to capture changes in the sound level within a page, indicating the children are +getting noisier or quieter potentially due to the effect of the previous action. +To aggregate the reward signal throughout the duration of a story segment, we average each of these metrics for each +video frame that belongs to this segment. Note that we use average instead of sum, as different story segments have +different durations. This gives us a state vector ⟨rgaze, rjump, rnoise, rnd⟩. +Finally, to aggregate these different metrics into a single reward function we use a weighed sum, and thus our reward +denoted by r is defined as: +r := α1 ∗ rgaze − α2 ∗ rjump − α3 ∗ rnoise + α4 ∗ rnd + ltl_reward +7 + +APP +(GUI) +Setup story & +StoryTeller +Choose +pause/play +story +Control video +Input action +Execute actions +on manual +Get Action +End of Page (EoP) +Pause (?) +(manual mode) +Play/Pause +ServoService +Execute action +Control servos (4) thru +VideoService +arduino. +Acceptspan+tiltfor +play/pause video +head and tilt for each +hand +Get Action +(auto mode) +Avg head angles (always +sent but only executed if +storyteller allows) +CamService +State (gaze, n_kids, movement) +AlgoService +Calc avg gaze +Calc actionbased on +Number of kids +camandmic +Avg Movement? +StatusLog +Noise Level +MicService +LogService +Record avg noise +Saveto DB +level (and derivative?)Robofriend TR +A PREPRINT +Figure 6: A simple python-based graphical user interface to interact with the robot +Where the ltl_reward will be described in details below, but conceptually corresponds to a set of pre-defined restraining +rules the robot should learn to obey. +We conclude the discussion of the reward by noting that these measurements serve as a proxy for the real reward (the +children’s attention), which we can not measure directly. +Robofriend States and Actions +As we described above, Robofriend reads a story, which is divided into segments, and chooses which action to perform +after every segment. Note that the order between the story segments is linear, and so there is no choice with regard to +which story segment to read next. The only choice is which action to perform after every story segment. +Robofriend’s actions correspond to different types of feedback it can give the children and are divided into: +Positive Feedback This action randomly chooses from a set of positive feedback sentences, such as “great job” or +“you are listening nicely”. +Negative Feedback This action chooses randomly from a set of (mildly) negative feedback sentences, such as “please +pay attention” or “please be quiet”. +Question This action chooses a random question relating to the story segment Robofriend just finished reading. +Robofriend does not attempt to extract an answer but merely pauses for an appropriate amount of time. +Continue Continue immediately to the next story segment. +Move head and arms Execute a series of random head and arm movements for a few seconds. +Note that all of these actions are applicable regardless of where Robofriend is in the story (the question action changes +as a function of it, but is always applicable). Furthermore, the only information Robofriend extracts from its sensors is +the reward signal. Thus, there is no notion of state for where we are in the story, and the only state information is the +state vector ⟨rgaze, rjump, rnoise, rnd⟩ described above. However, the LTL constraints used for the “restraining bolts” +do carry their own notion of state, as we discuss next. +Robofriend Restraining Bolt +The final part of Robofriend’s control architecture is the “restraining bolt” – the LTL constraint which the robot needs +to obey Giacomo et al. [2020]. That is, we specify LTL trajectory constraints over allowed and forbidden trajectories +for the robot, where in this case a trajectory is a sequence of actions. For example, we can specify the constraint to not +ask two questions in a row as +G(ask_question → X¬ask_question) +(that is, if the current action is ask question, then the next action is not ask question). This ensures the robot will not ask +questions constantly. +8 + +RoboBear ApP +RoboBear Controller Gui +Select story from drop-down and press start +Usemame: +admin +Session #: +1663055759 +manual +Choose Story: +AptForRent +Start Story! +Play/Pause +ExltRobofriend TR +A PREPRINT +For the preliminary evaluation of the robot, we tested the following restraining bolts which felt sensible to include, +where the overall constraint is a conjunction of the following: +Do not ask two questions in a row +G(ask_question → X¬ask_question) +Do not wave hands twice in a row +G(wave_hands → X¬wave_hands) +Eventually ask a question +F(ask_question) +Eventually wave hands +F(wave_hands) +Robofriend Learning Algorithm +Robofriend can use reinforcement learning to find a policy which optimizes expected sum of rewards, while still +having a high chance of conforming to the LTL constraints. Specifically, we use an Actor-Critic reinforcement learning +algorithm Konda and Tsitsiklis [1999] which is fed at each step the state of the children and predicts the next action +towards maximizing the expected sum of rewards throughout the episode, accounting also for the LTL constraints +modifying the rewards Giacomo et al. [2020]. +Bearing in mind these type of algorithms often need a massive amount of diverse enough training data to converge, +and with the specific problem setting complexity and training availability, we added an option to conduct imitation +learning of the policy network based on an external "wizard of oz" feedback given in real-time by a supervision/teacher. +Essentially, during the first trials of the robot, a teacher could manually select an appropriate action, and the algorithm +would use that as a labeled training set to conduct initial training of its policy network. This allows to both give a better +baseline to start the learning process as well as avoid unwanted situations in the very beginning of the training process. +Ethical Considerations +During the construction of Robofriend, many ethical considerations came up. We now discuss these in detail. +The first consideration is privacy, which is a potential problem, especially when children are involved. Therefore, we +designed Robofriend in a way that maximizes the privacy of the children. First, Robofriend does not keep any recorded +images. All images are processed online to detect faces (without any attempt to associate faces with names – which the +robot does not have anyway). The only data Robofriend records involve a count of how many faces it detected in any +given frame, as well as the direction of their gaze. +Another potential concern is that Robofriend will replace human contact with a teacher. As previously mentioned, +Robofriend is meant to be a tool to be used by a daycare teacher, so that part of the class can listen to a story, while the +teacher engages more personally with the remaining children. This can help the teacher devote more individual time to +children who experience challenges attending to stories with background distractions, or to supplement activities for +those who experience challenges with social interaction. Thus, Robofriend serves as a supplement for the teaching staff, +and not as a replacement for the teacher. +Conclusion +We have described the construction and software architecture for Robofriend, a robotic story-telling teddy bear. In +future work we intend to examine how children react to Robofriend, as well as its ability to learn to engage children in +the story. +References +P. K. Kuhl. Early language acquisition: cracking the speech code. Nat Rev Neurosci, 5(11):831–843, 2004. +Lebdeva GC Cardillo and P. K. Kuhl. Individual differences in infant speech perception predict language and pre-reading +skills through age 5. In Annual Meeting of the Society for Developmental and Behavioral Pediatrics, 2009. +C Moon, H. Lagercrantz, and P. K. Kuhl. Language experienced in utero affects vowel perception after birth: a +two-country study. ACTA Paediatra, 102(2):156–160, 2013. +9 + +Robofriend TR +A PREPRINT +JS Hutton, T Horowitz-Kraus, AL Mendelsohn, T DeWitt, and SK Holland. Home reading environment and brain +activation in preschool children listening to stories. Pediatrics, 136(3):466–478, 2015. +E Twait, R Farah, N. Shamir, and T. Horowitz-Kraus. Dialogic reading intervention in preschoolers is related to greater +cognitive control: an eeg study. ACTA Paediatra, 108(11):1993–2000, 2019. +Giuseppe De Giacomo, Luca Iocchi, Marco Favorito, and Fabio Patrizi. Restraining bolts for reinforcement learning +agents. In AAAI, pages 13659–13662. AAAI Press, 2020. +Kaipeng Zhang, Zhanpeng Zhang, Zhifeng Li, and Yu Qiao. Joint face detection and alignment using multitask cascaded +convolutional networks. IEEE signal processing letters, 23(10):1499–1503, 2016. +Raimondas Zemblys, Diederick C Niehorster, and Kenneth Holmqvist. gazenet: End-to-end eye-movement event +detection with deep neural networks. Behavior research methods, 2018. +Vijay Konda and John Tsitsiklis. Actor-critic algorithms. Advances in neural information processing systems, 12, 1999. +10 + diff --git a/adAzT4oBgHgl3EQfnP01/content/tmp_files/load_file.txt b/adAzT4oBgHgl3EQfnP01/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6784b1891f960252fb021e85f44647434a9c4b67 --- /dev/null +++ b/adAzT4oBgHgl3EQfnP01/content/tmp_files/load_file.txt @@ -0,0 +1,213 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf,len=212 +page_content='ROBOFRIEND: AN ADPATIVE STORYTELLING ROBOTIC TEDDY BEAR – TECHNICAL REPORT A PREPRINT Ido Glanz∗ Technion Autonomous Systems Program Technion – Israel Institute of Technology Matan Weksler∗ Technion Autonomous Systems Program Technion – Israel Institute of Technology Erez Karpas Faculty of Data and Decision Sciences Technion – Israel Institute of Technology karpase@technion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='il Tzipi Horowitz-Kraus Faculty of Education in Science and Technology Technion – Israel Institute of Technology tzipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='kraus@ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='technion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='il ABSTRACT In this paper we describe Robofriend, a robotic teddy bear for telling stories to young children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Robofriend adapts its behavior to keep the childrens’ attention using reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Keywords Robotics · Education · Reinforcement Learning Introduction Language exposure at an early stage of development is critical for the facilitation of brain networks associated with language Kuhl [2004], Cardillo and Kuhl [2009], Moon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' [2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Storytelling is one form of language exposure, which was found to be associated with a greater engagement not only in language processing but also in visualization and cognitive abilities in children Hutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' [2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Interestingly, it was suggested that it is not the storytelling itself that is related to these improvements, but it is the interaction during the stories that amplify these abilities in children Twait et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' A recent study demonstrated how a group of 4–6-year-old children attending storytelling sessions interactively vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' a group attending non-interactively (storytelling sessions on the screen), shared greater cognitive and language abilities Twait et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Hence, a question was raised regarding this positive effect during interactive (dialogic) storytelling – is the positive effect due to the human interaction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' or due to the interactive nature of the storytelling?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' in other words, will an interactive robot during storytelling result in similar results as the human-based interactive condition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' To study this question, we designed Robofriend (shown in Figure 1) – a robotic teddy bear that reads young children stories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Robofriend is constructed by taking a regular teddy bear and inserting a tablet in its belly, as well as a rudimentary skeleton, motors and sensors that allow it to move its head and arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Robofriend can read a story to a small group of children, with the robot’s main objective being to engage the children, keeping their attention on the story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Thus, although our main motivation for designing Robofriend is the scientific study described above, Robofriend can also serve as a tool that a teacher in a daycare class can use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Robofriend can read a story to one group of children while the teacher engages with the other children in the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Each story that Robofriend can tell is divided into prerecorded video segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Typically, each segment will correspond to showing a still image of one page in the printed book, with a human reading the text on the page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Note that Robofriend does not perform any text to speech, the segments are all prerecorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' At the end of each segment, Robofriend chooses which action to perform out of several actions it has available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Possible actions include asking a simple question about the story (there is a set of prerecorded questions for each segment of the story, Robofriend chooses one of these ∗ denotes equal contribution arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='01576v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='RO] 4 Jan 2023 Robofriend TR A PREPRINT Item Use Quantity Price Bear doll 1 US$30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='00 Display Play video 1 US$200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='00 Camera Monitor kids 1 US$79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='00 Arduino + wiring kit Control Servos 1 US$37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='00 Servo Move head 4 US$24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='00 Speakers Playing Sound 1 US$20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='00 Total: US$462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='00 Table 1: Robofriend Bill of Materials randomly), giving positive feedback (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=', “very good children, I see you are paying attention”), or negative feedback (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=', “children, are you listening to the story?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' As previously stated, the objective of Robofriend is to keep the childrens’ attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' The first step to optimizing something is to measure it, or at least some proxy of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Robofriend uses a camera to measure some things, which can serve as a proxy for engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' First, Robofriend uses computer vision to detect the faces of the children, and the direction of their gazes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' We remark that these faces are anonymous – Robofriend does not try to associate faces to identities in any way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' From these face detection, we extract several measurements: how many faces are looking at the robot, how focused are they on the story (using their relative gaze) as well as how “jumpy” the faces are (an "excitement" metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Aside from the visual attributes, we also monitor the noise level as and its momentary change (its derivative) to serve as supportive metrics capturing the children’s state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' These are aggregated into a reward signal for each camera video frame, and aggregated throughout each story segment to produce a state and reward for each segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Having defined the rewards, we can now try to optimize our objective – the total sum of rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Of course, we do not know in advance what is the right action to take after each story segment, nor do we have a model for how each action will affect the children’s engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Therefore, we chose to use reinforcement learning to control Robofriend’s actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' However, because young children are involved, we do not want to allow the robot to explore sequences of actions we know are not beneficial for the children (for example, always using the negative feedback action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Therefore, we adopt the approach of using LTL “restraining bolts” Giacomo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' [2020], and manually encode what are the allowed trajectories for Robofriend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' In the remainder of this paper, we describe the design of Robofriend in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' We also describe our preliminary evaluation of the robot at a local daycare center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Finally, we conclude with some lessons learned and a discussion of the ethical considerations that arose in this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Robofriend Design We now describe the design of Robofriend in more detail, starting with the mechanical build.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Robofriend Mechanical Build As previously stated Robofriend is constructed by taking a large, 1m tall, teddy bear, and instrumenting it to be able to move its head and arms, play videos and sound, and look at the children it is reading the story to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' First, we inserted an aluminum skeleton into the robot to support the other devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' To do so, we removed the majority of the stuffing and decoupled the head momentarily to mount the camera in the bear’s nose and create a fixture for the servo motors to connect to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' We then mounted a 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='3 inch display, which the robot displays the story segments on, as well as 4 servo motors which are used to move the head and arms (one for each arm and 2 for the the head pan and tilt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' As mentioned above, a camera was placed into the teddy bear’s nose, to monitor the children and measure the reward signal, finally, speakers were connected to play sound, see figure 2 for a schematic diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' This hardware was controlled from a PC, which was connected directly to the camera, display, and speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' The servo motors are controlled by an Arduino Uno (see Figure 3), which was connected to the PC as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' The controller for the servo motors runs in a separate process on the Arduino, following instructions from the PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' The Bill of Materials (BOM) for Robofriend is shown in Table 1, while a detailed BOM with links to each item is available online at: shorturl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='at/efBZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Overall, the total cost to construct Robofriend was less than US$500, making it fairly accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' 2 Robofriend TR A PREPRINT Figure 1: Robofriend in Home Testing 3 Robofriend TR A PREPRINT Figure 2: Hardware schematic block diagram Robofriend Code Architecture To operate and coordinate the different algorithms, hardware and user interactions, a proprietary python software stack was developed and will be briefly described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' The code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='com/IdoMatan/ RoboFriend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' The code architecture is based on RabbitMQ, which is a ROS-like publisher-subscriber framework that implements asynchronous parallel process control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' RabbitMQ allows us to simultaneously control the robot’s servos, camera, screen and any other needed peripherals, as well as to run the algorithms we will describe later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Figure 5 shows the schematic structure of our software, showing the processes and the messages that are passed between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' The main process is the StoryTeller, which coordinates the flow among the other processes and displays the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' This process runs a loop which plays the next story segment, then calls the algorithm service to get the next action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' This loop repeats until the story ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Throughout this loop, the robot moves its head, aiming to center its viewing angle so to center all faces in the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' This flow is illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' The other services are either timer-based (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' send a frame every N milliseconds) or event-driven, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=', a page-ended message would trigger a next action calculation in the algorithm-service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' All metrics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' actions and useful metadata (not 4 Teddy bear inputs Camera Mic Monitoring Teddy bear outputs Dashboard Speakers PC / Laptop 12in TFT screen Database (local) Arduino Uno Micro-controller Servo Servo Servo Servo left hand Right hand Head tilt Head pan Teddy bear control miroRobofriend TR A PREPRINT Figure 3: Arduino sub-system video footage) were logged in real-time in a local Postgres database,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' allowing both post-analysis of the trial as well as live monitoring using a Grafana dashboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Figure 6 shows the simple GUI implemented where a user can run the app, choose one of the supported stories, an operation mode (which will be discussed later) and start and stop the story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Robofriend Algorithm Having described the mechanical construction of Robofriend and the code architecture, we can now discuss the algorithm which is used to control it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' As previously mentioned, our high-level control algorithm involved using reinforcement learning with “restraining bolts” Giacomo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' [2020], to avoid the robot following trajectories which we know will not be good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' We begin by describing our sensing and reward function, then we describe the actions which are available to the robot, and finally, we describe the constraints which were used as the “restraining bolts” in our preliminary evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' 5 9 DIGITAL (PWM 8 IUNO XL RX ARDUINO POWER ANALOG VRobofriend TR A PREPRINT Figure 4: Software Operational Flow Robofriend Sensing and Reward Function As previously mentioned, the reward is based on using computer vision to detect the children’s faces and gaze direction, and on measuring the noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Specifically, we used an MTCNN neural-network-based face detector to detect the faces of the children within the frame Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' [2016], followed by a gaze estimation step for each using GazeNet Zemblys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' [2018] to generate a gaze vector relative to the camera lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' This results in the following metrics: Number of Faces The number of detected faces by the MTCNN face detection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' A change in the number of faces would likely indicate a child walking away or not looking at the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Average relative gaze (attention) For each detected face (denoted by index i), a gaze vector θi, φi is predicted by the GazeNet algorithm, where θi is the lateral (left/right) angle and φi is the vertical (up/down angle) – both angles are relative to the center of the frame (the camera lens center).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Based on these measurements, we define the gaze component of the reward as: rgaze := 1 n n � i=1 cos(θi) Roughly speaking, our attention metric, ranging from 0-1, corresponds to how focused the children are on the robot as opposed to looking around the room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Excitement Using consecutive frames we are able to calculate a per-face jumpiness metric corresponding to how still the children are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' This is another proxy to their attention and engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Formally, let us denote the positions of (centers of) faces detected in the first image by xi, yi (for i = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' n), and the positions of the faces detected in the next image by x′ j, y′ j (for j = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' As the faces do not have identities associated with them, we must first align the faces in the first image to the faces in the next image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' We do so greedily by finding, for each face position xi, yi in the first image, the closest face position x′ j, y′ j (using Euclidean distance) in the next image (which is also under a feasible max possible distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' We then define the jumpiness for face i by 6 JsonConfigfile (storypages, possible actions) Start playing video - Page 1 Choose Action Algorithm - Policy Maker GUI App Grafana Dashboard Init RabbitMQ Start playing video -Page 2 Init services ChooseAuto/Manual Choose Action (checkbox) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Choose story (start playing) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Get Action if in manual mode Start playing video - Page N Train cycleRobofriend TR A PREPRINT Figure 5: Software Architecture Diagram this Euclidean distance, and the total jumpiness is the sum of jumpiness for each face, that is: rjump := n � i=1 min j � (xi − x′ j)2 + (yi − y′ j)2 Noise Level The average noise level over a 1 second period as measured by the microphone inside the teddy bear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' This is a proxy for how much the children are talking to each other instead of listening to the story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' We denote this by rnoise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Deviation of Noise Level The derivative of the noise level, again averaged every second, denoted by rnd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' The motivation behind this metric is to capture changes in the sound level within a page, indicating the children are getting noisier or quieter potentially due to the effect of the previous action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' To aggregate the reward signal throughout the duration of a story segment, we average each of these metrics for each video frame that belongs to this segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Note that we use average instead of sum, as different story segments have different durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' This gives us a state vector ⟨rgaze, rjump, rnoise, rnd⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Finally, to aggregate these different metrics into a single reward function we use a weighed sum, and thus our reward denoted by r is defined as: r := α1 ∗ rgaze − α2 ∗ rjump − α3 ∗ rnoise + α4 ∗ rnd + ltl_reward 7 APP (GUI) Setup story & StoryTeller Choose pause/play story Control video Input action Execute actions on manual Get Action End of Page (EoP) Pause (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=') (manual mode) Play/Pause ServoService Execute action Control servos (4) thru VideoService arduino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Acceptspan+tiltfor play/pause video head and tilt for each hand Get Action (auto mode) Avg head angles (always sent but only executed if storyteller allows) CamService State (gaze, n_kids, movement) AlgoService Calc avg gaze Calc actionbased on Number of kids camandmic Avg Movement?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' StatusLog Noise Level MicService LogService Record avg noise Saveto DB level (and derivative?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' )Robofriend TR A PREPRINT Figure 6: A simple python-based graphical user interface to interact with the robot Where the ltl_reward will be described in details below, but conceptually corresponds to a set of pre-defined restraining rules the robot should learn to obey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' We conclude the discussion of the reward by noting that these measurements serve as a proxy for the real reward (the children’s attention), which we can not measure directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Robofriend States and Actions As we described above, Robofriend reads a story, which is divided into segments, and chooses which action to perform after every segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Note that the order between the story segments is linear, and so there is no choice with regard to which story segment to read next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' The only choice is which action to perform after every story segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Robofriend’s actions correspond to different types of feedback it can give the children and are divided into: Positive Feedback This action randomly chooses from a set of positive feedback sentences, such as “great job” or “you are listening nicely”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Negative Feedback This action chooses randomly from a set of (mildly) negative feedback sentences, such as “please pay attention” or “please be quiet”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Question This action chooses a random question relating to the story segment Robofriend just finished reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Robofriend does not attempt to extract an answer but merely pauses for an appropriate amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Continue Continue immediately to the next story segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Move head and arms Execute a series of random head and arm movements for a few seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Note that all of these actions are applicable regardless of where Robofriend is in the story (the question action changes as a function of it, but is always applicable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Furthermore, the only information Robofriend extracts from its sensors is the reward signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Thus, there is no notion of state for where we are in the story, and the only state information is the state vector ⟨rgaze, rjump, rnoise, rnd⟩ described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' However, the LTL constraints used for the “restraining bolts” do carry their own notion of state, as we discuss next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Robofriend Restraining Bolt The final part of Robofriend’s control architecture is the “restraining bolt” – the LTL constraint which the robot needs to obey Giacomo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' That is, we specify LTL trajectory constraints over allowed and forbidden trajectories for the robot, where in this case a trajectory is a sequence of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' For example, we can specify the constraint to not ask two questions in a row as G(ask_question → X¬ask_question) (that is, if the current action is ask question, then the next action is not ask question).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' This ensures the robot will not ask questions constantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' 8 RoboBear ApP RoboBear Controller Gui Select story from drop-down and press start Usemame: admin Session #: 1663055759 manual Choose Story: AptForRent Start Story!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Play/Pause ExltRobofriend TR A PREPRINT For the preliminary evaluation of the robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' we tested the following restraining bolts which felt sensible to include,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' where the overall constraint is a conjunction of the following: Do not ask two questions in a row G(ask_question → X¬ask_question) Do not wave hands twice in a row G(wave_hands → X¬wave_hands) Eventually ask a question F(ask_question) Eventually wave hands F(wave_hands) Robofriend Learning Algorithm Robofriend can use reinforcement learning to find a policy which optimizes expected sum of rewards,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' while still having a high chance of conforming to the LTL constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Specifically, we use an Actor-Critic reinforcement learning algorithm Konda and Tsitsiklis [1999] which is fed at each step the state of the children and predicts the next action towards maximizing the expected sum of rewards throughout the episode, accounting also for the LTL constraints modifying the rewards Giacomo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Bearing in mind these type of algorithms often need a massive amount of diverse enough training data to converge, and with the specific problem setting complexity and training availability, we added an option to conduct imitation learning of the policy network based on an external "wizard of oz" feedback given in real-time by a supervision/teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Essentially, during the first trials of the robot, a teacher could manually select an appropriate action, and the algorithm would use that as a labeled training set to conduct initial training of its policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' This allows to both give a better baseline to start the learning process as well as avoid unwanted situations in the very beginning of the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Ethical Considerations During the construction of Robofriend, many ethical considerations came up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' We now discuss these in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' The first consideration is privacy, which is a potential problem, especially when children are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Therefore, we designed Robofriend in a way that maximizes the privacy of the children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' First, Robofriend does not keep any recorded images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' All images are processed online to detect faces (without any attempt to associate faces with names – which the robot does not have anyway).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' The only data Robofriend records involve a count of how many faces it detected in any given frame, as well as the direction of their gaze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Another potential concern is that Robofriend will replace human contact with a teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' As previously mentioned, Robofriend is meant to be a tool to be used by a daycare teacher, so that part of the class can listen to a story, while the teacher engages more personally with the remaining children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' This can help the teacher devote more individual time to children who experience challenges attending to stories with background distractions, or to supplement activities for those who experience challenges with social interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Thus, Robofriend serves as a supplement for the teaching staff, and not as a replacement for the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' Conclusion We have described the construction and software architecture for Robofriend, a robotic story-telling teddy bear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' In future work we intend to examine how children react to Robofriend, as well as its ability to learn to engage children in the story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfnP01/content/2301.01576v1.pdf'} +page_content=' References P.' metadata={'source': 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--- /dev/null +++ b/atAzT4oBgHgl3EQfnP1v/content/tmp_files/2301.01577v1.pdf.txt @@ -0,0 +1,545 @@ +Comparison of Shock-Boundary Layer Interactions in +Adiabatic and Isothermal Supersonic Turbine Cascades +Hugo F. S. Lui∗, Tulio R. Ricciardi † and William R. Wolf‡ +University of Campinas, UNICAMP, Campinas, SP, 13083-860, Brazil +Carlos A. J. B. Junior§ +École Nationale Supérieure d’Arts et Métiers, ENSAM, Paris, 75013, France +Wall-resolved large eddy simulations are employed to investigate the shock-boundary layer +interactions (SBLIs) in a supersonic turbine cascade. An analysis of the suction side separation +bubbles forming due to the SBLIs is presented for adiabatic and isothermal (cooled) walls. Flow +snapshots indicate that the separation bubble contracts and expands in a similar fashion for +both thermal boundary conditions. However, the skin-friction coefficient distributions reveal +a downstream displacement of the separation region when cooling is applied. The separation +bubble is also smaller for this setup compared to the adiabatic one. A steeper pressure rise +is observed for the isothermal wall downstream of the incident oblique shock, and this occurs +because the incident shock wave gets closer to the blade surface when cooling is applied. The +Reynolds stresses are computed to investigate the effects of wall temperature on the turbulence +activity. While the levels of the tangential stresses are similar for the cases analyzed, those for +the wall-normal component are higher for the cooled wall. +I. Introduction +S +upersonic fluid machinery are applied in high-speed propulsion and power generation systems due to their high +power density [1]. In supersonic turbines, inlet shock waves are formed and interact with the boundary layers of +neighboring blades. The shock-boundary layer interactions (SBLIs) can increase the aerodynamic drag due to flow +separation and induce higher heat transfer rates to the blade surface. They can also be a source of flow unsteadiness, +where multiple frequencies are excited due to motion of the incident and reflected shock waves, breathing of the +separation bubble, besides the incoming turbulent boundary layer. Typically, the shock wave motion leads to strong +pressure fluctuations that can compromise the system’s structural integrity [2–5]. +Most studies of SBLIs have considered adiabatic wall conditions and, thus, the effects of surface heat transfer are not +fully explored. Schülein [6] used non-intrusive techniques to perform heat transfer and skin-friction measurements in the +impingement of an incident oblique shock wave on a flat plate with isothermal wall conditions. Their results show that +within the separation region, the heat flux increases in the streamwise direction, while the skin-friction decreases. Jaunet +et al. [7] investigated experimentally the impact of the wall temperature on a shock-induced boundary layer separation. +They observed that the interaction length considerably increases when the wall temperature is raised. Bernardini et al. +[8] and Volpiani et al. [9] carried out direct numerical simulations (DNS) to investigate the wall temperature effects on +the physics of SBLIs. Results revealed that wall cooling significantly reduces the size of the separation bubble and +interaction scales, while the opposite behavior is noticed in the case of wall heating. +In the present work, a high-order overset compressible large eddy simulation (LES) methodology is employed to +investigate the flow in a supersonic turbine cascade with two different wall thermal boundary conditions. These include +an adiabatic and a cooled walls, where the wall to inlet temperature ratio is set as 𝑇𝑤/𝑇∞ = 0.75. First, the numerical +methodology is described including the grid details and flow configurations. Spanwise and time averaged pressure +and skin-friction coefficients, as well as the mean flow fields, are presented to assess the effect of cooling on the size +and form of the separation bubble. Then, flow snapshots are analyzed to investigate the features of the separation +bubbles and the shear layer dynamics at different instants of the SBLI. Finally, the effects of the wall thermal boundary +conditions on the turbulence activity are analyzed by assessing the Reynolds stress distributions. +∗Graduate Student, School of Mechanical Engineering, E-mail: hugo.slui@gmail.com. +†Postdoctoral Research Associate, National Center for Supercomputing Applications, University of Illinois Urbana-Champaign, E-mail: +tricci@illinois.edu. +‡Associate Professor, School of Mechanical Engineering, Department of Energy, E-mail: wolf@fem.unicamp.br. +§Research Engineer, Laboratoire de Dynamique de Fluides (Dynfluid), E-mail: junior.junqueira@ensam.eu. +1 +arXiv:2301.01577v1 [physics.flu-dyn] 4 Jan 2023 + +II. Numerical Methodology +The present wall-resolved large eddy simulations solve the compressible Navier-Stokes equations in a curvilinear +coordinate system. The fluid is assumed to be a calorically perfect gas, where the molecular viscosity 𝜇 is considered +to depend on the local temperature through the nondimensional Sutherland’s law. The spatial discretization of the +governing equations is performed using a sixth-order accurate finite-difference compact scheme [10] implemented on a +staggered grid. A sixth-order compact interpolation scheme is also used to obtain flow quantities on the different nodes +of the staggered grid configuration. +Two grids are employed in the present simulations: one is a body-fitted O-grid block which surrounds the airfoil and +the other is an H-grid block used to enforce the pitchwise periodicity of the cascade. In the O-grid, the time integration +of the equations is carried out by the implicit second-order scheme of Beam and Warming [11]. This method overcomes +the stiffness problem arising from the wall-resolving boundary layer mesh. In the background H-grid block, a third-order +Runge-Kutta scheme is used for time advancement of the governing equations. A fourth-order Hermite interpolation +scheme [12, 13] is used to exchange information between grid blocks in an overlapping zone. Further details about the +numerical procedure can be found in [13]. +Due to the non-dissipative characteristics of the compact finite-difference schemes, numerical instabilities may +arise from mesh non-curvature, interpolation between the overset grids, and boundary conditions. To preserve stability +of the numerical simulations, the high wavenumber compact filter presented by Lele [14] is applied in flow regions +far away from solid boundaries at each time step. A shock capturing scheme is also employed to capture the shock +waves forming in the present flows. In order to introduce minimal numerical dissipation in the vicinity of the shocks, +without damping the small scales of turbulence, the localized artificial diffusivity (LAD) method [15] is employed to +compute the artificial bulk viscosity and thermal conductivity. The approach LAD-D2-0 proposed by Kawai et al. [16] +is employed here with no artificial shear viscosity. In order to transition the boundary layers, we apply a body forcing +on the RHS of the Navier-Stokes equations, as described by Sansica [17]. Here, an unsteady actuation with a random +spanwise treatment is assumed and the amplitude of the disturbances are chosen experimentally in order to guarantee a +bypass transition with minimal flow disturbance. More details of the numerical procedure can be found in [5]. +III. Flow and Mesh Configurations +This section shows details of the flow configuration studied and describes the computational grid used in the LES +calculations. Figure 1 (a) presents the geometrical parameters and flow conditions. The inlet Mach number is set as 𝑀 += 2.0 and the Reynolds number based on the inlet velocity 𝑈∞ and axial blade chord is 𝑅𝑒 = 200,000. The ratio of +specific heats is chosen as 𝛾 = 1.31, the Prandlt number is 𝑃𝑟 = 0.747 and the ratio of the Sutherland constant over inlet +temperature is set as 𝑆𝜇/𝑇∞ = 0.07182. These conditions are chosen based on previous studies [4, 5, 18]. +(a) +(b) +Fig. 1 +Schematics of (a) flow configuration and geometrical parameters, and (b) computational domain skipping +every 5 grid points. +Figure 1 (b) displays a schematic of the overset grid employed in the LES along with the implemented boundary +conditions. The O-grid block has 1200 × 280 × 144 points and is embedded in the background Cartesian grid block of +size 960 × 280 × 72. Therefore, the grid has approximately 68, 000, 000 points. Depending on the case, adiabatic or +isothermal boundary conditions are applied along the blade surface. For the latter, the wall to inlet temperature ratio is +𝑇𝑤/𝑇∞ = 0.75, representing a cooled wall. Supersonic inflow boundary conditions are used to set the inlet conditions. +For the outflow, a boundary condition based on the Navier-Stokes characteristic boundary condition (NSCBC) [19] +is employed. A damping sponge is also applied near the inflow and outflow boundaries to minimize reflections of +disturbances [10, 20]. Periodic boundary conditions are used in the 𝑦-direction of the background grid, according to Fig. +2 + +Adiabatic or isothermal +_no-slip wall +M= 2.0 +Qout +20° +Re = 200,000 +Supersonic Inflow +Periodic +NSCBC +g = 0.7ca += 1.31 +Pr = 0.747 +span = 0.12ca +y +0 = u?o +Inflow Sponge +Periodic +Outflow Sponge +>C +-1 +0 +2 +3 +41 (a), in order to simulate a linear cascade of blades and periodic boundary conditions are also applied in the spanwise +direction, to enforce a statistically homogeneous flow along the span. +For the adiabatic wall case, the grid resolution in terms of wall units is kept in the range given by 6 < Δ𝑠+ < 25, +0.1 < Δ𝑛+ < 0.3, and 3 < Δ𝑧+ < 9, where 𝑠, 𝑛 and 𝑧 represent the streamwise, wall-normal and spanwise flow +coordinates. For the isothermal wall simulation, the near-wall grid spacing ranges from 15 < Δ𝑠+ < 60, 0.2 < Δ𝑛+ < 0.6, +and 6 < Δ𝑧+ < 19. These numbers are computed for regions where the boundary layers are fully developed and in +equilibrium, away from the tripping and recirculation regions. It is worthwhile to mention that the same computational +grid is used for both cases, but higher values in terms of wall units are obtained for the isothermal wall case due to a +inherent reduction of the viscous length scales caused by cooling. +The simulation is initialized with a uniform flow and statistics are computed after the initial transients are discarded. +In the simulations, a variable time step is computed based on an inviscid CFL parameter of 0.8. The body-force tripping +is applied at 0.22 < 𝑥/𝑐𝑎𝑥 < 0.27 for the suction side, and at 0.10 < 𝑥/𝑐𝑎𝑥 < 0.15 for the pressure side. The wall +normal height of the body-foce region is 𝛿 = 0.001𝑐𝑎𝑥 and the actuation changes every Δ𝑡 ≈ 0.003 in a spanwise-random +fashion. +IV. Results +This section presents results obtained by the LES computed for adiabatic and isothermal (cooled) wall boundary +conditions. Flow quantities are collected for 4 flow through times, based on the inlet velocity and blade axial chord. +Figure 2 shows iso-surfaces of 𝑄-criterion colored by the 𝑢-velocity component together with a background view of +density gradient magnitude, |∇𝜌|. The top and bottom rows present results for the adiabatic and cooled wall cases, +respectively. +(a) +(b) +(c) +(d) +Fig. 2 +Iso-surfaces of 𝑄-criterion colored by 𝑢-velocity component for the adiabatic (top) and cooled (bottom) +wall cases. The background plane displays the shock waves by visualizing the density gradient magnitude |∇𝜌|. +In Figs. 2 (a) and (c), we can observe the complex shock structure across the turbine passage. The detached oblique +shock waves generated at the airfoil leading edges interact with the boundary layers of the neighboring blades and are +reflected across the cascade. On the pressure side, the incident shock wave becomes normal to the wall and, then, a +Mach reflection is formed, while an oblique shock reflection is generated on the suction side. To highlight the effect +3 + +of cooling on the SBLI, a detailed view of the flow field can be seen in Figs. 2 (b) and (d), where one can observe +differences between the lengths of the separation bubbles, especially on the suction side. For the cooled wall, a smaller +recirculation region is noticed. +(a) Skin friction coefficient 𝑐 𝑓 +(b) Pressure coefficient 𝑐𝑝. +Fig. 3 +Mean skin-friction and pressure coefficient distributions for the adiabatic (black) and cooled (blue) wall +cases. The distributions are shown only along the suction side. +(a) +(b) +(c) +(d) +Fig. 4 +Time-averaged contours of normalized 𝑢-velocity (top) and temperature (bottom) for the adiabatic (left) +and cooled (right) wall cases. The black lines display the shock waves visualized by pressure gradient magnitude. +The black dashed lines show the sonic line. +The mean skin-friction coefficient distribution 𝑐 𝑓 = +𝜏𝑤 +0.5𝜌∞𝑈2∞ is provided in Fig. 3(a) for the blade suction side. +This plot shows the presence of a separation bubble characterized by locations where 𝑐 𝑓 < 0, which is delimited by a +horizontal dashed line. The effect of cooling on the size of the recirculation region is evident. For the isothermal case, +one can observe a downstream displacement of the separation region compared to the adiabatic wall setup. On the other +hand, the reattachment locations are similar for both cases. Hence, the cooled wall depicts a smaller separation bubble. +For the adiabatic wall case, the time-averaged characteristic length of the separation bubble is ⟨𝐿𝑆𝐵⟩ = 0.16𝑐𝑎𝑥 and it is +observed along 0.70 < 𝑥/𝑐𝑎𝑥 < 0.86. For the cooled wall, ⟨𝐿𝑆𝐵⟩ = 0.10𝑐𝑎𝑥 and it is formed on 0.75 < 𝑥/𝑐𝑎𝑥 < 0.85. +4 + +Adiabatic +0.006 +Isothermal +0.004 +0.002 +0.000 +-0.002 +0.6 +0.7 +0.8 +0.9 +1.0 +0.5-0.05 +Adiabatic +Isothermal +-0.10 +-0.15 +-0.20 +0.5 +0.6 +0.7 +0.8 +0.9 +1.00.0 0.4 0.8 1.2 1.6 2.0 +0.6 +0.7 +0.8 +0.90.0 0.4 0.8 1.2 1.6 2.0 +0.6 +0.7 +0.8 +0.90.8 1.1 1.3 1.6 +0.6 +0.7 +0.8 +0.9 +X0.8 0.9 1.0 1.1 +0.6 +0.7 +0.8 +0.9 +XAfter a small negative skin-friction coefficient plateau, a similar recovery is observed downstream of the reattachment +location for both cases. +Figure 3 (b) plots the mean pressure coefficient 𝑐 𝑝 = +𝑝−𝑝∞ +0.5𝜌∞𝑈2∞ along the airfoil chord. For both adiabatic and cold +wall cases, it is possible to note two pressure rises: the first occurs near the separation point due to the compression +waves formed upstream of the separation bubble, and the second takes place near the reattachment location as a result of +the incident shock impingement and the turbulence amplification mechanism [21]. For the cooled wall setup, a steeper +variation of 𝑐 𝑝 is observed, especially for the second pressure rise. +To highlight the influence of the wall thermal boundary conditions on the size and shape of the separation bubbles, +the mean (spanwise and time averaged) 𝑢-velocity contours are presented in Figs. 4 (a) and (b), for the adiabatic and +isothermal cases, respectively. Here, the velocity component is normalized by the inlet speed of sound. These figures +reinforce the findings observed in the friction coefficient distributions. The main effect of wall cooling is to reduce the +viscous length scales near the wall [8, 9] which in turn affects the shock penetration, as shown in 4 (a) and (b). One can +see that the impinging shock penetrates deeper in the boundary layer for the cooled wall case due to the displacement of +the sonic line (displayed as a dashed line) towards the wall. This effect is responsible for the steeper variation in the +pressure coefficient observed in Fig. 3(b). One can also see that, for the cooled wall, the incident shock reaches further +downstream compared to the adiabatic case. +Figures 4 (c) and (d) show the mean temperature fields for the the adiabatic and cold wall boundary conditions, +respectively. The values are presented normalized by the inlet temperature. For the former case, one can observe that a +region of maximum temperature occurs within the separation bubble. On the other hand, when cooling is applied, higher +temperature values are observed in the free shear layer, downstream the bubble. For the adiabatic wall, friction from the +shear stresses near the wall and around the bubble are converted into heat which is transferred along the boundary layer +and inside the bubble. This causes the near-wall flow to reach higher temperatures. However, heat from the flow is +transferred to the blade in the isothermal case, which has a lower temperature than the surrounding flow. For the cooled +wall case, the maximum temperature values are observed along the free shear layer, behind the bubble, due to strong +shearing effects that cause aerodynamic heating. +Fig. 5 +Temporal variation of the suction side separation bubble length 𝐿𝑆𝐵 for the adiabatic (top) and cooled +(bottom) wall cases. +The temporal evolution of the separation bubble length 𝐿𝑆𝐵 is shown in Fig. 5 for the adiabatic and isothermal walls. +The instantaneous length of the bubble is defined as the distance between the instantaneous reattachment and separation +locations. One can observe that the separation region undergoes a contraction/expansion motion for both cases. The +excursions from the mean appear to be similar for both cases. A spectral analysis of this signal should provide further +information on the frequency scales related to the bubble motion and such analysis should be conducted in future work, +once longer signals are collected for statistical convergence of the lower frequencies of interest. +To highlight the 2D structure of the suction side separation bubble and shear layer at different time instants, snapshots +of 𝑧-vorticity are displayed in Fig. 6 for both thermal boundary conditions. These snapshots correspond to the instants +indicated by the letters “a-d” in Fig. 5. The region enclosed by the green line shows the separation region and the black +lines display the impinging shocks. In addition, the mean separation and reattachment positions are indicated by the +orange and cyan squares, respectively. For both cases, when the bubble suffers a contraction, the instantaneous separation +(reattachment) point moves downstream (upstream) with respect to its mean value, as can be visualized in Figs. 6(a) and +(c). On the other hand, when the bubble undergoes an expansion, one can observe the upstream (downstream) movement +of the instantaneous separation (reattachment) point with respect to its mean position. This indicates that the bubble has +5 + +0.20 +0.1( +Adiabatic +Isothermal +0.05 +3 +t(a) +(b) +(c) +(d) +Fig. 6 +Spanwise 𝑧-vorticity contours at different time instants for the adiabatic (top) and cooled (bottom) wall +cases. The green line delimits the bubble while the black line shows the incident shock wave. +a breathing pattern, but its central position does not have large excursions from the mean. Figure 6 also shows that the +shear layer downstream of the bubble is more diffused for the adiabatic case, while more concentrated vorticity values +are observed when cooling is applied. These findings corroborate the maximum temperature values observed in Figs. +4(c) and (d). For example, in the adiabatic case, the shear layer around the bubble creates a zone of intense heating. +The effects of the thermal boundary conditions on the turbulence properties, are investigated by the tangential and +wall-normal Reynolds stresses, ⟨𝑢𝑡𝑢𝑡⟩ and ⟨𝑢𝑛𝑢𝑛⟩, respectively, and the turbulent kinetic energy (TKE) are presented in +Fig. 7. In this figure, the top and bottom rows display results for the adiabatic and isothermal walls, respectively. In +Figs. 7 (a) and (d), it can be seen that the highest fluctuations of ⟨𝑢𝑡𝑢𝑡⟩ are observed just upstream of the shock-bubble +interaction for both cases, with similar fluctuation values. The amplification of ⟨𝑢𝑡𝑢𝑡⟩ is associated with the development +of the shear layer [21]. The peak values of ⟨𝑢𝑛𝑢𝑛⟩ are found along the free shear layer downstream of the bubble. The +magnitude of ⟨𝑢𝑛𝑢𝑛⟩ decreases when cooling is applied. In Figs. 7 (c) and (f), one can observe that the turbulent kinetic +energy combines the trends observed from the ⟨𝑢𝑡𝑢𝑡⟩ and ⟨𝑢𝑛𝑢𝑛⟩ components. In addition, before the SBLI, we can +notice a downstream displacement of the maximum turbulence amplification location for the cooled wall case. This +occurs due to the higher shock penetration discussed previously. +V. Conclusions +Wall-resolved large eddy simulations are employed to investigate thermal effects in a supersonic turbine cascade. +Simulations are performed for adiabatic and isothermal boundary conditions, where in the latter case the blade is +cooled. For the present flow configurations, oblique shock waves are generated at the leading edges of the airfoils, and +they interact with the boundary layers of the neighboring blades. A study of the shock-boundary layer interactions is +presented for the blade suction side, where an incident oblique shock reflects on the wall leading to the formation of a +separation bubble. +The impact of the thermal boundary conditions on the separation bubbles is investigated. The distributions of mean +skin-friction show that the separation bubble is considerably smaller for the cooled wall compared to the adiabatic case. +Pressure coefficient distributions show that a steeper pressure rise occurs downstream the incident shock wave for the +cooled wall. For this case, cooling induces the formation of a thinner boundary layer and the sonic line forms closer to +the wall. Results in terms of mean velocity contours reveal that the more pronounced pressure rise occurs due to the +6 + +-0.04 +-100 +-140 +-180 +-220 +-260 +-0.08 +-300 +-0.12 +-0.16 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +X-0.04 +-100 +-140 +-180 +-220 +-260 +-0.08 +-300 +-0.12 +-0.16 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +X-0.04 +-100 +-140 +-180 +-220 +-260 +-0.08 +-300 +-0.12 +-0.16 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +X-0.04 +-100 +-140 +-180 +-220 +-260 +-0.08 +-300 +-0.12 +-0.16 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +X(a) +(b) +(c) +(d) +(e) +(f) +Fig. 7 +Turbulence quantities on the suction side for the adiabatic (top) and cooled (bottom) cases: ⟨𝑢𝑡𝑢𝑡⟩ (left), +⟨𝑢𝑛𝑢𝑛⟩ (middle) and TKE (right). +higher penetration of the incident shock in the isothermal (cooled) wall. Maximum temperature values are observed +along the bubble, for the adiabatic case, and the free shear layer, for the cooled wall. In the former case, aerodynamic +heating is transferred to the bubble due to a surrounding shear layer. For the latter case, intense shearing is observed +along the free shear layer, behind the bubble, and leads to high temperatures. +An analysis of the instantaneous separation and reattachment locations demonstrates that the separation bubbles +have a breathing pattern of contractions and expansions. For the contraction motions, the instantaneous separation +point moves downstream while the reattachment point moves upstream. The other way around is observed for the +expansion motions. The tangential Reynolds stress distributions reach maximum values just upstream the shock-bubble +interactions, being similar for both the adiabatic and isothermal walls. However, due to the higher shock penetration of +the isothermal wall, the peaks appear more downstream along the blade chord. The wall-normal Reynolds stresses +reach maximum amplitudes downstream the SBLI and they are more pronounced for the adiabatic wall. In future work, +further analysis of the SBLI dynamics will be provided for both suction and pressure side boundary layers. +Acknowledgments +The authors acknowledge the financial support received from Fundação de Amparo à Pesquisa do Estado de São Paulo, +FAPESP, under grants No. 2013/08293-7, 2019/26196-5 and 2021/06448-0. The authors also thank Conselho Nacional +de Desenvolvimento Científico e Tecnológico, CNPq, for supporting this research under grants No. 407842/2018-7 and +308017/2021-8. This work was granted access to the HPC resources of IDRIS under the allocation 2021-A0112A12067 +made by GENCI. +References +[1] Paniagua, G., Iorio, M., Vinha, N., and Sousa, J., “Design and analysis of pioneering high supersonic axial turbines,” +International Journal of Mechanical Sciences, Vol. 89, 2014, pp. 65 – 77. +[2] Babinsky, H., and Harvey, J., Shock Wave-Boundary-Layer Interactions, Cambridge Aerospace Series, Cambridge University +Press, 2011. +[3] Gaitonde, D. V., “Progress in shock wave/boundary layer interactions,” Progress in Aerospace Sciences, Vol. 72, 2015, pp. +80–99. +[4] Lui, H., Wolf, W. 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A., “Approximation of radiation boundary conditions,” Journal of Computational Physics, Vol. 41, +No. 1, 1981, pp. 115 – 135. +[21] Fang, J., Zheltovodov, A. A., Yao, Y., Moulinec, C., and Emerson, D. R., “On the turbulence amplification in shock-wave/turbulent +boundary layer interaction,” Journal of Fluid Mechanics, Vol. 897, 2020, p. A32. +8 + diff --git a/atAzT4oBgHgl3EQfnP1v/content/tmp_files/load_file.txt b/atAzT4oBgHgl3EQfnP1v/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..346bb4df1cee9e1627d58c2716b616cc5a0687bd --- /dev/null +++ b/atAzT4oBgHgl3EQfnP1v/content/tmp_files/load_file.txt @@ -0,0 +1,590 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf,len=589 +page_content='Comparison of Shock-Boundary Layer Interactions in Adiabatic and Isothermal Supersonic Turbine Cascades Hugo F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Lui∗, Tulio R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Ricciardi † and William R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Wolf‡ University of Campinas, UNICAMP, Campinas, SP, 13083-860, Brazil Carlos A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Junior§ École Nationale Supérieure d’Arts et Métiers, ENSAM, Paris, 75013, France Wall-resolved large eddy simulations are employed to investigate the shock-boundary layer interactions (SBLIs) in a supersonic turbine cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' An analysis of the suction side separation bubbles forming due to the SBLIs is presented for adiabatic and isothermal (cooled) walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Flow snapshots indicate that the separation bubble contracts and expands in a similar fashion for both thermal boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' However, the skin-friction coefficient distributions reveal a downstream displacement of the separation region when cooling is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The separation bubble is also smaller for this setup compared to the adiabatic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' A steeper pressure rise is observed for the isothermal wall downstream of the incident oblique shock, and this occurs because the incident shock wave gets closer to the blade surface when cooling is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The Reynolds stresses are computed to investigate the effects of wall temperature on the turbulence activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' While the levels of the tangential stresses are similar for the cases analyzed, those for the wall-normal component are higher for the cooled wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Introduction S upersonic fluid machinery are applied in high-speed propulsion and power generation systems due to their high power density [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' In supersonic turbines, inlet shock waves are formed and interact with the boundary layers of neighboring blades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The shock-boundary layer interactions (SBLIs) can increase the aerodynamic drag due to flow separation and induce higher heat transfer rates to the blade surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' They can also be a source of flow unsteadiness, where multiple frequencies are excited due to motion of the incident and reflected shock waves, breathing of the separation bubble, besides the incoming turbulent boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Typically, the shock wave motion leads to strong pressure fluctuations that can compromise the system’s structural integrity [2–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Most studies of SBLIs have considered adiabatic wall conditions and, thus, the effects of surface heat transfer are not fully explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Schülein [6] used non-intrusive techniques to perform heat transfer and skin-friction measurements in the impingement of an incident oblique shock wave on a flat plate with isothermal wall conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Their results show that within the separation region, the heat flux increases in the streamwise direction, while the skin-friction decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Jaunet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' [7] investigated experimentally the impact of the wall temperature on a shock-induced boundary layer separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' They observed that the interaction length considerably increases when the wall temperature is raised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Bernardini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' [8] and Volpiani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' [9] carried out direct numerical simulations (DNS) to investigate the wall temperature effects on the physics of SBLIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Results revealed that wall cooling significantly reduces the size of the separation bubble and interaction scales, while the opposite behavior is noticed in the case of wall heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' In the present work, a high-order overset compressible large eddy simulation (LES) methodology is employed to investigate the flow in a supersonic turbine cascade with two different wall thermal boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' These include an adiabatic and a cooled walls, where the wall to inlet temperature ratio is set as 𝑇𝑤/𝑇∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' First, the numerical methodology is described including the grid details and flow configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Spanwise and time averaged pressure and skin-friction coefficients, as well as the mean flow fields, are presented to assess the effect of cooling on the size and form of the separation bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Then, flow snapshots are analyzed to investigate the features of the separation bubbles and the shear layer dynamics at different instants of the SBLI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Finally, the effects of the wall thermal boundary conditions on the turbulence activity are analyzed by assessing the Reynolds stress distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' ∗Graduate Student, School of Mechanical Engineering, E-mail: hugo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='slui@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' †Postdoctoral Research Associate, National Center for Supercomputing Applications, University of Illinois Urbana-Champaign, E-mail: tricci@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' ‡Associate Professor, School of Mechanical Engineering, Department of Energy, E-mail: wolf@fem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='unicamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' §Research Engineer, Laboratoire de Dynamique de Fluides (Dynfluid), E-mail: junior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='junqueira@ensam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='eu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='01577v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='flu-dyn] 4 Jan 2023 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Numerical Methodology The present wall-resolved large eddy simulations solve the compressible Navier-Stokes equations in a curvilinear coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The fluid is assumed to be a calorically perfect gas, where the molecular viscosity 𝜇 is considered to depend on the local temperature through the nondimensional Sutherland’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The spatial discretization of the governing equations is performed using a sixth-order accurate finite-difference compact scheme [10] implemented on a staggered grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' A sixth-order compact interpolation scheme is also used to obtain flow quantities on the different nodes of the staggered grid configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Two grids are employed in the present simulations: one is a body-fitted O-grid block which surrounds the airfoil and the other is an H-grid block used to enforce the pitchwise periodicity of the cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' In the O-grid, the time integration of the equations is carried out by the implicit second-order scheme of Beam and Warming [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' This method overcomes the stiffness problem arising from the wall-resolving boundary layer mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' In the background H-grid block, a third-order Runge-Kutta scheme is used for time advancement of the governing equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' A fourth-order Hermite interpolation scheme [12, 13] is used to exchange information between grid blocks in an overlapping zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Further details about the numerical procedure can be found in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Due to the non-dissipative characteristics of the compact finite-difference schemes, numerical instabilities may arise from mesh non-curvature, interpolation between the overset grids, and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' To preserve stability of the numerical simulations, the high wavenumber compact filter presented by Lele [14] is applied in flow regions far away from solid boundaries at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' A shock capturing scheme is also employed to capture the shock waves forming in the present flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' In order to introduce minimal numerical dissipation in the vicinity of the shocks, without damping the small scales of turbulence, the localized artificial diffusivity (LAD) method [15] is employed to compute the artificial bulk viscosity and thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The approach LAD-D2-0 proposed by Kawai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' [16] is employed here with no artificial shear viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' In order to transition the boundary layers, we apply a body forcing on the RHS of the Navier-Stokes equations, as described by Sansica [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Here, an unsteady actuation with a random spanwise treatment is assumed and the amplitude of the disturbances are chosen experimentally in order to guarantee a bypass transition with minimal flow disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' More details of the numerical procedure can be found in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Flow and Mesh Configurations This section shows details of the flow configuration studied and describes the computational grid used in the LES calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Figure 1 (a) presents the geometrical parameters and flow conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The inlet Mach number is set as 𝑀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='0 and the Reynolds number based on the inlet velocity 𝑈∞ and axial blade chord is 𝑅𝑒 = 200,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The ratio of specific heats is chosen as 𝛾 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='31, the Prandlt number is 𝑃𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='747 and the ratio of the Sutherland constant over inlet temperature is set as 𝑆𝜇/𝑇∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='07182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' These conditions are chosen based on previous studies [4, 5, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 1 Schematics of (a) flow configuration and geometrical parameters, and (b) computational domain skipping every 5 grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Figure 1 (b) displays a schematic of the overset grid employed in the LES along with the implemented boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The O-grid block has 1200 × 280 × 144 points and is embedded in the background Cartesian grid block of size 960 × 280 × 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Therefore, the grid has approximately 68, 000, 000 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Depending on the case, adiabatic or isothermal boundary conditions are applied along the blade surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For the latter, the wall to inlet temperature ratio is 𝑇𝑤/𝑇∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='75, representing a cooled wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Supersonic inflow boundary conditions are used to set the inlet conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For the outflow, a boundary condition based on the Navier-Stokes characteristic boundary condition (NSCBC) [19] is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' A damping sponge is also applied near the inflow and outflow boundaries to minimize reflections of disturbances [10, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Periodic boundary conditions are used in the 𝑦-direction of the background grid, according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 2 Adiabatic or isothermal _no-slip wall M= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='0 Qout 20° Re = 200,000 Supersonic Inflow Periodic NSCBC g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='7ca = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='31 Pr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='747 span = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='12ca y 0 = u?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='o Inflow Sponge Periodic Outflow Sponge >C 1 0 2 3 41 (a), in order to simulate a linear cascade of blades and periodic boundary conditions are also applied in the spanwise direction, to enforce a statistically homogeneous flow along the span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For the adiabatic wall case, the grid resolution in terms of wall units is kept in the range given by 6 < Δ𝑠+ < 25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='1 < Δ𝑛+ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='3, and 3 < Δ𝑧+ < 9, where 𝑠, 𝑛 and 𝑧 represent the streamwise, wall-normal and spanwise flow coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For the isothermal wall simulation, the near-wall grid spacing ranges from 15 < Δ𝑠+ < 60, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='2 < Δ𝑛+ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='6, and 6 < Δ𝑧+ < 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' These numbers are computed for regions where the boundary layers are fully developed and in equilibrium, away from the tripping and recirculation regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' It is worthwhile to mention that the same computational grid is used for both cases, but higher values in terms of wall units are obtained for the isothermal wall case due to a inherent reduction of the viscous length scales caused by cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The simulation is initialized with a uniform flow and statistics are computed after the initial transients are discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' In the simulations, a variable time step is computed based on an inviscid CFL parameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The body-force tripping is applied at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='22 < 𝑥/𝑐𝑎𝑥 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='27 for the suction side, and at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='10 < 𝑥/𝑐𝑎𝑥 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='15 for the pressure side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The wall normal height of the body-foce region is 𝛿 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='001𝑐𝑎𝑥 and the actuation changes every Δ𝑡 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='003 in a spanwise-random fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Results This section presents results obtained by the LES computed for adiabatic and isothermal (cooled) wall boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Flow quantities are collected for 4 flow through times, based on the inlet velocity and blade axial chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Figure 2 shows iso-surfaces of 𝑄-criterion colored by the 𝑢-velocity component together with a background view of density gradient magnitude, |∇𝜌|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The top and bottom rows present results for the adiabatic and cooled wall cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 2 Iso-surfaces of 𝑄-criterion colored by 𝑢-velocity component for the adiabatic (top) and cooled (bottom) wall cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The background plane displays the shock waves by visualizing the density gradient magnitude |∇𝜌|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 2 (a) and (c), we can observe the complex shock structure across the turbine passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The detached oblique shock waves generated at the airfoil leading edges interact with the boundary layers of the neighboring blades and are reflected across the cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' On the pressure side, the incident shock wave becomes normal to the wall and, then, a Mach reflection is formed, while an oblique shock reflection is generated on the suction side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' To highlight the effect 3 of cooling on the SBLI, a detailed view of the flow field can be seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 2 (b) and (d), where one can observe differences between the lengths of the separation bubbles, especially on the suction side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For the cooled wall, a smaller recirculation region is noticed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' (a) Skin friction coefficient 𝑐 𝑓 (b) Pressure coefficient 𝑐𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 3 Mean skin-friction and pressure coefficient distributions for the adiabatic (black) and cooled (blue) wall cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The distributions are shown only along the suction side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 4 Time-averaged contours of normalized 𝑢-velocity (top) and temperature (bottom) for the adiabatic (left) and cooled (right) wall cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The black lines display the shock waves visualized by pressure gradient magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The black dashed lines show the sonic line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The mean skin-friction coefficient distribution 𝑐 𝑓 = 𝜏𝑤 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='5𝜌∞𝑈2∞ is provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 3(a) for the blade suction side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' This plot shows the presence of a separation bubble characterized by locations where 𝑐 𝑓 < 0, which is delimited by a horizontal dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The effect of cooling on the size of the recirculation region is evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For the isothermal case, one can observe a downstream displacement of the separation region compared to the adiabatic wall setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' On the other hand, the reattachment locations are similar for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Hence, the cooled wall depicts a smaller separation bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For the adiabatic wall case, the time-averaged characteristic length of the separation bubble is ⟨𝐿𝑆𝐵⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='16𝑐𝑎𝑥 and it is observed along 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='70 < 𝑥/𝑐𝑎𝑥 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For the cooled wall, ⟨𝐿𝑆𝐵⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='10𝑐𝑎𝑥 and it is formed on 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='75 < 𝑥/𝑐𝑎𝑥 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 4 Adiabatic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='006 Isothermal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='5-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='05 Adiabatic Isothermal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='00.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='9 XAfter a small negative skin-friction coefficient plateau, a similar recovery is observed downstream of the reattachment location for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Figure 3 (b) plots the mean pressure coefficient 𝑐 𝑝 = 𝑝−𝑝∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='5𝜌∞𝑈2∞ along the airfoil chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For both adiabatic and cold wall cases, it is possible to note two pressure rises: the first occurs near the separation point due to the compression waves formed upstream of the separation bubble, and the second takes place near the reattachment location as a result of the incident shock impingement and the turbulence amplification mechanism [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For the cooled wall setup, a steeper variation of 𝑐 𝑝 is observed, especially for the second pressure rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' To highlight the influence of the wall thermal boundary conditions on the size and shape of the separation bubbles, the mean (spanwise and time averaged) 𝑢-velocity contours are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 4 (a) and (b), for the adiabatic and isothermal cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Here, the velocity component is normalized by the inlet speed of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' These figures reinforce the findings observed in the friction coefficient distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The main effect of wall cooling is to reduce the viscous length scales near the wall [8, 9] which in turn affects the shock penetration, as shown in 4 (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' One can see that the impinging shock penetrates deeper in the boundary layer for the cooled wall case due to the displacement of the sonic line (displayed as a dashed line) towards the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' This effect is responsible for the steeper variation in the pressure coefficient observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' One can also see that, for the cooled wall, the incident shock reaches further downstream compared to the adiabatic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Figures 4 (c) and (d) show the mean temperature fields for the the adiabatic and cold wall boundary conditions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The values are presented normalized by the inlet temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For the former case, one can observe that a region of maximum temperature occurs within the separation bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' On the other hand, when cooling is applied, higher temperature values are observed in the free shear layer, downstream the bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For the adiabatic wall, friction from the shear stresses near the wall and around the bubble are converted into heat which is transferred along the boundary layer and inside the bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' This causes the near-wall flow to reach higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' However, heat from the flow is transferred to the blade in the isothermal case, which has a lower temperature than the surrounding flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For the cooled wall case, the maximum temperature values are observed along the free shear layer, behind the bubble, due to strong shearing effects that cause aerodynamic heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 5 Temporal variation of the suction side separation bubble length 𝐿𝑆𝐵 for the adiabatic (top) and cooled (bottom) wall cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The temporal evolution of the separation bubble length 𝐿𝑆𝐵 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 5 for the adiabatic and isothermal walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The instantaneous length of the bubble is defined as the distance between the instantaneous reattachment and separation locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' One can observe that the separation region undergoes a contraction/expansion motion for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The excursions from the mean appear to be similar for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' A spectral analysis of this signal should provide further information on the frequency scales related to the bubble motion and such analysis should be conducted in future work, once longer signals are collected for statistical convergence of the lower frequencies of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' To highlight the 2D structure of the suction side separation bubble and shear layer at different time instants, snapshots of 𝑧-vorticity are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 6 for both thermal boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' These snapshots correspond to the instants indicated by the letters “a-d” in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The region enclosed by the green line shows the separation region and the black lines display the impinging shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' In addition, the mean separation and reattachment positions are indicated by the orange and cyan squares, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For both cases, when the bubble suffers a contraction, the instantaneous separation (reattachment) point moves downstream (upstream) with respect to its mean value, as can be visualized in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 6(a) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' On the other hand, when the bubble undergoes an expansion, one can observe the upstream (downstream) movement of the instantaneous separation (reattachment) point with respect to its mean position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' This indicates that the bubble has 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='1( Adiabatic Isothermal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='05 3 t(a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 6 Spanwise 𝑧-vorticity contours at different time instants for the adiabatic (top) and cooled (bottom) wall cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The green line delimits the bubble while the black line shows the incident shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' a breathing pattern, but its central position does not have large excursions from the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Figure 6 also shows that the shear layer downstream of the bubble is more diffused for the adiabatic case, while more concentrated vorticity values are observed when cooling is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' These findings corroborate the maximum temperature values observed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 4(c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For example, in the adiabatic case, the shear layer around the bubble creates a zone of intense heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The effects of the thermal boundary conditions on the turbulence properties, are investigated by the tangential and wall-normal Reynolds stresses, ⟨𝑢𝑡𝑢𝑡⟩ and ⟨𝑢𝑛𝑢𝑛⟩, respectively, and the turbulent kinetic energy (TKE) are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' In this figure, the top and bottom rows display results for the adiabatic and isothermal walls, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 7 (a) and (d), it can be seen that the highest fluctuations of ⟨𝑢𝑡𝑢𝑡⟩ are observed just upstream of the shock-bubble interaction for both cases, with similar fluctuation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The amplification of ⟨𝑢𝑡𝑢𝑡⟩ is associated with the development of the shear layer [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The peak values of ⟨𝑢𝑛𝑢𝑛⟩ are found along the free shear layer downstream of the bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The magnitude of ⟨𝑢𝑛𝑢𝑛⟩ decreases when cooling is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 7 (c) and (f), one can observe that the turbulent kinetic energy combines the trends observed from the ⟨𝑢𝑡𝑢𝑡⟩ and ⟨𝑢𝑛𝑢𝑛⟩ components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' In addition, before the SBLI, we can notice a downstream displacement of the maximum turbulence amplification location for the cooled wall case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' This occurs due to the higher shock penetration discussed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Conclusions Wall-resolved large eddy simulations are employed to investigate thermal effects in a supersonic turbine cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Simulations are performed for adiabatic and isothermal boundary conditions, where in the latter case the blade is cooled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For the present flow configurations, oblique shock waves are generated at the leading edges of the airfoils, and they interact with the boundary layers of the neighboring blades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' A study of the shock-boundary layer interactions is presented for the blade suction side, where an incident oblique shock reflects on the wall leading to the formation of a separation bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The impact of the thermal boundary conditions on the separation bubbles is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The distributions of mean skin-friction show that the separation bubble is considerably smaller for the cooled wall compared to the adiabatic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Pressure coefficient distributions show that a steeper pressure rise occurs downstream the incident shock wave for the cooled wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For this case, cooling induces the formation of a thinner boundary layer and the sonic line forms closer to the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Results in terms of mean velocity contours reveal that the more pronounced pressure rise occurs due to the 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content='04 100 140 180 220 260 0.' metadata={'source': 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incident shock in the isothermal (cooled) wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Maximum temperature values are observed along the bubble, for the adiabatic case, and the free shear layer, for the cooled wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' In the former case, aerodynamic heating is transferred to the bubble due to a surrounding shear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For the latter case, intense shearing is observed along the free shear layer, behind the bubble, and leads to high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' An analysis of the instantaneous separation and reattachment locations demonstrates that the separation bubbles have a breathing pattern of contractions and expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' For the contraction motions, the instantaneous separation point moves downstream while the reattachment point moves upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The other way around is observed for the expansion motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The tangential Reynolds stress distributions reach maximum values just upstream the shock-bubble interactions, being similar for both the adiabatic and isothermal walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' However, due to the higher shock penetration of the isothermal wall, the peaks appear more downstream along the blade chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The wall-normal Reynolds stresses reach maximum amplitudes downstream the SBLI and they are more pronounced for the adiabatic wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' In future work, further analysis of the SBLI dynamics will be provided for both suction and pressure side boundary layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' Acknowledgments The authors acknowledge the financial support received from Fundação de Amparo à Pesquisa do Estado de São Paulo, FAPESP, under grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 2013/08293-7, 2019/26196-5 and 2021/06448-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' The authors also thank Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, for supporting this research under grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' 407842/2018-7 and 308017/2021-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' This work was granted access to the HPC resources of IDRIS under the allocation 2021-A0112A12067 made by GENCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfnP1v/content/2301.01577v1.pdf'} +page_content=' References [1] 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a/atE2T4oBgHgl3EQfvwi4/content/tmp_files/2301.04095v1.pdf.txt b/atE2T4oBgHgl3EQfvwi4/content/tmp_files/2301.04095v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..545388cb1784683255c5fa866be104631b164d5b --- /dev/null +++ b/atE2T4oBgHgl3EQfvwi4/content/tmp_files/2301.04095v1.pdf.txt @@ -0,0 +1,2087 @@ +Optimal randomized multilevel Monte Carlo +for repeatedly nested expectations +Yasa Syed +Department of Statistics +Rutgers University +Guanyang Wang +Department of Statistics +Rutgers University +Abstract +The estimation of repeatedly nested expectations is a challenging +problem that arises in many real-world systems. However, existing +methods generally suffer from high computational costs when the +number of nestings becomes large. Fix any non-negative integer D for +the total number of nestings. Standard Monte Carlo methods typically +cost at least O(ε−(2+D)) and sometimes O(ε−2(1+D)) to obtain an +estimator up to ε-error. More advanced methods, such as multilevel +Monte Carlo, currently only exist for D = 1. +In this paper, we +propose a novel Monte Carlo estimator called READ, which stands +for “Recursive Estimator for Arbitrary Depth.” Our estimator has +an optimal computational cost of O(ε−2) for every fixed D under +1 +arXiv:2301.04095v1 [stat.CO] 10 Jan 2023 + +suitable assumptions, and a nearly optimal computational cost of +O(ε−2(1+δ)) for any 0 < δ < 1 +2 under much more general assumptions. +Our estimator is also unbiased, which makes it easy to parallelize. The +key ingredients in our construction are an observation of the problem’s +recursive structure and the recursive use of the randomized multilevel +Monte Carlo method. +Keywords: nested expectation, optimal cost, randomized Multilevel +Monte Carlo, unbiased estimator +1 +Introduction +Monte Carlo methods are a class of algorithms that use random sampling to +estimate quantities of interest, such as integrals or expected values. When the +estimand can be expressed as an expectation, for example Eπ[g(X)], these +methods work by generating independent random samples X1, . . . , Xn from π, +and using the average �n +i=1 g(Xi)/n as an estimator. Monte Carlo estimators +are unbiased and converge at a rate of n−1/2, regardless of the dimension of the +samples. This dimension-independent convergence rate makes Monte Carlo +methods a powerful tool for approximating high-dimensional integrations, as +they do not suffer from the curse of dimensionality that plagues deterministic +numeric integration methods. +However, the above analysis implicitly assumes the integrand g can be +pointwisely evaluated, which may not be possible in many situations of +2 + +practical interest. In this paper, we study the problem of estimating repeatedly +nested expectations (RNE), which means the integrand depends on a sequence +of other functions and conditional expectations. Specifically, fix any positive +integer D for the total number of nestings, and {gd}D +d=0 for a family of real- +valued functions. Let (y(0), . . . , y(D)) be a finite-time stochastic process with +underlying joint distribution π, and let y(0:d) denote the vector (y(0), . . . , y(d)) +for every d ≤ D. The RNE is defined as: +γ0 = E +� +g0 +� +y(0), γ1 +� +y(0)��� +, +(1) +where {γi}D−1 +i=1 is recursively defined as: +γd(y(0:d−1)) = E +� +gd +� +y(0:d), γd+1 +� +y(0:d)�� +| y(0:d−1)� +, +(2) +and +γD(y(0:D−1)) = E +� +gD +� +y(0:D)� +| y(0:D−1)� +. +(3) +Estimating RNEs is a fundamental problem that covers a variety of +real-world applications, where the quantity of interest depends on multiple +stages or decision points. For example, when gd(y(0:d), u) := max{y(d), u} for +0 ≤ d ≤ D − 1 and gD(y(0:D)) = y(D), the quantity γ0 stands for the expected +utility of the optimal strategy in a D-horizon optimal stopping problem – a +central problem in financial modeling. Other applications include Bayesian +experimental design Goda et al. [2022], portfolio risk management Gordy and +Juneja [2010] and probabilistic programs Rainforth [2018]. +3 + +However, estimating RNEs is challenging. +As shown in formulas (1) +– (3), we are interested in the expectation of g0, which depends on the +random variable y(0) and γ1(y(0)) – a conditional expectation of g1 given +y(0). Then g1 further depends on a random variable y(1) and γ2(y(0), y(1)) +which is a conditional expectation of g2 given y(0) and y(1). This procedure +is recursively defined until it reaches the deepest depth, D. Since γ1(y(0)) +(and also γ2, γ3, . . .) cannot be directly evaluated in most practical cases, +estimating RNEs cannot be handled by standard Monte Carlo methods. +The most natural way to estimate RNEs is by nesting Monte Carlo (NMC) +estimators. In the D = 1 case, this method works by first sampling inde- +pendent and identically distributed (i.i.d.) copies y(0) +1 , y(0) +2 , . . . , y(0) +N0 according +to the distribution of y(0). For each fixed y(0) +i , one further samples N1 i.i.d. +y(1) +1 , . . . , y(1) +N1 according to π(y(1) | y(0) +i ), and uses the standard estimator +ˆγ1(y(0) +i ) := �N1 +i=j g1(y(0) +i , y(1) +j )/N1 to estimate γ1(y(0) +i ). The final estimator is +another standard Monte Carlo estimator which uses the estimated ˆγ1(y(0) +i ) to +replace the intractable γ1(y(0) +i ), i.e., +IN0,N1 = 1 +N0 +N0 +� +i=1 +g0(y(0) +i , ˆγ1(y(0) +i ). +This nested estimator can be easily extended to the general D case, albeit the +notations become more complex. Roughly, one still samples N0 i.i.d. copies +according to π(y(0)), and for each fixed trajectory y(0:d−1), the user generates +Nd samples i.i.d. from π(y(d) | y(0:d−1)) all the way to depth D and then form +the nested estimator from the deepest depth to the shallower depths. The +4 + +construction details are referred to Section 3.2 of Rainforth et al. [2018]. +After suitably allocating the number of samples (Ni)D +i=0 for each depth, +the root-mean-square error (rMSE) of the nested Monte Carlo method is +known to converge to 0 at a rate of N −1/(2D+2) or N −1/(D+2) Rainforth et al. +[2018], depending on the regularity conditions of the functions giD +i=1, where +N = �D +i=1 Ni is the total number of samples used to form a nested estimator. +This convergence rate diminishes exponentially with D, meaning that NMC +estimators do not have the same dimension-free convergence rate as standard +Monte Carlo estimators. As a result, NMC methods require at least O(ε−(2+D)) +and sometimes O(ε−2(1+D)) samples to get an estimator within ε of the true +value, while standard Monte Carlo estimators require only O(ε−2) samples. +Although there are a few cases mentioned in Rainforth et al. [2018] where +the canonical O(N −1/2) rate can be achieved for NMC methods, the problem +of estimating RNEs with a dimension-free convergence rate remains largely +open. +In the special case D = 1, advanced Monte Carlo methods have been pro- +posed Giles [2018], Giles and Goda [2019], Giles and Haji-Ali [2019] based on +the celebrated multilevel Monte Carlo (MLMC) methods Heinrich [2001], Giles +[2008]. These estimators achieve up to ε-rMSE with cost O(ε−2 log(1/ε)2) or +O(ε−2) under varying conditions, comparing favorably with the NMC estima- +tor. However, existing methods cannot be directly generalized to solve the +general D case. Meanwhile, implementing these methods requires users to pre- +5 + +specify the precision level ε and conduct preliminary experiments/calculations +to carefully estimate/bound the parameters in the MLMC algorithm (see, e.g., +Theorem 1 of Giles and Goda [2019]). Therefore, existing MLMC estimators +seem to be harder to implement and less amendable to our original problem, +which has a recursive structure. +In this work, we propose the READ, a novel Monte Carlo estimator for +the RNE problem with an arbitrary number of nestings D. Our construction +is interesting in the following three aspects. Firstly, under suitable regularity +conditions similar to those in Rainforth et al. [2018], the rMSE of our estimator +has an optimal convergence rate N −1/2 Heinrich and Sindambiwe [1999] +regardless of D. Equivalently, our method costs in expectation O(ε−2) to +get an estimator up to ε-rMSE. Under much more general assumptions, our +method still achieves a nearly-optimal cost of O(ε−2(1+δ)) for any 0 < δ < 1 +2 +to get an estimator up to ε-mean-absolute-error (MAE). +It is worth mentioning that most of our effort is devoted to designing +unbiased estimators of γ0 in (1) with finite computational cost and finite +variance (or finite (2-δ)-th moment under more general assumptions). Af- +ter developing such an unbiased estimator, we can simulate independent +copies of these estimators and average them. The N − 1 +2 convergence rate and +O(ε−2) computational cost are then immediate corollaries of the bias-variance +decomposition formula. +Therefore, another appealing property of READ, in contrast to existing +6 + +methods, is that it admits no estimation bias. Unbiasedness implies these +estimators can be implemented in parallel processors without requiring any +communication between them. Designing unbiased estimators has recently +attracted much interest in statistics, operations research, and machine learning +communities for its potential for parallelization. Our methods add to the rich +body of works of Glynn and Rhee [2014], Rhee and Glynn [2015], Blanchet +and Glynn [2015], Jacob et al. [2020], Biswas et al. [2019], Wang et al. [2021], +Wang and Wang [2022], Kahale [2022]. +Finally, our algorithm for constructing READ relies on the randomized +multilevel Monte Carlo (rMLMC) method McLeish [2011], Rhee and Glynn +[2015], Blanchet and Glynn [2015], but it is significantly different from previous +applications of this method. Most existing rMLMC applications Rhee and +Glynn [2015], Vihola [2018], Goda et al. [2022] have a non-randomized version +with similar or better computational cost guarantees, leading some to believe +that every problem solved by rMLMC also has a natural non-randomized +counterpart. However, our work seems to suggest that this belief may not +always hold true. The rMLMC framework is well-suited to the recursive +structure of RNEs, and can be used as a subroutine in our method. In +contrast, the non-randomized MLMC cannot be easily applied to the general +case of D > 1. This suggests that the rMLMC framework may be more widely +applicable than previously thought. +The rest of this paper is organized as follows: in the remainder of this +7 + +section, we discuss related works, set up our notation, and introduce our +technical assumptions. In Section 2, we introduce our algorithm and show +that it attains the optimal and nearly optimal computational cost under +two different assumptions, respectively. In Section 3, we demonstrate the +empirical performance of our method on a toy example. We conclude this +paper with a short discussion in Section 4. Proof details are deferred to the +Appendix. +1.1 +Related work +Our algorithm design strategy mainly follows the randomized multilevel Monte +Carlo (rMLMC) framework McLeish [2011], Rhee and Glynn [2015], Blanchet +and Glynn [2015]. Our algorithm is inspired by the unbiased optimal stopping +estimator Zhou et al. [2022], which develops estimators for the optimal +stopping problem by recursively calling the rMLMC algorithm. We extend +the methodology in Zhou et al. [2022] both in scope and depth. Our method +works with a more general class of problems formulated by Rainforth et al. +[2018], which includes the optimal stopping problem as a special case, and +provides more precise results under practical assumptions. +Throughout this paper, we will assume the functions {gd}D−1 +d=0 are all +continuous and the process π can be perfectly simulated. When D = 1 and +g0 is discontinuous, progresses have been made by Broadie et al. [2011] and +Giles and Haji-Ali [2019, 2022]. When the underlying distribution is itself +8 + +challenging, users have to first use MCMC to approximately sample from π. +The case of D = 1 and challenging π is considered in Wang and Wang [2022]. +1.2 +Notations +Now we introduce our notations. Many of our notations follow those used in +Rainforth et al. [2018], which first formally defines the RNEs. Throughout this +paper, we preserve the letter D for the total number of nestings. We denote +by π the underlying joint distribution of a finite-time, real-valued stochastic +process (y(0), . . . , y(D)). For every 0 ≤ i ≤ j ≤ D, we use the shorthand +notation y(i:j) to denote the vector (y(i), . . . , y(j)). The conditional distribution +of y(d:D) given the value of y(0:d−1) is denoted by πd:D(· | y(0:d−1)). The marginal +distribution of y(d) conditioning on y(0:d−1) is denoted by πd(· | y(0:d−1)). We +adopt the convention that y(0:−1) = ∅, and therefore π0 stands for the +(unconditioned) marginal distribution of y(0). +Let Π be any probability +distribution on some probability space, and Z be some random variable on +the same space, then we use ∥Z∥Π,m to denote the Lm–norm of Z under Π, +i.e., +� +EΠ[|Z|m] +�1/m. The geometric distribution with parameter r is denoted +by Geo(r). We also define pr(n) := P[Geo(r) = n] = r(1 − r)n for every +n ∈ {0, 1, 2, . . . , }. For every 0 ≤ d < D, the function gd introduced in (1) – (2) +maps from Rd+2 to R. The function gD in (3) maps from RD+1 to R. For i.i.d. +random variables X1, . . . , Xn, we denote their summation by Sn := �n +i=1 Xi. +When n is an even number, we denote by SO +n/2 := X1 + X3 + . . . + Xn−1 and +9 + +SE +n/2 := X2 + X4 + . . . + Xn the summations of their odd and even terms, +respectively. +1.3 +Assumptions +Throughout this paper, we assume that we can access a simulator S. The +simulator can take any trajectory y(0:d−1) with 0 ≤ d ≤ D as input, and +outputs y(d) which follows the distribution πd(· | y(0:d−1)). In particular, S can +take ∅ as input and simulates y(0) ∼ π0. Calling S recursively for D +1 times +generates one complete sample path. This assumption enables us to sample +from any marginal or conditional distribution perfectly. This assumption is +also standard and is posed explicitly or implicitly in nearly all the existing +works concerning the estimation of nested expectations, see Giles and Goda +[2019], Goda et al. [2022], Zhou et al. [2022] for examples. +Fix a function f : Rk+2 → R, we say f satisfies the last-component +bounded Lipschitz condition (LBL) if there exists L < ∞ such that: +sup +y(0:k)|f(y(0:k), x) − f(y(0:k), z)| ≤ L|x − z|. +(4) +We say f satisfies the last-component bounded second-derivative condition +(LBS) if f has continuous second-order derivative on its last component, and +there exists C < ∞ such that +sup +y(0:k+1) +��∂2 +k+1f(y(0:k+1)) +�� < C, +(5) +10 + +where ∂2 +k+1f := (∂2f)/(∂y(k+1))2 stands for the second-order partial derivative +for the last component of f. These assumptions (and their variants) are also +posed in related works such as Rainforth [2018], Blanchet and Glynn [2015], +Giles [2018]. +2 +Algorithm, estimator, and theoretical re- +sults +Now we are ready to present our main results. As discussed in Section 1, we +will be focusing on designing a Monte Carlo estimator which is unbiased, has +a finite computational cost, and has finite variance or (2-δ)-th moment under +different assumptions. Then our estimator with O(ε−2) or O(ε−2(1+δ)) cost +can be directly obtained by averaging over these unbiased estimators. +2.1 +Preliminary analysis +One of the challenges in estimating the RNEs is the difficulty of estimating +γ1(y(0)). Users typically first estimate γ1(y(0)) and then use these estimators +to estimate γ0. For the time being, we are temporarily adding the assumption +that users can simulate unbiased estimators ˆγ1(y(0)) of γ1(y(0)) for every fixed +y(0) with finite computational cost. This assumption will be removed in +Section 2.2. It easily holds when D = 1, as users can repeatedly simulate +y(1) +i +∼ π1(· | y(0)) and it follows from the problem definition that each +11 + +g1(y(0), y(1) +i ) is unbiased for γ1(y(0)). +In the general case of D > 1, this +assumption is far from trivial, as γ1(y(0)) is itself a nested expectation with +a nesting depth of D − 1. Nevertheless, as we will see in Section 2.2, this +assumption helps us to capture and reduce the intrinsic difficulty of the +problem and, therefore, will guide us to design the general algorithm. +With this extra assumption, constructing unbiased estimators of (1) is +equivalent to constructing unbiased estimators of g0(y(0), γ1(y(0))). Even with +access to unbiased estimators of γ1(y(0)), the intuitive estimator g0 +� +y(0), ˆγ1(y(0)) +� +is still biased, as in general E[g0 +� +y(0), ˆγ1(y(0)) +� +| y(0)] ̸= g0(y(0), E[ˆγ1(y(0)) | +y(0)]). To eliminate this bias, we turn to the rMLMC method Blanchet and +Glynn [2015], which will be briefly reviewed below. +The rMLMC method uses the Law of Large Numbers (LLN) and rewrites +g0 as the following telescoping summation. +g0(y(0), γ1(y(0))) = E +� +g0 +� +y(0), lim +k→∞ +Sk +k +� +| y(0) +� += +∞ +� +n=1 +E +� +g0 +� +y(0), S2n +2n +� +| y(0) +� +− E +� +g0 +� +y(0), S2n−1 +2n−1 +� +| y(0) +� +, +where Sk = �k +i=1 ˆγ1,i(y(0)) is the summation of i.i.d. copies of ˆγ1(y(0)). To +unbiasedly estimate the infinite sum, the rMLMC algorithm first samples +y(0) ∼ π0, then samples a random N ∼ Geo(r), finally generates 2N unbiased +estimators {ˆγ1,i(y(0))}2N +i=0 of γ1(y(0)) and estimates γ0 by R0 := ∆N/pr(N), +12 + +where ∆n is defined as: +∆n := g0 +� +y(0), S2n +2n +� +− 1 +2 +� +g0 +� +y(0), SE +2n−1 +2n−1 +� ++ g0 +� +y(0), SO +2n−1 +2n−1 +�� +for n ≥ 1 and ∆0 := g0(y(0), ˆγ1,1(y(0))). +The next theorem justifies the theoretical properties of R0: +Theorem 2.1. With all the notations as above, suppose g0 : R2 → R satisfies +LBS condition defined in (5), and ∥ˆγ1(y(0))∥π,m < ∞ for some m ≥ 4. Then +for any r ∈ (1/2, 3/4), the estimator R0 := ∆N/pr(N) has expectation γ0, +finite variance, and finite expected computational cost. +Theorem 2.1 will not be proved directly, as it is a special case of our +Theorem 2.2. For now, we use the following heuristic calculation to justify +the unbiasedness of ˆγ0: +E[R0 | y(0)] = E +� +E +� +R0 | N, y(0)�� += +∞ +� +n=0 +E +� ∆n +pr(n)pr(n) | y(0) +� += +∞ +� +n=0 +E +� +g0 +� +y(0), S2n +2n +� +| y(0) +� +− E +� +g0 +� +y(0), S2n−1 +2n−1 +� +| y(0) +� += g0(y(0), γ1(y(0))). +Therefore E[R0] = E[g0(y(0), γ1(y(0)))] = γ0 by (1). More technical discussions +such as the range of r, other possible regularity conditions on g0, and the +moment guarantees of γ0 will all be deferred after Theorem 2.2. +13 + +2.2 +Recursive rMLMC algorithm for general D +Theorem 2.1 is useful to solve our original problem (without the extra as- +sumption) in two ways. First, Theorem 2.1 already solves the case where +D = 1, as our extra assumption automatically holds. It states that if g0 has +a bounded second derivative on its last component, and g1(y(0), y(1)) has at +least finite fourth moment under π, then R0 is unbiased, has finite variance, +and finite expected computational cost. More importantly, Theorem 2.1 tells +us that the original problem of estimating an RNE with a depth of D can be +solved if we can unbiasedly estimate γ1(y(0)) for fixed y(0), which is another +RNE with a depth of D − 1. Therefore, we have successfully reduced the +number of nestings by one. This observation motivates us to come up with +an algorithm for the general D case, as explained below. +We first go one step further to illustrate the D = 2 case. When D = 2, +estimating γ1(y(0)) again reduces the case we have analyzed in Section 2.1. +To be precise, since g2(y(0:2)) is unbiased for γ2(y(0:1)) if y(2) ∼ π2(· | y(0:1)), +one can first sample y(1) ∼ π1(· | y(0)), then simulate N ∼ Geo(r) and 2N +samples {y(2) +i }2N +i=1 from π2(· | y(0:1)). Let ˆγ2,i(y(0:1)) := g2(y(0:1), y(2) +i ), our +estimator of γ1(y(0)) is then constructed in the same way as Section 2.1, i.e., +R1(y(0)) := ∆N/pr(N) with +∆n := g1 +� +y(0:1), S2n +2n +� +− 1 +2 +� +g1 +� +y(0:1), SE +2n−1 +2n−1 +� ++ g1 +� +y(0:1), SO +2n−1 +2n−1 +�� +, +14 + +where S2n, SE +2n−1, SO +2n−1 are the summation of every, even, and odd terms in +{γ2,i(y(0:1))}, respectively. The same procedure of simulating R1(y(0)) can be +repeated independently. Therefore we can sample another geometrically dis- +tributed random variable N ′ ∼ Geo(r′), and generate R1,i(y(0)) := ∆N′/pr(N ′) +independently. Since each R1,i(y(0)) is unbiased for γ1(y(0)), one can again use +the method described in Section 2.1 to form our final estimator for γ0. After +checking R1(y(0)) satisfies the finite fourth-moment assumption, Theorem 2.1 +can be applied which implies our estimator is unbiased, has finite variance +and finite cost (for the D = 2 case). +The general case works in the same way. A key observation is that, due +to the nested structure of the problem, Theorem 2.1 not only states that an +unbiased estimator of γ0 can be constructed if one can unbiasedly estimate +γ1(y(0)) for every y(0), but also directly implies that an unbiased estimator of +γd(y(0:d−1)) can be constructed if one can unbiasedly estimate γd+1(y(0:d)) for +every y(0:d). Therefore, we can estimate γ0 in a backward, inductive manner. +To begin, we consider the deepest depth of the problem, fixing any +y(0:D−1). An unbiased estimator of γD(y(0:D−1)) can be directly constructed +as gD(y(0:D−1), y(D)), where y(D) ∼ πD(· | y(0:D−1)). For 0 ≤ d ≤ D − 1, if we +assume that users can generate unbiased estimators of γd+1(y(0:d)) for every +y(0:d), then we can obtain an unbiased estimator of γd(y(0:d−1)) by sampling +one y(d), generating Nd ∼ Geo(rd) and 2Nd unbiased estimators of γd+1(y(0:d)), +and applying the method described in Section 2.1. This process continues +15 + +until we reach d = 0, at which point we have an unbiased estimator of γ0. +The parameters (r0, r1, . . . , rD−1) will be carefully chosen and depend on the +regularity assumptions of (g0, g1, . . . , gD−1). These choices will be discussed +in more detail later. +Our algorithm for constructing READ is described in Algorithm 1. It is +written as a recursive algorithm, though it could also be equivalently written +in an iterative form with much more cumbersome notations. Algorithm +1 takes a depth index, a trajectory, a simulator, and parameters for the +geometric distribution as inputs, and outputs one unbiased estimator of +γd(H). In particular, with inputs {depth = 0, trajectory = ∅, parameters += (r0, r1, . . . , rD−1)}, it outputs READ – an unbiased estimator of the RNE +defined in (1). The logic of Algorithm 1 is precisely the same as we just +discussed. To estimate γd(y(0:d−1)), the algorithm first checks the value of +d. When d = D, the problem becomes straightforward. When d < D, the +algorithm samples y(d), appends y(d) to the trajectory, samples Nd, and calls +itself 2Nd times with depth d + 1 and new trajectory {y(0:d)} to get 2Nd +unbiased estimators of γd+1(y(0:d)). Finally, we split these 2Nd estimators into +even and odd terms and apply the method described earlier in Section 2.1. +The algorithm is guaranteed to stop as the depth will eventually reach the +deepest depth D. +16 + +Algorithm 1 A recursive rMLMC algorithm for RNEs +Input: Depth index d ∈ {0, ..., D}. Trajectory history H = {y0, ..., yd−1} +or ∅. A simulator S. Parameters rd, ..., rD−1 determined by conditions on +{gd}D−1 +d=0 (see Theorem 2.2, 2.4). +Output: An unbiased estimator of γd(H) +if d = D then +Sample one y(D) ∼ πD (· | H); +Return: RD := gD +� +y(0:D)� +. +else +Sample y(d) ∼ πd (· | H); +Update the trajectory H ← H ∪ +� +y(d)� +; +Sample Nd ∼ Geo(rd); +Call Algorithm 2 for 2Nd times with inputs (d + 1; H; S; rd+1, ..., rD−1), +and label the observations as Rd+1(y(0:d))(1), ..., Rd+1(y(0:d)) +� +2Nd� +; +Calculate S2Nd, SE +2Nd−1, SO +2Nd−1 defined in Section 1.2; +Calculate +� +note ∆0 := gd +� +y(0:d), Rd+1(y(0:d))(1) +�� +: +∆Nd = gd +� +y(0:d), S2Nd +2Nd +� +− +1 +2 +� +gd +� +y(0:d), SO +2Nd−1 +2Nd−1 +� ++ gd +� +y(0:d), SE +2Nd−1 +2Nd−1 +�� +; +Return: Rd := ∆Nd/prd(Nd). +end if +17 + +2.3 +Theoretical guarantees +We now discuss the computational costs of Algorithm 1 and the statistical +properties of READ. +Our theoretical results depend on the smoothness +conditions of {gd}D−1 +d=0 , so we will consider two cases where {gd}D−1 +d=0 satisfies +the LBS and LBL assumptions separately. +2.3.1 +The LBS case +The following theorem shows, under the LBS assumption, the computational +cost and the variance of READ can be controlled simultaneously. +Theorem 2.2. Suppose for every d ∈ {0, 1, . . . , D − 1}, the function gd +satisfies the LBS assumption defined in (5), and rd := 1 − 2−kd satisfies +kd ∈ +� +1, +2d+1 +2d+1−1 +� +. +Moreover, suppose ∥gD(y(0:D))∥π,2D+1 < ∞. +Then for +every 0 ≤ d ≤ D, the output Rd(y(0:d−1)) of Algorithm 1 with inputs {depth += d, trajectory = y(0:d−1), S, parameters (rd, . . . , rD−1)} has the following +properties: +• For almost surely every fixed y(0:d−1), +E +� +Rd(y(0:d−1)) | y(0:d−1)� += γd(y(0:d−1)). +• The expected computational cost of Rd is finite. +• The output has finite 2d+1-th moment, i.e., +Eπ +� +|Rd(y(0:d−1))|2d+1� +< ∞ for 0 ≤ d ≤ D. +18 + +Theorem 2.2 states for π-almost surely every y(0:d−1), the expectation of +the output Rd conditioning on the input is unbiased for γd(y(0:d−1)). The +computational cost has a finite expectation, and the output has a finite +2d+1-th moment 1. The detailed proof of Theorem 2.2 will be provided in the +appendix. Here, we highlight two special cases. First, Theorem 2.2 shows that +READ, the output R0 of Algorithm 1 when given input {depth = 0, trajectory += ∅, S, parameters = (r0, . . . , rD−1)}, has the desired properties. Specifically, +it is an unbiased estimator for γ0 with finite expected computational cost and +finite variance. Second, Theorem 2.2 recovers Theorem 2.1 when D = 1. +Let R0,1, R0,2, . . . , be the i.i.d. outcomes by repeatedly implementing +Algorithm 1. Since each R0,i is unbiased and has a finite variance, the standard +Central Limit Theorem (CLT) implies that √n(�n +i=1 R0,i/n − γ0) → N(0, 1) +in distribution. This means that the estimator �n +i=1 R0,i/n converges to +γ0 at a rate of n−1/2, which compares favorably with the rates obtained +by NMC estimators in Rainforth [2018]. This rate is optimal in the sense +that it matches the minimax lower bound over all the Monte Carlo methods +(Theorem 2.1 of Heinrich and Sindambiwe [1999]). The next corollary shows +that, by repeatedly implementing Algorithm 1, it is possible to obtain an +unbiased estimator for γ0 with at most ε2-MSE within O(ε−2) computational +1Readers should notice that the expectation of Rd(y(0:d−1)) is calculated under the +conditional distribution πd:D(· | y(0:d−1)). The computational cost and the 2d+1-th moment +are calculated under the joint distribution π. When the input depth = 0, these two +underlying distributions coincide. +19 + +cost. +Corollary 2.3. With all the assumptions the same as Theorem 2.2, for any +ε > 0, we can construct an estimator R with expected computational cost +O(ε−2) such that the mean squared error E[(R − γ0)2] ≤ ε2. +Proof of Corollary 2.3. Calling Algorithm 1 independently for n times with +{depth = 0, trajectory = ∅, S, parameters = (r0, . . . , rD−1)} yield i.i.d. +unbiased estimators R0,1, ..., R0,n for γ0. Let our estimator be R := 1 +n +�n +i=1 R(i) +0 . +Then, +E[(R − γ0)2] = E +� +� +� +1 +n +n +� +i=1 +R0,i − γ0 +�2� +� = 1 +nVar(R0). +Thus noting that Var(R0) < ∞ by Theorem 2.2, taking n = Var(R0)/ε2 +samples ensures R has up to ε2-MSE. Finally, let C := C(D) < ∞ be +the expected computational cost for implementing Algorithm 1 once. The +expected computational cost for constructing R is then C · Var(R0)/ε2. +The ε2-MSE of R can be easily translated to other performance metrics +via standard inequalities. For example, for any δ, Markov’s inequality implies +the absolute error |R − γ0| is less than ε/ +√ +δ with probability at least 1 − δ. +Next, we discuss the assumptions of Theorem 2.2. +The assumptions +require the first D functions {gd}D−1 +d=0 all satisfy the LBS condition, and the +final function gD has finite 2D+1-th moment under π. The LBS assumption +also appears in the work of NMC estimators (see the second part of Theorem +3 in Rainforth et al. [2018]). The moment assumption of gD is not required +20 + +in Rainforth et al. [2018]. Nevertheless, it is a mild assumption that holds in +most practical applications. It covers all the cases where gD is bounded or +has a moment generating function (including the uniform, Gaussian, Poisson, +or exponential distributions), which implies E[|gD|k] < ∞ for every k. As +we will see in our proofs, these assumptions help us to establish the third +conclusion of Theorem 2.2 in a backward inductive way. For example, 2D+1-th +moment assumption on gD and the LBS assumption on gD−1 implies RD−1 +has finite 2D-th moment. More generally, the finiteness of the 2d+1-th moment +of Rd follows from the LBS assumption on gd−1 and the 2d+2-th moment of +Rd+1 (which is the conclusion of the previous inductive step). Eventually, +we conclude R0 has a finite variance. Finally, we want to emphasize our +moment assumption on gD is not ‘trajectory-dependent’. We require gD +has finite 2D+1-th moment under the joint distribution π of y(0:D), which is +much weaker than gD has a uniformly bounded finite 2D+1-th moment under +π0:D−1(· | y(0:D−1)) for every fixed trajectory y(0:D−1). +Finally, the parameters {rd}D=1 +d=0 reflect the trade-off between the variance +and computation cost. Since 2Nd calls are required for each d, standard +calculation shows that E[2Nd] = rd/(2rd − 1) when rd > 0.5, and +∞ if +rd ≤ 0.5. Therefore, every rd has to be strictly greater than 0.5 to ensure a +finite expected computational cost. Meanwhile, we cannot guarantee finite +variance or unbiasedness of READ when rd becomes too large. Our range for +rd in Theorem 2.2 follows from a careful calculation in our proof to ensure +21 + +unbiasedness, finite computational cost, and variance simultaneously. +2.3.2 +The LBL case +The assumptions in Theorem 2.2 guarantees READ enjoys the optimal con- +vergence rate and computational cost. However, the second-order derivative +assumption also rules out many functions of practical interest, such as max +and min. In this section, we study the theoretical properties of Algorithm 1 +and READ under weaker smoothness and moment assumptions. Our result is +summarized below: +Theorem 2.4. Fix any 0 < δ < 1/2. Suppose for every d ∈ {0, 1, . . . , D −1}, +the function gd satisfies the LBL assumption defined in (4), and rd := 1−2−kd +satisfies +kd ∈ +� +1, +�2d+2 − 3δ +2d+3 − 3δ +� �2d+1 − δ +2d − δ +�� +. +Moreover, suppose ∥gD(y(0:D))∥π,2 < ∞. Then for every 0 ≤ d ≤ D, the output +Rd(y(0:d−1)) of Algorithm 1 with inputs {depth = d, trajectory = y(0:d−1), S, +parameters (rd, . . . , rD−1)} has the following properties: +• For almost surely every fixed y(0:d−1), +E +� +Rd(y(0:d−1)) | y(0:d−1)� += γd(y(0:d−1)). +• The expected computational cost of Rd is finite. +• The output has finite (2 − δ/2d)-th moment, i.e., +Eπ +� +|Rd(y(0:d−1))|(2−δ/2d)� +< ∞ for 0 ≤ d ≤ D. +22 + +Comparing Theorem 2.2, which requires the LBS assumption for {gd}D−1 +d=0 +and finite 2D+1-th moment for gD, with Theorem 2.4, which only requires the +LBL assumption for {gd}D−1 +d=1 and finite second moment for gD, we see that +the latter has more general assumptions but at the cost of not guaranteeing +that READ has a finite variance (though it is still unbiased and has a finite +expected computational cost). However, the loss of moment guarantees can +be minimal, as for any small δ, one can still choose suitable parameters such +that READ has finite (2 − δ)-th moment. +Again, let R0,1, R0,2, . . . , be the i.i.d. outcomes by repeatedly implement- +ing Algorithm 1. There are some more technical subtleties when analyzing the +convergence rate of �n +i=1 R0,i/n−γ0 as the CLT cannot be applied. Neverthe- +less, we use the Marcinkiewicz-Zygmund generalized law of large numbers (see +Theorem A.4 in the Appendix), which shows n−1E[|�n +i=1 Xi|p] → 0 if {Xi}n +i=1 +are i.i.d., centered random variables with finite p-th moment for p ∈ [1, 2). +Our result is the following: +Corollary 2.5. With all the setting the same as Theorem 2.4, let R0,1, R0,2, . . . , +be the i.i.d. outcomes by repeatedly implementing Algorithm 1, we have: +• E[|�n +i=1 R0,i/n − γ0|] = o(n−1/(2(1+δ))). +• We can construct an estimator R with expected computational cost +O(ε−2(1+δ)) such that the mean absolute error E[|R − γ0|] < ε. +Proof of Corollary 2.5. Applying Theorem A.4 with p = 2 − δ, Xi = R0,i − γ0 +23 + +and Jensen’s inequality, we have: +E +������ +n +� +i=1 +R0,i/n − γ0 +����� +� += n−1E +������ +n +� +i=1 +Xi +����� +� +≤ n−1 +� +E +������ +n +� +i=1 +Xi +����� +p��1/p += o(n−1+1/p) = o(n−1/(2(1+δ))), +which proves the first part. The last step follows from +1 − 1/(2 − δ) > 1/(2 + 2δ) for δ ∈ (0, 1/2). Setting n = ε−2(1+δ) and the +second part immediately follows. +Although we are not able to recover the optimal n−1/2 convergence rate +under this more general assumption, our convergence rate is still near-optimal +as it can be as close to n−1/2 as we want and does not depend on D. At +the same time, although we replace the MSE by MAE due to the moment +constraint, one can still use Markov’s inequality to show the absolute error +|R − γ0| is less than ε/δ with probability at least 1 − δ. +As the max function satisfies the LBL assumption, our results here include +the optimal stopping problem as a special case. Our results complement +the work of Zhou et al. [2022], where the authors use rMLMC to design an +estimator with O(ε−2) computational cost under stronger assumptions (see +their Assumption 4). We have a slightly worse cost of O(ε−2(1+δ)) but holds +under more general assumptions. +24 + +3 +Numerical experiments +We consider the following simple example with known ground-truth. Suppose +the process (y(0), y(1), y(2)) satisfies y(0) ∼ N(π/2, 1), y(1) ∼ N(y(0), 1), y(2) ∼ +N(y(1), 1). Define g0(y(0), z) := sin +� +y(0) + z +� +, g1(y(0:1), z) := sin +� +y(1) − z +� +, and +g2(y(0:2)) := y(2). The target quantity γ0 defined (1) is a nested expectation +with D = 2. One can use the formula EZ∼N(µ,σ2)[sin(Z)] = sin(µ) exp(−σ2/2) +to analytically calculate γ0 = exp(−1/2) ≈ 0.6065. Now we compare our +READ estimator with the NMC estimator in Rainforth et al. [2018]. +For the NMC estimator, users first specify N0, N1, N2. Then we sample +N0 copies of y(0), N1 copies of y(1) for each fixed y(0), and N2 copies of y(2) for +each fixed y(0:1), and use these samples to form the NMC estimator. Following +Rainforth [2018], we consider two ways of allocating (N0, N1, N2). The first +estimator NMC1 is to choose N0 = N1 = N2, the second NMC2 is to choose +N0 = N 2 +1 = N 2 +2. Both methods have cost n = N0N1N2. For READ, notice that +all the assumptions in Theorem 2.2 are satisfied, therefore for r0 ∈ (1/2, 3/4) +and r1 ∈ (1/2, 1 − 2−4/3), and so READ estimator generated by Algorithm 1 +should be unbiased and of finite variance. Since the computational cost gets +lower when each ri gets larger, we choose r0 = 0.74 and r1 = 0.6 (close to the +upper-end of their respective ranges above) to facilitate the computational +efficiency. Therefore, implementing Algorithm 1 once has an expected sample +size/computational cost (r1/(2r1 − 1)) (r2/(2r2 − 1)) ≈ 4.625. +Our comparison result is summarized in Figure 1. The slopes of the blue, +25 + +red, and green lines, which correspond to the empirical convergence rate of +READ, NMC1, NMC2, equals −0.97, −0.33, −0.53, respectively. They match +well with the theoretical predictions n−1 in Corollary 2.3 for READ, n−1/3 for +NMC1, and n−1/2 for NMC2 in Theorem 3 of Rainforth et al. [2018]. +We also repeatedly call Algorithm 1 for 106 times and plot the running +averages of our estimates in Figure 2. Our estimator becomes more accurate +(closer to the red dashed line) when we increase the number of repetitions. In +addition, for each k ∈ (1, 2, . . . , 106), we also calculate the standard deviation +(sd) of the first k repetitions and use Mean ±1.96 sd to form the 95% confidence +interval. It is also clear from Figure 2 that our confidence intervals always +include the ground-truth, suggesting the high accuracy of our method. In +contrast, constructing confidence intervals of NMC estimators are much more +time-consuming. +4 +Further discussions +Here we provide some remarks for practical implementation and discuss +some potential generalizations. The users need to specify the parameters +{rd}D−1 +d=0 when implementing Algorithm 1. Larger values of ri lead to a shorter +time for each implementation but potentially larger variance. When some +ri is not chosen according to Theorem 2.2 or 2.4, the algorithm can still +be implemented, but the variance may be infinite. The trade-off between +the values of {rd} and the fluctuations of the resulting estimator is problem- +26 + +−8 +−6 +−4 +−2 +2 +4 +6 +8 +log(Sample Size) +log(MSE) +method +NMC1 +NMC2 +READ +Figure 1: The comparison on the empirical MSEs of estimating the RNE +among READ (blue), NMC1 (red), and NMC2 (green). All the logarithms +are of the base 10. Each method’s empirical errors are calculated based on 20 +independent repetitions. +27 + +0.58 +0.60 +0.62 +0.64 +0 +250000 +500000 +750000 +1000000 +Number of estimators +Running Average +Figure 2: The trace plot (solid blue curve) of the running averages of READ. +The blue dotted curves are the 95% confidence intervals. The red dashed line +is the ground truth exp(−1/2). +28 + +specific. In practice, knowing how many repetitions are sufficient is important +to provide an accurate estimator. One possible way is to bound the variance +of READ and use Corollary 2.3 to choose a sufficiently large n. But this bound +can be problem-specific and very conservative. Instead, we follow Glynn and +Rhee [2014] and suggest the following adaptive stopping rule: users first specify +a precision-level ε and a small δ%. When repeatedly implementing Algorithm +1, users calculate the empirical (1 − δ%) confidence interval [Lδ(k), Uδ(k)] for +first k repetitions in the same way as Section 3 for every k. Users can stop +when the width of the confidence interval is less than 2ε. The validity of this +stopping rule is proven in Glynn and Whitt [1992]. +There are two directions for extensions. First, each y(i) in our paper is +assumed to be univariate for concreteness. Extending this assumption to +multivariate should be straightforward. Moreover, the computational cost +of Algorithm 1 scales linearly with the dimensionality of y(i), which may +be another appealing property of our method. Second, in this paper, we +only consider the ‘fix D’ regime and construct estimators with optimal/near- +optimal convergence rate for total sample size n (or computational cost +for ε). On the other hand, the cost of Algorithm 1 scales exponentially +with D. Therefore, although our algorithm is more efficient than the NMC +estimator for every fixed D, both methods are not practically useful when +D becomes very large. Indeed, the poor scaling with D frequently happens +in related literature such as Glasserman and Yu [2004], Zanger [2013] and +29 + +seems unavoidable. An interesting direction would be constructing proper +modifications of Algorithm 1 under extra practically relevant assumptions. +For example, if we know the ‘influence’ of γd on γ0 decays exponentially or +double-exponentially when d increases. Then it may be sufficient to truncate +the depth to ˜D := log(1/ε) or +� +log(1/ε). We hope to report progress in our +future work. +References +N. Biswas, P. E. Jacob, and P. 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Unbiased optimal +stopping via the MUSE. Stochastic Processes and their Applications, 2022. +33 + +A +Auxiliary results +The following theorem is from pg. 150 [Gut, 2005]. +Theorem A.1. Let p ≥ 1. Suppose that X1, X2, ...Xn are independent r.v.s. +with mean 0 such that E|Xk|p < ∞ for all k, and let Sn := �n +i=1 Xi denote +the partial sums. Then there exist constants A∗ +p, B∗ +p depending only on p such +that +A∗ +pE +� n +� +k=1 +X2 +k +�p/2 +≤ E|Sn|p ≤ B∗ +pE +� n +� +k=1 +X2 +k +�p/2 +. +The following corollary to the above theorem is from pg. 151 [Gut, 2005], +Corollary 8.2. +Corollary A.2. Let p ≥ 1. Suppose that X, X1, X2...Xn are i.i.d. r.v.s. with +mean 0 such that E|X|p < ∞, and let Sn := �n +i=1 Xi denote the partial sums. +Then there exists a constant Bp depending only on p, such that +E|Sn|p ≤ +� +� +� +� +� +� +� +Bpnp/2E|X|p, +p > 2 +BpnE|X|p, +1 ≤ p ≤ 2. +The following lemma is instrumental in the proofs for the theoretical +guarantees of our algorithm under both the LBS and LBL assumptions. +Lemma A.3. Let (Z1, Z2) be a 2-stage stochastic process and there exists p ≥ +1, such that E[|Z2|p] < ∞. Conditioning on Z1, sample i.i.d. Z2(1), ..., Z2(n). +34 + +Then, +E +������ +1 +n +n +� +i=1 +Z2(i) − E[Z2 | Z1] +����� +p� +≤ +� +� +� +� +� +� +� +B′ +p +E[|Z2|p] +np/2 +p > 2 +B′ +p +E[|Z2|p] +np−1 +1 ≤ p ≤ 2 +Proof. Let p > 2. For arbitrary fixed Z1 = z1, define ¯Z2(i) := Z2(i)−E[Z2(i) | +Z1 = z1], and apply Corollary A.2 on the i.i.d. mean 0 random variables ¯Z2(i) +under the probability distribution π(· | Z1 = z1), we have +E +������ +1 +n +n +� +i=1 +Z2(i) − E[Z2 | Z1] +����� +p� += +� +Ω +E +������ +1 +n +n +� +i=1 +Z2(i) − E[Z2 | Z1] +����� +p +| Z1 = z1 +� +π1(dz1) += 1 +npE[ +����� +n +� +i=1 +¯Z2(i) +����� +p +| Z1 = z1]π1(dz1) +≤ Bp +np/2E[| ¯Z2(1)|p | Z1 = z1]π1(dz1) +≤ B′ +p +np/2E[|Z2(1)|p | Z1 = z1]π1(dz1) += B′ +p +np/2E[|Z2(1)|p|]. +The second inequality follows from the inequality (a+b)p ≤ 2p−1(|a|p+|b|p) +and the monotonicity of a random variable’s Lp norm : +E[|X − E[X]|p] ≤ 2p−1(E[|X|p] + |E[X]|p) ≤ 2pE[|X|p] +For the case 1 ≤ p ≤ 2, the calculation is identical to the above, except +replace the B′ +p/np/2 with B′ +p/np−1 from Corollary A.2. +The following theorem is the Marcinkiewicz-Zygmund law of large numbers +from pg. 311 [Gut, 2005], which gives us the O +� +ε−2(1+δ)� +sampling complexity +for the LBL case for 0 < δ < 1/2. +35 + +Theorem A.4. Suppose that X, X1, X2, ... are i.i.d. r.v.s., and set Sn = +�n +k=1 Xk, n ≥ 1. If E|X|p < ∞ and EX = 0 when 1 ≤ p < 2, then +E +���� +Sn +n1/p +���� +p += E|Sn|p +n +n→∞ +−→ 0. +36 + +B +Proof of Theorem 2.2 +Proof. Case 1: d = D +When d = D, Algorithm 1 samples one y(D) ∼ πD and outputs RD(y(0:D−1)) := +gD(y(0:D)). We first prove our output RD(y(0:D−1)) has a finite expectation +for almost surely every fixed y(0:D−1), then its expectation equals γD(y(0:D−1)) +follows directly from the algorithm design. To show the first point, notice +that the expectation of |RD(y(0:D−1))| given y(0:D−1) equals the conditional +expectation E[|gD(y(0:D))| | y(0:D−1)]. Since E[|gD|] < ∞ by assumption, we +have E[|gD(y(0:D))| | y(0:D−1)] < ∞ almost surely. Therefore Algorithm 1 is +unbiased when d = D for almost surely every input y(0:D−1). Furthermore, it +has computational cost 1, and the output has finite 2D+1-st moment. +Case 2: 0 ≤ d ≤ D − 1 +Now that our base case is proven, we proceed via backwards induction. +Suppose (a), (b), (c) are satisfied for d+1 where 0 ≤ d ≤ D−1. Conditioning +on y(0:d−1), we sample y(d) ∼ πd and Nd ∼ Geo(rd). Algorithm 1 will call itself +independently for 2Nd times, each with input {Depth index: d + 1, Trajectory +History: H = y(0:d), Parameters: rd+1, · · · , rD−1}. This gives us i.i.d. samples +Rd+1(y(0:d))(1), ..., Rd+1(y(0:d))(2Nd) which are used to compute the following: +S2Nd = Rd+1(y(0:d))(1) + Rd+1(y(0:d))(2) + · · · + Rd+1(y(0:d))(2Nd), +SO +2Nd−1 = Rd+1(y(0:d))(1) + Rd+1(y(0:d))(3) + · · · + Rd+1(y(0:d))(2Nd − 1), +SE +2Nd−1 = Rd+1(y(0:d))(2) + Rd+1(y(0:d))(4) + · · · + Rd+1(y(0:d))(2Nd). +37 + +Then Algorithm 1 returns as output Rd(y(0:d)) = ∆Nd/prd(Nd), where ∆Nd +is the antithetic quantity in Algorithm 1. By the inductive hypothesis on +d + 1, we have for almost surely every y(0:d): +Eπd+1:D[Rd+1(y(0:d)) | y(0:d)] = γd+1(y(0:d)) +and +Eπ +� +|Rd+1(y(0:d))|2d+2� +< +� +D +� +i=d+1 +˜Ci +� +��gD(y(0:D)) +��2D+1 +π,2D+1. +We will start with showing Rd(y(0:d−1)) has a finite computational cost and a +finite 2d+1-th moment, and then show the unbiasedness. +Finite cost: +To show the computational cost, recall that implementing Algorithm 1 with +input depth d requires 2Nd calls of Algorithm 1 with input depth d + 1. Since +Nd ∼ Geo(rd) with rd > 0.5, calling Algorithm 1 with input depth d has an +expected cost: +rd +2rd − 1 × the expected cost of Algorithm 1 with input depth d + 1, +where +rd +2rd−1 = E[2Nd] < ∞. By our inductive hypothesis, the second term in +the above product is finite, therefore the expected cost of Algorithm 1 with +input depth d is also finite. +Finite 2d+1-th moment: +Next we show Rd+1(y(0:d)) has a finite 2d+1-th moment. For every fixed +positive integer n, doing a Taylor expansions for gd at (y(0:d), γd+1) with +38 + +respect to the last component gives us: +gd +� +y(0:d), S2n +2n +� += gd(y(0:d), γd+1) + ∂d+2gd(y(0:d), γd+1) +�S2n +2n − γd+1 +� ++ 1 +2∂2 +d+2gd(y(0:d), ξ(n)) +�S2n +2n − γd+1 +�2 +gd +� +y(0:d), SO +2n−1 +2n−1 +� += gd(y(0:d), γd+1) + ∂d+2gd(y(0:d), γd+1) +�SO +2n−1 +2n−1 − γd+1 +� ++ 1 +2∂2 +d+2gd(y(0:d), ξO(n − 1)) +�SO +2n−1 +2n−1 − γd+1 +�2 +gd +� +y(0:d), SE +2n−1 +2n−1 +� += gd(y(0:d), γd+1) + ∂d+2gd(y(0:d), γd+1) +�SE +2n−1 +2n−1 − γd+1 +� ++ 1 +2∂2 +d+2gd(y(0:d), ξE(n − 1)) +�SE +2n−1 +2n−1 − γd+1 +�2 +, +with ξ(n) between γd+1 and S2n/2n, ξO(n−1) between γd+1 and SO +2n−1/2n−1, +ξE(n − 1) between γd+1 and S2n−1/2n−1. +Thus, we have: +∆n = gd +� +y(0:d), S2n +2n +� +− 1 +2 +� +gd +� +y(0:d), SO +2n−1 +2n−1 +� ++ gd +� +y(0:d), SE +2n−1 +2n−1 +�� += ∂d+2gd(y(0:d), γd+1) +�S2n +2n − γd+1 +� ++ 1 +2∂2 +d+2gd(y(0:d), ξ(n)) +�S2n +2n − γd+1 +�2 +− 1 +2 +� +∂d+2gd(y(0:d), γd+1) +�SO +2n−1 +2n−1 − γd+1 +� ++ 1 +2∂2 +d+2gd(y(0:d), ξO(n − 1)) +�SO +2n−1 +2n−1 − γd+1 +�2 ++ ∂d+2gd(y(0:d), γd+1) +�SE +2n−1 +2n−1 − γd+1 +� ++ 1 +2∂2 +d+2gd(y(0:d), ξE(n − 1)) +�SE +2n−1 +2n−1 − γd+1 +�2 � += 1 +2∂2 +d+2gd(y(0:d), ξ(n)) +�S2n +2n − γd+1 +�2 +− 1 +2 +� +1 +2∂2 +d+2gd(y(0:d), ξO(n − 1)) +�SO +2n−1 +2n−1 − γd+1 +�2 ++ 1 +2∂2 +d+2gd(y(0:d), ξE(n − 1)) +�SE +2n−1 +2n−1 − γd+1 +�2 � +. +39 + +By the LBS assumption which assumes |∂2 +d+2gd(y(0:d), z))| < Kd for every +(y(0:d), z), and our inductive hypothesis: γd+1 = E[Rd+1(y(0:d)) | y(0:d)] which +allows us to use Lemma A.3 with Z1 = y(0:d), Z2 = Rd+1(y(0:d)), we have: +∥∆n∥π,2d+1 ≤ Kd +����� +�S2n +2n − γd+1 +�2����� +π,2d+1 ++ Kd +����� +�SO +2n−1 +2n−1 − γd+1 +�2����� +π,2d+1 +≤ KdB′ +2d+2 +� +� +Eπ +� +|Rd+1(y(0:d))|2d+2� +2(2d+1n) +� +� +1/2d+1 ++ KdB′ +2d+2 +� +� +Eπ +� +|Rd+1(y(0:d))|2d+2� +2(2d+1(n−1)) +� +� +1/2d+1 += KdB′ +2d+2 +� +� +Eπ +� +|Rd+1(y(0:d))|2d+2� +2(2d+1n) +� +� +1/2d+1 ++ 2KdB′ +2d+2 +� +� +Eπ +� +|Rd+1(y(0:d))|2d+2� +2(2d+1n) +� +� +1/2d+1 += 3KdB′ +2d+2 +� +� +Eπ +� +|Rd+1(y(0:d))|2d+2� +2(2d+1n) +� +� +1/2d+1 +. +Therefore, in total: +Eπ +� +|∆n|2d+1� +≤ Dd +Eπ +� +|Rd+1(y(0:d))|2d+2� +2(2d+1n) +, +with Dd = (3KdB′ +2d+2)(2d+1). +The result claimed in (c) is obtained as follows. We have for (1−rd) = 2−kd +40 + +for some kd ∈ +� +1, +2d+1 +2d+1−1 +� +: +Eπ +� +|Rd(y(0:d−1))|2d+1� += +∞ +� +n=0 +Eπ +� +|∆n|2d+1� +prd(n)2d+1−1 +≤ +DdEπ +� +|Rd+1(y(0:d))|2d+2� +r2d+1−1 +d +∞ +� +n=0 +1 +22d+1n(1 − rd)(2d+1−1)n +≤ Cd +� +D +� +i=d+1 +˜Ci +� +��g2 +D(y(0:D)) +��2D +π,2D +∞ +� +n=0 +� +1 +22d+1−kd(2d+1−1) +�n += Cd +� +D +� +i=d+1 +˜Ci +� +��g2 +D(y(0:D)) +��2D +π,2D +� +2(2d+1−kd(2d+1−1)) +2(2d+1−kd(2d+1−1)) − 1 +� += +� D +� +i=d +˜Ci +� +��g2 +D(y(0:D)) +��2D +π,2D, +here the second inequality follows from our inductive hypothesis. The +choice kd ∈ +� +1, +2d+1 +2d+1−1 +� +is crucial. It ensures 2(2d+1)(1 − rd)(2d+1−1) > 1, and +in turn ensures the infinite summation of the above geometric series is finite. +Unbiasedness: +Now we show the unbiasedness of Rd(y(0:d−1)). Firstly, since we have just +shown Rd(y(0:d−1)) has a finite 2(d + 1)-th moment under π, it directly implies +|Rd(y(0:d−1))| has a finite first moment, which further implies +E[|Rd(y(0:d−1))| | y(0:d−1)] +is finite for π-almost surely y(0:d−1). +We fix y(0:d−1) from now on, and we will write Eπd:D[·] as a shorthand +notation for E[· | y(0:d−1)] . Recall that we have output Rd = ∆Nd/prd(Nd), +41 + +with +∆Nd = gd +� +y(0:d), S2Nd +2Nd +� +− 1 +2 +� +gd +� +y(0:d), SO +2Nd−1 +2Nd−1 +� ++ gd +� +y(0:d), SE +2Nd−1 +2Nd−1 +�� +. +Then, +Eπd:D[Rd(y(0:d−1))] = Eπd:D +� +E +� ∆Nd +prd(Nd) | Nd +�� += Eπd:D +� ∞ +� +n=0 +∆n +prd(n)prd(n) +� += Eπd:D +� ∞ +� +n=0 +∆n +� +(⋆⋆⋆) += +∞ +� +n=0 +Eπd:D[∆n] += +∞ +� +n=0 +Eπd:D +� +gd +� +y(0:d), S2n +2n +� +− 1 +2 +� +gd +� +y(0:d), SO +2n−1 +2n−1 +� ++ gd +� +y(0:d), SE +2n−1 +2n−1 +��� += Eπd:D +� +gd +� +y(0:d), lim +n→∞ +S2n +2n +�� += Eπd:D +� +gd +� +y(0:d), γd+1(y(0:d)) +�� += γd(y(0:d−1)). +All the above calculations are straightforward except for (⋆ ⋆ ⋆), which +swaps the order of expectation and summation. +Therefore we complete +this proof of unbiasedness by justifying the swap in (⋆ ⋆ ⋆). To justify the +swap, it suffices to show � +n E[|∆n|] < ∞. Notice that Rd(y(0:d)) can be +equivalently written as �∞ +n=1 ∆nI(Nd = n)/prd(n) where Nd independent +with {∆i}. Calculating Eπd:D[|Rd(y(0:d))|] yields: +42 + +Eπd:D[|Rd(y(0:d))|] = Eπd:D +������ +∞ +� +n=1 +∆nI(Nd = n) +prd(n) +����� +� += Eπd:D +� ∞ +� +n=1 +���� +∆nI(Nd = n) +prd(n) +���� +� +only one term in the summation is non-zero += +∞ +� +n=1 +Eπd:D +����� +∆nI(Nd = n) +prd(n) +���� +� +every term is non-negative += +∞ +� +n=1 +Eπd:D +����� +∆n +prd(n) +���� +� +E [I(Nd = n)] +independence between N and {∆i} += +∞ +� +n=1 +Eπd:D [|∆n|] . +Since we already know Eπd:D[|Rd(y(0:d−1))|] < ∞, this justifies our swap (⋆⋆⋆). +C +Proof of Theorem 2.4 +The proof strategy of Theorem 2.4 is very similar to Theorem 2.2. We start +with a backward induction. +Proof. Case 1: d = D +When d = D, Algorithm 1 samples one y(D) ∼ πD and outputs RD(y(0:D−1)) := +gD(y(0:D)). Again, we first prove our output RD(y(0:D−1)) has a finite expec- +tation for almost surely every fixed y(0:D−1), then its expectation equals +γD(y(0:D−1)) follows directly from the algorithm design. To show the first +43 + +point, notice that the expectation of |RD(y(0:D−1))| equals the conditional +expectation E[|gD(y(0:D))| | y(0:D−1)]. Since E[|gD|] < ∞ by assumption, we +have E[|gD(y(0:D))| | y(0:D−1)] < ∞ almost surely. Therefore Algorithm 1 is +unbiased when d = D for almost surely every input y(0:D−1). Furthermore, it +has computational cost 1, and the output has finite +� +2 − +δ +2D +� +-th moment. +Case 2: 0 ≤ d ≤ D − 1 +Now that our base case is proven, we proceed via backwards induction. +Let δd := δ/2d for every d ∈ {0, 1, . . . , D}. Suppose unbiasedness, finite +(2 − δd+1)-th moment, and finite expected computational cost are all sat- +isfied for d + 1 where 0 ≤ d ≤ D − 1. Then Algorithm 1 will call itself +independently for 2Nd times, each with input {Depth index: d + 1, Trajectory +History: H = y(0:d), Parameters: rd+1, · · · , rD−1}. This gives us i.i.d. samples +Rd+1(y(0:d))(1), ..., Rd+1(y(0:d))(2Nd) which are used to compute the following: +S2Nd = Rd+1(y(0:d))(1) + Rd+1(y(0:d))(2) + · · · + Rd+1(y(0:d))(2Nd), +SO +2Nd−1 = Rd+1(y(0:d))(1) + Rd+1(y(0:d))(3) + · · · + Rd+1(y(0:d))(2Nd − 1), +SE +2Nd−1 = Rd+1(y(0:d))(2) + Rd+1(y(0:d))(4) + · · · + Rd+1(y(0:d))(2Nd). +Then Algorithm 1 returns as output Rd(y(0:d)) = ∆Nd/prd(Nd), where ∆Nd +is defined in Algorithm 1. By the inductive hypothesis on d + 1, we have for +almost surely every y(0:d): +E[Rd+1(y(0:d)) | y(0:d)] = γd+1(y(0:d)) +44 + +and +Eπ +� +|Rd+1(y(0:d))|2−δd+1� +< +� D +� +i=d +˜Ci +� +��gD(y(0:D)) +��2−δd+1 +π,2 +. +We will start with showing Rd(y(0:d−1)) has a finite computational cost +and a finite (2 − δd)-th moment, and then show the unbiasedness. +Finite cost: +To show the computational cost, recall that implementing Algorithm 1 with +input depth d requires 2Nd calls of Algorithm 1 with input depth d + 1. It +suffices to check rd > 0.5, which reduces to check the the upper bound for kd +(defined in Theorem 2.4) staisfies +�2d+2 − 3δ +2d+3 − 3δ +� �2d+1 − δ +2d − δ +� +> 1. +Let t := 2d and the above product becomes: +4t − 3δ +8t − 3δ +2t − δ +t − δ = 8t2 + 3δ2 − 10δ +8t2 + 3δ2 − 11δ > 1. +Since Nd ∼ Geo(rd) with rd > 0.5, calling Algorithm 1 with input depth d +has an expected cost: +rd +2rd − 1 × the expected cost of Algorithm 1 with input depth d + 1, +where +rd +2rd−1 = E[2Nd] < ∞. By our inductive hypothesis, the second term in +the above product is finite, therefore the expected cost of Algorithm 1 with +input depth d is also finite. +Finite (2 − δd)-th moment: +Next we show Rd has a finite (2−δd)-th moment. By the uniform Ld-Lipschitz +45 + +property of gd: +|∆n| ≤ 1 +2 +����gd +� +y(0:d), S2n +2n +� +− gd +� +y(0:d), SO +2n−1 +2n−1 +����� + 1 +2 +����gd +� +y(0:d), S2n +2n +� +− gd +� +y(0:d), SE +2n−1 +2n−1 +����� +≤ Ld +2 +���� +SO +2n−1 +2n−1 − SE +2n−1 +2n−1 +���� . +Therefore, for any fixed 1 ≤ p < 2, applying triangle inequality under the +norm ∥·∥π,p, and applying Lemma A.3 with Z1 = y(0:d), Z2 = Rd+1(y(0:d)), we +have: +∥∆n∥π,p ≤ Ld +2 +���� +SO +2n−1 +2n−1 − SE +2n−1 +2n−1 +���� +π,p +≤ Ld +2 +���� +SO +2n−1 +2n−1 − γd+1 +���� +π,p ++ Ld +2 +����γd+1 − SE +2n−1 +2n−1 +���� +π,p +≤ Ld +�BpEπ[|Rd+1(y(0:d))|p] +2(n−1)(p−1) +�1/p +, +exponentiate both sides by p yields, +Eπ[|∆n|p] ≤ Lp +dBpEπ[|Rd+1(y(0:d))|p] +2(n−1)(p−1) +≤ C(d, p)Eπ[|Rd+1(y(0:d))|p] +2(p−1)n +, +where C(d, p) = Lp +dBp21−p. +Recall that δd = δ/2d, let us choose qd = 2 − (δd + δd+1)/2. Since +� +1, +�qd − 1 +qd +� �2 − δd +1 − δd +�� += +� +1, +�2d+2 − 3δ +2d+3 − 3δ +� �2d+1 − δ +2d − δ +�� +, +by definition of kd in the Theorem statement we have +kd < +�qd − 1 +qd +� �2 − δd +1 − δd +� +. +46 + +Now we estimate the (2 − δd)-th moment of Rd. An important trick in +the calculation below is that we are not going to use the above estimate +of Eπ[|∆n|p] directly on p = 2 − δd. +Instead, we will first use H¨older’s +inequality, and then bound Eπ[|∆n|qd](2−δd)/qd via the above estimate. It turns +out the first way gives us an order of 2−n(1−δd), while the latter is of order +2−n(qd−1)(2−δd)/qd. Since the function (x − 1)(2 − δd)/x is increasing with x +when x > 1, and equals 1−δd when x = 2−δd, we gain an extra factor 2−Ω(1)n +by choosing qd > 2 − δd and use H¨older’s inequality, which is important for +establishing our main result. The detailed calculation is below: +Eπ[|Rd(y(0:d−1))|2−δd] ≤ +∞ +� +n=0 +Eπ[|∆n|2−δd] +prd(n)1−δd +≤ +∞ +� +n=0 +� +Eπ +� +|∆n| +2−δd· +qd +2−δd +�� 2−δd +qd +prd(n)1−δd +H¨older’s inequality +≤ +1 +r1−δd +d +∞ +� +n=0 +�C(d, qd)Eπ[|Rd+1(y(0:d))|qd] +2(qd−1)n +� 2−δd +qd +1 +(1 − rd)(1−δd)n +estimate of Eπ[|∆n|p] += C′(d) +��Rd+1(y(0:d)) +��2−δd +π,qd +∞ +� +n=0 +� +1 +2 +(qd−1) +qd +(2−δd)−kd(1−δd) +�n +here C′(d) = C(d, qd)(2−δd)/qd +r1−δd +d +≤ C′(d) +��Rd+1(y(0:d)) +��2−δd +π,2−δd+1 +∞ +� +n=0 +� +1 +2 +(qd−1) +qd +(2−δd)−kd(1−δd) +�n +since qd < 2 − δd+1 +≤ C′(d) +� +D +� +i=d+1 +˜Ci +� +��gD(y(0:D)) +��2−δd +π,2 +� +� +2 +(qd−1) +qd +(2−δd)−kd(1−δd) +2 +(qd−1) +qd +(2−δd)−kd(1−δd) − 1 +� +� +inductive hypothesis += +� D +� +i=d +˜Ci +� +��gD(y(0:D)) +��2−δd +π,2 , +47 + +and note the RHS is still finite given the assumption of our theorem on gD. +Again, as we can see in the proof, the choice of kd and qd is crucial for our +calculation. It ensures (qd−1) +qd +(2 − δd) − kd(1 − δd) > 0, and in turn ensures +the above summation of the geometric series converges. +Unbiasedness: +The proof of unbiasedness of our estimator in this case is identical to the +LBS case, however we still require a justification of the existence of a finite +conditional expectation of Rd(y(0:d−1)). By what we have just proven, +Eπ[|Rd(y(0:d−1))|] ≤ +� +Eπ +� +|Rd(y(0:d−1))|2− δ +2d +��1/(2− δ +2d ) +< ∞. +Given Eπ[|Rd(y(0:d−1))|] < ∞, we immediately have Eπd:D[Rd(y(0:d−1))] exists +for almost surely every y(0:d−1). +48 + diff --git a/atE2T4oBgHgl3EQfvwi4/content/tmp_files/load_file.txt b/atE2T4oBgHgl3EQfvwi4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..09ce440bfc4f090495833d884075b8c3ca5034f1 --- /dev/null +++ b/atE2T4oBgHgl3EQfvwi4/content/tmp_files/load_file.txt @@ -0,0 +1,902 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf,len=901 +page_content='Optimal randomized multilevel Monte Carlo for repeatedly nested expectations Yasa Syed Department of Statistics Rutgers University Guanyang Wang Department of Statistics Rutgers University Abstract The estimation of repeatedly nested expectations is a challenging problem that arises in many real-world systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' However, existing methods generally suffer from high computational costs when the number of nestings becomes large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Fix any non-negative integer D for the total number of nestings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Standard Monte Carlo methods typically cost at least O(ε−(2+D)) and sometimes O(ε−2(1+D)) to obtain an estimator up to ε-error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' More advanced methods, such as multilevel Monte Carlo, currently only exist for D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' In this paper, we propose a novel Monte Carlo estimator called READ, which stands for “Recursive Estimator for Arbitrary Depth.” Our estimator has an optimal computational cost of O(ε−2) for every fixed D under 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='04095v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='CO] 10 Jan 2023 suitable assumptions, and a nearly optimal computational cost of O(ε−2(1+δ)) for any 0 < δ < 1 2 under much more general assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Our estimator is also unbiased, which makes it easy to parallelize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The key ingredients in our construction are an observation of the problem’s recursive structure and the recursive use of the randomized multilevel Monte Carlo method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Keywords: nested expectation, optimal cost, randomized Multilevel Monte Carlo, unbiased estimator 1 Introduction Monte Carlo methods are a class of algorithms that use random sampling to estimate quantities of interest, such as integrals or expected values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' When the estimand can be expressed as an expectation, for example Eπ[g(X)], these methods work by generating independent random samples X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , Xn from π, and using the average �n i=1 g(Xi)/n as an estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Monte Carlo estimators are unbiased and converge at a rate of n−1/2, regardless of the dimension of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' This dimension-independent convergence rate makes Monte Carlo methods a powerful tool for approximating high-dimensional integrations, as they do not suffer from the curse of dimensionality that plagues deterministic numeric integration methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' However, the above analysis implicitly assumes the integrand g can be pointwisely evaluated, which may not be possible in many situations of 2 practical interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' In this paper, we study the problem of estimating repeatedly nested expectations (RNE), which means the integrand depends on a sequence of other functions and conditional expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Specifically, fix any positive integer D for the total number of nestings, and {gd}D d=0 for a family of real- valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Let (y(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , y(D)) be a finite-time stochastic process with underlying joint distribution π, and let y(0:d) denote the vector (y(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , y(d)) for every d ≤ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The RNE is defined as: γ0 = E � g0 � y(0), γ1 � y(0)��� , (1) where {γi}D−1 i=1 is recursively defined as: γd(y(0:d−1)) = E � gd � y(0:d), γd+1 � y(0:d)�� | y(0:d−1)� , (2) and γD(y(0:D−1)) = E � gD � y(0:D)� | y(0:D−1)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' (3) Estimating RNEs is a fundamental problem that covers a variety of real-world applications, where the quantity of interest depends on multiple stages or decision points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' For example, when gd(y(0:d), u) := max{y(d), u} for 0 ≤ d ≤ D − 1 and gD(y(0:D)) = y(D), the quantity γ0 stands for the expected utility of the optimal strategy in a D-horizon optimal stopping problem – a central problem in financial modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Other applications include Bayesian experimental design Goda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2022], portfolio risk management Gordy and Juneja [2010] and probabilistic programs Rainforth [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 3 However, estimating RNEs is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' As shown in formulas (1) – (3), we are interested in the expectation of g0, which depends on the random variable y(0) and γ1(y(0)) – a conditional expectation of g1 given y(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Then g1 further depends on a random variable y(1) and γ2(y(0), y(1)) which is a conditional expectation of g2 given y(0) and y(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' This procedure is recursively defined until it reaches the deepest depth, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Since γ1(y(0)) (and also γ2, γ3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=') cannot be directly evaluated in most practical cases, estimating RNEs cannot be handled by standard Monte Carlo methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The most natural way to estimate RNEs is by nesting Monte Carlo (NMC) estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' In the D = 1 case, this method works by first sampling inde- pendent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=') copies y(0) 1 , y(0) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , y(0) N0 according to the distribution of y(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' For each fixed y(0) i , one further samples N1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' y(1) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , y(1) N1 according to π(y(1) | y(0) i ), and uses the standard estimator ˆγ1(y(0) i ) := �N1 i=j g1(y(0) i , y(1) j )/N1 to estimate γ1(y(0) i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The final estimator is another standard Monte Carlo estimator which uses the estimated ˆγ1(y(0) i ) to replace the intractable γ1(y(0) i ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=', IN0,N1 = 1 N0 N0 � i=1 g0(y(0) i , ˆγ1(y(0) i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' This nested estimator can be easily extended to the general D case, albeit the notations become more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Roughly, one still samples N0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' copies according to π(y(0)), and for each fixed trajectory y(0:d−1), the user generates Nd samples i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' from π(y(d) | y(0:d−1)) all the way to depth D and then form the nested estimator from the deepest depth to the shallower depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The 4 construction details are referred to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2 of Rainforth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' After suitably allocating the number of samples (Ni)D i=0 for each depth, the root-mean-square error (rMSE) of the nested Monte Carlo method is known to converge to 0 at a rate of N −1/(2D+2) or N −1/(D+2) Rainforth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2018], depending on the regularity conditions of the functions giD i=1, where N = �D i=1 Ni is the total number of samples used to form a nested estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' This convergence rate diminishes exponentially with D, meaning that NMC estimators do not have the same dimension-free convergence rate as standard Monte Carlo estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' As a result, NMC methods require at least O(ε−(2+D)) and sometimes O(ε−2(1+D)) samples to get an estimator within ε of the true value, while standard Monte Carlo estimators require only O(ε−2) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Although there are a few cases mentioned in Rainforth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2018] where the canonical O(N −1/2) rate can be achieved for NMC methods, the problem of estimating RNEs with a dimension-free convergence rate remains largely open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' In the special case D = 1, advanced Monte Carlo methods have been pro- posed Giles [2018], Giles and Goda [2019], Giles and Haji-Ali [2019] based on the celebrated multilevel Monte Carlo (MLMC) methods Heinrich [2001], Giles [2008].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' These estimators achieve up to ε-rMSE with cost O(ε−2 log(1/ε)2) or O(ε−2) under varying conditions, comparing favorably with the NMC estima- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' However, existing methods cannot be directly generalized to solve the general D case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Meanwhile, implementing these methods requires users to pre- 5 specify the precision level ε and conduct preliminary experiments/calculations to carefully estimate/bound the parameters in the MLMC algorithm (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=', Theorem 1 of Giles and Goda [2019]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Therefore, existing MLMC estimators seem to be harder to implement and less amendable to our original problem, which has a recursive structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' In this work, we propose the READ, a novel Monte Carlo estimator for the RNE problem with an arbitrary number of nestings D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Our construction is interesting in the following three aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Firstly, under suitable regularity conditions similar to those in Rainforth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2018], the rMSE of our estimator has an optimal convergence rate N −1/2 Heinrich and Sindambiwe [1999] regardless of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Equivalently, our method costs in expectation O(ε−2) to get an estimator up to ε-rMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Under much more general assumptions, our method still achieves a nearly-optimal cost of O(ε−2(1+δ)) for any 0 < δ < 1 2 to get an estimator up to ε-mean-absolute-error (MAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' It is worth mentioning that most of our effort is devoted to designing unbiased estimators of γ0 in (1) with finite computational cost and finite variance (or finite (2-δ)-th moment under more general assumptions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Af- ter developing such an unbiased estimator, we can simulate independent copies of these estimators and average them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The N − 1 2 convergence rate and O(ε−2) computational cost are then immediate corollaries of the bias-variance decomposition formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Therefore, another appealing property of READ, in contrast to existing 6 methods, is that it admits no estimation bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Unbiasedness implies these estimators can be implemented in parallel processors without requiring any communication between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Designing unbiased estimators has recently attracted much interest in statistics, operations research, and machine learning communities for its potential for parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Our methods add to the rich body of works of Glynn and Rhee [2014], Rhee and Glynn [2015], Blanchet and Glynn [2015], Jacob et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2020], Biswas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2019], Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2021], Wang and Wang [2022], Kahale [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Finally, our algorithm for constructing READ relies on the randomized multilevel Monte Carlo (rMLMC) method McLeish [2011], Rhee and Glynn [2015], Blanchet and Glynn [2015], but it is significantly different from previous applications of this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Most existing rMLMC applications Rhee and Glynn [2015], Vihola [2018], Goda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2022] have a non-randomized version with similar or better computational cost guarantees, leading some to believe that every problem solved by rMLMC also has a natural non-randomized counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' However, our work seems to suggest that this belief may not always hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The rMLMC framework is well-suited to the recursive structure of RNEs, and can be used as a subroutine in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' In contrast, the non-randomized MLMC cannot be easily applied to the general case of D > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' This suggests that the rMLMC framework may be more widely applicable than previously thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The rest of this paper is organized as follows: in the remainder of this 7 section, we discuss related works, set up our notation, and introduce our technical assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' In Section 2, we introduce our algorithm and show that it attains the optimal and nearly optimal computational cost under two different assumptions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' In Section 3, we demonstrate the empirical performance of our method on a toy example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' We conclude this paper with a short discussion in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Proof details are deferred to the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='1 Related work Our algorithm design strategy mainly follows the randomized multilevel Monte Carlo (rMLMC) framework McLeish [2011], Rhee and Glynn [2015], Blanchet and Glynn [2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Our algorithm is inspired by the unbiased optimal stopping estimator Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2022], which develops estimators for the optimal stopping problem by recursively calling the rMLMC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' We extend the methodology in Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2022] both in scope and depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Our method works with a more general class of problems formulated by Rainforth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2018], which includes the optimal stopping problem as a special case, and provides more precise results under practical assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Throughout this paper, we will assume the functions {gd}D−1 d=0 are all continuous and the process π can be perfectly simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' When D = 1 and g0 is discontinuous, progresses have been made by Broadie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2011] and Giles and Haji-Ali [2019, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' When the underlying distribution is itself 8 challenging, users have to first use MCMC to approximately sample from π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The case of D = 1 and challenging π is considered in Wang and Wang [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2 Notations Now we introduce our notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Many of our notations follow those used in Rainforth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2018], which first formally defines the RNEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Throughout this paper, we preserve the letter D for the total number of nestings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' We denote by π the underlying joint distribution of a finite-time, real-valued stochastic process (y(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , y(D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' For every 0 ≤ i ≤ j ≤ D, we use the shorthand notation y(i:j) to denote the vector (y(i), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , y(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The conditional distribution of y(d:D) given the value of y(0:d−1) is denoted by πd:D(· | y(0:d−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The marginal distribution of y(d) conditioning on y(0:d−1) is denoted by πd(· | y(0:d−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' We adopt the convention that y(0:−1) = ∅, and therefore π0 stands for the (unconditioned) marginal distribution of y(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Let Π be any probability distribution on some probability space, and Z be some random variable on the same space, then we use ∥Z∥Π,m to denote the Lm–norm of Z under Π, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=', � EΠ[|Z|m] �1/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The geometric distribution with parameter r is denoted by Geo(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' We also define pr(n) := P[Geo(r) = n] = r(1 − r)n for every n ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' For every 0 ≤ d < D, the function gd introduced in (1) – (2) maps from Rd+2 to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The function gD in (3) maps from RD+1 to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' For i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' random variables X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , Xn, we denote their summation by Sn := �n i=1 Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' When n is an even number, we denote by SO n/2 := X1 + X3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' + Xn−1 and 9 SE n/2 := X2 + X4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' + Xn the summations of their odd and even terms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='3 Assumptions Throughout this paper, we assume that we can access a simulator S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The simulator can take any trajectory y(0:d−1) with 0 ≤ d ≤ D as input, and outputs y(d) which follows the distribution πd(· | y(0:d−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' In particular, S can take ∅ as input and simulates y(0) ∼ π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Calling S recursively for D +1 times generates one complete sample path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' This assumption enables us to sample from any marginal or conditional distribution perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' This assumption is also standard and is posed explicitly or implicitly in nearly all the existing works concerning the estimation of nested expectations, see Giles and Goda [2019], Goda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2022], Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2022] for examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Fix a function f : Rk+2 → R, we say f satisfies the last-component bounded Lipschitz condition (LBL) if there exists L < ∞ such that: sup y(0:k)|f(y(0:k), x) − f(y(0:k), z)| ≤ L|x − z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' (4) We say f satisfies the last-component bounded second-derivative condition (LBS) if f has continuous second-order derivative on its last component, and there exists C < ∞ such that sup y(0:k+1) ��∂2 k+1f(y(0:k+1)) �� < C, (5) 10 where ∂2 k+1f := (∂2f)/(∂y(k+1))2 stands for the second-order partial derivative for the last component of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' These assumptions (and their variants) are also posed in related works such as Rainforth [2018], Blanchet and Glynn [2015], Giles [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 2 Algorithm, estimator, and theoretical re- sults Now we are ready to present our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' As discussed in Section 1, we will be focusing on designing a Monte Carlo estimator which is unbiased, has a finite computational cost, and has finite variance or (2-δ)-th moment under different assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Then our estimator with O(ε−2) or O(ε−2(1+δ)) cost can be directly obtained by averaging over these unbiased estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='1 Preliminary analysis One of the challenges in estimating the RNEs is the difficulty of estimating γ1(y(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Users typically first estimate γ1(y(0)) and then use these estimators to estimate γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' For the time being, we are temporarily adding the assumption that users can simulate unbiased estimators ˆγ1(y(0)) of γ1(y(0)) for every fixed y(0) with finite computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' This assumption will be removed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' It easily holds when D = 1, as users can repeatedly simulate y(1) i ∼ π1(· | y(0)) and it follows from the problem definition that each 11 g1(y(0), y(1) i ) is unbiased for γ1(y(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' In the general case of D > 1, this assumption is far from trivial, as γ1(y(0)) is itself a nested expectation with a nesting depth of D − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Nevertheless, as we will see in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2, this assumption helps us to capture and reduce the intrinsic difficulty of the problem and, therefore, will guide us to design the general algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' With this extra assumption, constructing unbiased estimators of (1) is equivalent to constructing unbiased estimators of g0(y(0), γ1(y(0))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Even with access to unbiased estimators of γ1(y(0)), the intuitive estimator g0 � y(0), ˆγ1(y(0)) � is still biased, as in general E[g0 � y(0), ˆγ1(y(0)) � | y(0)] ̸= g0(y(0), E[ˆγ1(y(0)) | y(0)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' To eliminate this bias, we turn to the rMLMC method Blanchet and Glynn [2015], which will be briefly reviewed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The rMLMC method uses the Law of Large Numbers (LLN) and rewrites g0 as the following telescoping summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' g0(y(0), γ1(y(0))) = E � g0 � y(0), lim k→∞ Sk k � | y(0) � = ∞ � n=1 E � g0 � y(0), S2n 2n � | y(0) � − E � g0 � y(0), S2n−1 2n−1 � | y(0) � , where Sk = �k i=1 ˆγ1,i(y(0)) is the summation of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' copies of ˆγ1(y(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' To unbiasedly estimate the infinite sum, the rMLMC algorithm first samples y(0) ∼ π0, then samples a random N ∼ Geo(r), finally generates 2N unbiased estimators {ˆγ1,i(y(0))}2N i=0 of γ1(y(0)) and estimates γ0 by R0 := ∆N/pr(N), 12 where ∆n is defined as: ∆n := g0 � y(0), S2n 2n � − 1 2 � g0 � y(0), SE 2n−1 2n−1 � + g0 � y(0), SO 2n−1 2n−1 �� for n ≥ 1 and ∆0 := g0(y(0), ˆγ1,1(y(0))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The next theorem justifies the theoretical properties of R0: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' With all the notations as above, suppose g0 : R2 → R satisfies LBS condition defined in (5), and ∥ˆγ1(y(0))∥π,m < ∞ for some m ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Then for any r ∈ (1/2, 3/4), the estimator R0 := ∆N/pr(N) has expectation γ0, finite variance, and finite expected computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='1 will not be proved directly, as it is a special case of our Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' For now, we use the following heuristic calculation to justify the unbiasedness of ˆγ0: E[R0 | y(0)] = E � E � R0 | N, y(0)�� = ∞ � n=0 E � ∆n pr(n)pr(n) | y(0) � = ∞ � n=0 E � g0 � y(0), S2n 2n � | y(0) � − E � g0 � y(0), S2n−1 2n−1 � | y(0) � = g0(y(0), γ1(y(0))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Therefore E[R0] = E[g0(y(0), γ1(y(0)))] = γ0 by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' More technical discussions such as the range of r, other possible regularity conditions on g0, and the moment guarantees of γ0 will all be deferred after Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2 Recursive rMLMC algorithm for general D Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='1 is useful to solve our original problem (without the extra as- sumption) in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' First, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='1 already solves the case where D = 1, as our extra assumption automatically holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' It states that if g0 has a bounded second derivative on its last component, and g1(y(0), y(1)) has at least finite fourth moment under π, then R0 is unbiased, has finite variance, and finite expected computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' More importantly, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='1 tells us that the original problem of estimating an RNE with a depth of D can be solved if we can unbiasedly estimate γ1(y(0)) for fixed y(0), which is another RNE with a depth of D − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Therefore, we have successfully reduced the number of nestings by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' This observation motivates us to come up with an algorithm for the general D case, as explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' We first go one step further to illustrate the D = 2 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' When D = 2, estimating γ1(y(0)) again reduces the case we have analyzed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' To be precise, since g2(y(0:2)) is unbiased for γ2(y(0:1)) if y(2) ∼ π2(· | y(0:1)), one can first sample y(1) ∼ π1(· | y(0)), then simulate N ∼ Geo(r) and 2N samples {y(2) i }2N i=1 from π2(· | y(0:1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Let ˆγ2,i(y(0:1)) := g2(y(0:1), y(2) i ), our estimator of γ1(y(0)) is then constructed in the same way as Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=', R1(y(0)) := ∆N/pr(N) with ∆n := g1 � y(0:1), S2n 2n � − 1 2 � g1 � y(0:1), SE 2n−1 2n−1 � + g1 � y(0:1), SO 2n−1 2n−1 �� , 14 where S2n, SE 2n−1, SO 2n−1 are the summation of every, even, and odd terms in {γ2,i(y(0:1))}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The same procedure of simulating R1(y(0)) can be repeated independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Therefore we can sample another geometrically dis- tributed random variable N ′ ∼ Geo(r′), and generate R1,i(y(0)) := ∆N′/pr(N ′) independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Since each R1,i(y(0)) is unbiased for γ1(y(0)), one can again use the method described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='1 to form our final estimator for γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' After checking R1(y(0)) satisfies the finite fourth-moment assumption, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='1 can be applied which implies our estimator is unbiased, has finite variance and finite cost (for the D = 2 case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The general case works in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' A key observation is that, due to the nested structure of the problem, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='1 not only states that an unbiased estimator of γ0 can be constructed if one can unbiasedly estimate γ1(y(0)) for every y(0), but also directly implies that an unbiased estimator of γd(y(0:d−1)) can be constructed if one can unbiasedly estimate γd+1(y(0:d)) for every y(0:d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Therefore, we can estimate γ0 in a backward, inductive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' To begin, we consider the deepest depth of the problem, fixing any y(0:D−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' An unbiased estimator of γD(y(0:D−1)) can be directly constructed as gD(y(0:D−1), y(D)), where y(D) ∼ πD(· | y(0:D−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' For 0 ≤ d ≤ D − 1, if we assume that users can generate unbiased estimators of γd+1(y(0:d)) for every y(0:d), then we can obtain an unbiased estimator of γd(y(0:d−1)) by sampling one y(d), generating Nd ∼ Geo(rd) and 2Nd unbiased estimators of γd+1(y(0:d)), and applying the method described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' This process continues 15 until we reach d = 0, at which point we have an unbiased estimator of γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The parameters (r0, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , rD−1) will be carefully chosen and depend on the regularity assumptions of (g0, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , gD−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' These choices will be discussed in more detail later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Our algorithm for constructing READ is described in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' It is written as a recursive algorithm, though it could also be equivalently written in an iterative form with much more cumbersome notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Algorithm 1 takes a depth index, a trajectory, a simulator, and parameters for the geometric distribution as inputs, and outputs one unbiased estimator of γd(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' In particular, with inputs {depth = 0, trajectory = ∅, parameters = (r0, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , rD−1)}, it outputs READ – an unbiased estimator of the RNE defined in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The logic of Algorithm 1 is precisely the same as we just discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' To estimate γd(y(0:d−1)), the algorithm first checks the value of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' When d = D, the problem becomes straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' When d < D, the algorithm samples y(d), appends y(d) to the trajectory, samples Nd, and calls itself 2Nd times with depth d + 1 and new trajectory {y(0:d)} to get 2Nd unbiased estimators of γd+1(y(0:d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Finally, we split these 2Nd estimators into even and odd terms and apply the method described earlier in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The algorithm is guaranteed to stop as the depth will eventually reach the deepest depth D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 16 Algorithm 1 A recursive rMLMC algorithm for RNEs Input: Depth index d ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=', D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Trajectory history H = {y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=', yd−1} or ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' A simulator S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Parameters rd, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=', rD−1 determined by conditions on {gd}D−1 d=0 (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Output: An unbiased estimator of γd(H) if d = D then Sample one y(D) ∼ πD (· | H);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Return: RD := gD � y(0:D)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' else Sample y(d) ∼ πd (· | H);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Update the trajectory H ← H ∪ � y(d)� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Sample Nd ∼ Geo(rd);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Call Algorithm 2 for 2Nd times with inputs (d + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' rd+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=', rD−1), and label the observations as Rd+1(y(0:d))(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=', Rd+1(y(0:d)) � 2Nd� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Calculate S2Nd, SE 2Nd−1, SO 2Nd−1 defined in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Calculate � note ∆0 := gd � y(0:d), Rd+1(y(0:d))(1) �� : ∆Nd = gd � y(0:d), S2Nd 2Nd � − 1 2 � gd � y(0:d), SO 2Nd−1 2Nd−1 � + gd � y(0:d), SE 2Nd−1 2Nd−1 �� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Return: Rd := ∆Nd/prd(Nd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' end if 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='3 Theoretical guarantees We now discuss the computational costs of Algorithm 1 and the statistical properties of READ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Our theoretical results depend on the smoothness conditions of {gd}D−1 d=0 , so we will consider two cases where {gd}D−1 d=0 satisfies the LBS and LBL assumptions separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='1 The LBS case The following theorem shows, under the LBS assumption, the computational cost and the variance of READ can be controlled simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Suppose for every d ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , D − 1}, the function gd satisfies the LBS assumption defined in (5), and rd := 1 − 2−kd satisfies kd ∈ � 1, 2d+1 2d+1−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Moreover, suppose ∥gD(y(0:D))∥π,2D+1 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Then for every 0 ≤ d ≤ D, the output Rd(y(0:d−1)) of Algorithm 1 with inputs {depth = d, trajectory = y(0:d−1), S, parameters (rd, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , rD−1)} has the following properties: For almost surely every fixed y(0:d−1), E � Rd(y(0:d−1)) | y(0:d−1)� = γd(y(0:d−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The expected computational cost of Rd is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The output has finite 2d+1-th moment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=', Eπ � |Rd(y(0:d−1))|2d+1� < ∞ for 0 ≤ d ≤ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 18 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2 states for π-almost surely every y(0:d−1), the expectation of the output Rd conditioning on the input is unbiased for γd(y(0:d−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The computational cost has a finite expectation, and the output has a finite 2d+1-th moment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The detailed proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2 will be provided in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Here, we highlight two special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' First, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2 shows that READ, the output R0 of Algorithm 1 when given input {depth = 0, trajectory = ∅, S, parameters = (r0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , rD−1)}, has the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Specifically, it is an unbiased estimator for γ0 with finite expected computational cost and finite variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Second, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2 recovers Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='1 when D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Let R0,1, R0,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , be the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' outcomes by repeatedly implementing Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Since each R0,i is unbiased and has a finite variance, the standard Central Limit Theorem (CLT) implies that √n(�n i=1 R0,i/n − γ0) → N(0, 1) in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' This means that the estimator �n i=1 R0,i/n converges to γ0 at a rate of n−1/2, which compares favorably with the rates obtained by NMC estimators in Rainforth [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' This rate is optimal in the sense that it matches the minimax lower bound over all the Monte Carlo methods (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='1 of Heinrich and Sindambiwe [1999]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The next corollary shows that, by repeatedly implementing Algorithm 1, it is possible to obtain an unbiased estimator for γ0 with at most ε2-MSE within O(ε−2) computational 1Readers should notice that the expectation of Rd(y(0:d−1)) is calculated under the conditional distribution πd:D(· | y(0:d−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The computational cost and the 2d+1-th moment are calculated under the joint distribution π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' When the input depth = 0, these two underlying distributions coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 19 cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' With all the assumptions the same as Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2, for any ε > 0, we can construct an estimator R with expected computational cost O(ε−2) such that the mean squared error E[(R − γ0)2] ≤ ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Calling Algorithm 1 independently for n times with {depth = 0, trajectory = ∅, S, parameters = (r0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , rD−1)} yield i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' unbiased estimators R0,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=', R0,n for γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Let our estimator be R := 1 n �n i=1 R(i) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Then, E[(R − γ0)2] = E � � � 1 n n � i=1 R0,i − γ0 �2� � = 1 nVar(R0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Thus noting that Var(R0) < ∞ by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2, taking n = Var(R0)/ε2 samples ensures R has up to ε2-MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Finally, let C := C(D) < ∞ be the expected computational cost for implementing Algorithm 1 once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The expected computational cost for constructing R is then C · Var(R0)/ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The ε2-MSE of R can be easily translated to other performance metrics via standard inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' For example, for any δ, Markov’s inequality implies the absolute error |R − γ0| is less than ε/ √ δ with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Next, we discuss the assumptions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The assumptions require the first D functions {gd}D−1 d=0 all satisfy the LBS condition, and the final function gD has finite 2D+1-th moment under π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The LBS assumption also appears in the work of NMC estimators (see the second part of Theorem 3 in Rainforth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2018]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The moment assumption of gD is not required 20 in Rainforth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Nevertheless, it is a mild assumption that holds in most practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' It covers all the cases where gD is bounded or has a moment generating function (including the uniform, Gaussian, Poisson, or exponential distributions), which implies E[|gD|k] < ∞ for every k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' As we will see in our proofs, these assumptions help us to establish the third conclusion of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2 in a backward inductive way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' For example, 2D+1-th moment assumption on gD and the LBS assumption on gD−1 implies RD−1 has finite 2D-th moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' More generally, the finiteness of the 2d+1-th moment of Rd follows from the LBS assumption on gd−1 and the 2d+2-th moment of Rd+1 (which is the conclusion of the previous inductive step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Eventually, we conclude R0 has a finite variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Finally, we want to emphasize our moment assumption on gD is not ‘trajectory-dependent’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' We require gD has finite 2D+1-th moment under the joint distribution π of y(0:D), which is much weaker than gD has a uniformly bounded finite 2D+1-th moment under π0:D−1(· | y(0:D−1)) for every fixed trajectory y(0:D−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Finally, the parameters {rd}D=1 d=0 reflect the trade-off between the variance and computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Since 2Nd calls are required for each d, standard calculation shows that E[2Nd] = rd/(2rd − 1) when rd > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='5, and +∞ if rd ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Therefore, every rd has to be strictly greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='5 to ensure a finite expected computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Meanwhile, we cannot guarantee finite variance or unbiasedness of READ when rd becomes too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Our range for rd in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2 follows from a careful calculation in our proof to ensure 21 unbiasedness, finite computational cost, and variance simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2 The LBL case The assumptions in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2 guarantees READ enjoys the optimal con- vergence rate and computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' However, the second-order derivative assumption also rules out many functions of practical interest, such as max and min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' In this section, we study the theoretical properties of Algorithm 1 and READ under weaker smoothness and moment assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Our result is summarized below: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Fix any 0 < δ < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Suppose for every d ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , D −1}, the function gd satisfies the LBL assumption defined in (4), and rd := 1−2−kd satisfies kd ∈ � 1, �2d+2 − 3δ 2d+3 − 3δ � �2d+1 − δ 2d − δ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Moreover, suppose ∥gD(y(0:D))∥π,2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Then for every 0 ≤ d ≤ D, the output Rd(y(0:d−1)) of Algorithm 1 with inputs {depth = d, trajectory = y(0:d−1), S, parameters (rd, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , rD−1)} has the following properties: For almost surely every fixed y(0:d−1), E � Rd(y(0:d−1)) | y(0:d−1)� = γd(y(0:d−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The expected computational cost of Rd is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The output has finite (2 − δ/2d)-th moment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=', Eπ � |Rd(y(0:d−1))|(2−δ/2d)� < ∞ for 0 ≤ d ≤ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 22 Comparing Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2, which requires the LBS assumption for {gd}D−1 d=0 and finite 2D+1-th moment for gD, with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='4, which only requires the LBL assumption for {gd}D−1 d=1 and finite second moment for gD, we see that the latter has more general assumptions but at the cost of not guaranteeing that READ has a finite variance (though it is still unbiased and has a finite expected computational cost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' However, the loss of moment guarantees can be minimal, as for any small δ, one can still choose suitable parameters such that READ has finite (2 − δ)-th moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Again, let R0,1, R0,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , be the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' outcomes by repeatedly implement- ing Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' There are some more technical subtleties when analyzing the convergence rate of �n i=1 R0,i/n−γ0 as the CLT cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Neverthe- less, we use the Marcinkiewicz-Zygmund generalized law of large numbers (see Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='4 in the Appendix), which shows n−1E[|�n i=1 Xi|p] → 0 if {Xi}n i=1 are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=', centered random variables with finite p-th moment for p ∈ [1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Our result is the following: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' With all the setting the same as Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='4, let R0,1, R0,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , be the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' outcomes by repeatedly implementing Algorithm 1, we have: E[|�n i=1 R0,i/n − γ0|] = o(n−1/(2(1+δ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' We can construct an estimator R with expected computational cost O(ε−2(1+δ)) such that the mean absolute error E[|R − γ0|] < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Applying Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='4 with p = 2 − δ, Xi = R0,i − γ0 23 and Jensen’s inequality, we have: E ������ n � i=1 R0,i/n − γ0 ����� � = n−1E ������ n � i=1 Xi ����� � ≤ n−1 � E ������ n � i=1 Xi ����� p��1/p = o(n−1+1/p) = o(n−1/(2(1+δ))), which proves the first part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The last step follows from 1 − 1/(2 − δ) > 1/(2 + 2δ) for δ ∈ (0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Setting n = ε−2(1+δ) and the second part immediately follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Although we are not able to recover the optimal n−1/2 convergence rate under this more general assumption, our convergence rate is still near-optimal as it can be as close to n−1/2 as we want and does not depend on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' At the same time, although we replace the MSE by MAE due to the moment constraint, one can still use Markov’s inequality to show the absolute error |R − γ0| is less than ε/δ with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' As the max function satisfies the LBL assumption, our results here include the optimal stopping problem as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Our results complement the work of Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2022], where the authors use rMLMC to design an estimator with O(ε−2) computational cost under stronger assumptions (see their Assumption 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' We have a slightly worse cost of O(ε−2(1+δ)) but holds under more general assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 24 3 Numerical experiments We consider the following simple example with known ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Suppose the process (y(0), y(1), y(2)) satisfies y(0) ∼ N(π/2, 1), y(1) ∼ N(y(0), 1), y(2) ∼ N(y(1), 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Define g0(y(0), z) := sin � y(0) + z � , g1(y(0:1), z) := sin � y(1) − z � , and g2(y(0:2)) := y(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The target quantity γ0 defined (1) is a nested expectation with D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' One can use the formula EZ∼N(µ,σ2)[sin(Z)] = sin(µ) exp(−σ2/2) to analytically calculate γ0 = exp(−1/2) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='6065.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Now we compare our READ estimator with the NMC estimator in Rainforth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' For the NMC estimator, users first specify N0, N1, N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Then we sample N0 copies of y(0), N1 copies of y(1) for each fixed y(0), and N2 copies of y(2) for each fixed y(0:1), and use these samples to form the NMC estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Following Rainforth [2018], we consider two ways of allocating (N0, N1, N2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The first estimator NMC1 is to choose N0 = N1 = N2, the second NMC2 is to choose N0 = N 2 1 = N 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Both methods have cost n = N0N1N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' For READ, notice that all the assumptions in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2 are satisfied, therefore for r0 ∈ (1/2, 3/4) and r1 ∈ (1/2, 1 − 2−4/3), and so READ estimator generated by Algorithm 1 should be unbiased and of finite variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Since the computational cost gets lower when each ri gets larger, we choose r0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='74 and r1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='6 (close to the upper-end of their respective ranges above) to facilitate the computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Therefore, implementing Algorithm 1 once has an expected sample size/computational cost (r1/(2r1 − 1)) (r2/(2r2 − 1)) ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Our comparison result is summarized in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The slopes of the blue, 25 red, and green lines, which correspond to the empirical convergence rate of READ, NMC1, NMC2, equals −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='97, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='33, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='53, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' They match well with the theoretical predictions n−1 in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='3 for READ, n−1/3 for NMC1, and n−1/2 for NMC2 in Theorem 3 of Rainforth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' We also repeatedly call Algorithm 1 for 106 times and plot the running averages of our estimates in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Our estimator becomes more accurate (closer to the red dashed line) when we increase the number of repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' In addition, for each k ∈ (1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , 106), we also calculate the standard deviation (sd) of the first k repetitions and use Mean ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='96 sd to form the 95% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' It is also clear from Figure 2 that our confidence intervals always include the ground-truth, suggesting the high accuracy of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' In contrast, constructing confidence intervals of NMC estimators are much more time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 4 Further discussions Here we provide some remarks for practical implementation and discuss some potential generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The users need to specify the parameters {rd}D−1 d=0 when implementing Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Larger values of ri lead to a shorter time for each implementation but potentially larger variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' When some ri is not chosen according to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='4, the algorithm can still be implemented, but the variance may be infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The trade-off between the values of {rd} and the fluctuations of the resulting estimator is problem- 26 −8 −6 −4 −2 2 4 6 8 log(Sample Size) log(MSE) method NMC1 NMC2 READ Figure 1: The comparison on the empirical MSEs of estimating the RNE among READ (blue), NMC1 (red), and NMC2 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' All the logarithms are of the base 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Each method’s empirical errors are calculated based on 20 independent repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='64 0 250000 500000 750000 1000000 Number of estimators Running Average Figure 2: The trace plot (solid blue curve) of the running averages of READ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The blue dotted curves are the 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The red dashed line is the ground truth exp(−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 28 specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' In practice, knowing how many repetitions are sufficient is important to provide an accurate estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' One possible way is to bound the variance of READ and use Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='3 to choose a sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' But this bound can be problem-specific and very conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Instead, we follow Glynn and Rhee [2014] and suggest the following adaptive stopping rule: users first specify a precision-level ε and a small δ%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' When repeatedly implementing Algorithm 1, users calculate the empirical (1 − δ%) confidence interval [Lδ(k), Uδ(k)] for first k repetitions in the same way as Section 3 for every k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Users can stop when the width of the confidence interval is less than 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The validity of this stopping rule is proven in Glynn and Whitt [1992].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' There are two directions for extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' First, each y(i) in our paper is assumed to be univariate for concreteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Extending this assumption to multivariate should be straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Moreover, the computational cost of Algorithm 1 scales linearly with the dimensionality of y(i), which may be another appealing property of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Second, in this paper, we only consider the ‘fix D’ regime and construct estimators with optimal/near- optimal convergence rate for total sample size n (or computational cost for ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' On the other hand, the cost of Algorithm 1 scales exponentially with D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Therefore, although our algorithm is more efficient than the NMC estimator for every fixed D, both methods are not practically useful when D becomes very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Indeed, the poor scaling with D frequently happens in related literature such as Glasserman and Yu [2004], Zanger [2013] and 29 seems unavoidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' An interesting direction would be constructing proper modifications of Algorithm 1 under extra practically relevant assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' For example, if we know the ‘influence’ of γd on γ0 decays exponentially or double-exponentially when d increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Then it may be sufficient to truncate the depth to ˜D := log(1/ε) or � log(1/ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' We hope to report progress in our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' References N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Biswas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Jacob, and P.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Let p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Suppose that X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='Xn are independent r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' with mean 0 such that E|Xk|p < ∞ for all k, and let Sn := �n i=1 Xi denote the partial sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Then there exist constants A∗ p, B∗ p depending only on p such that A∗ pE � n � k=1 X2 k �p/2 ≤ E|Sn|p ≤ B∗ pE � n � k=1 X2 k �p/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The following corollary to the above theorem is from pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 151 [Gut, 2005], Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Let p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Suppose that X, X1, X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='Xn are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' with mean 0 such that E|X|p < ∞, and let Sn := �n i=1 Xi denote the partial sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Then there exists a constant Bp depending only on p, such that E|Sn|p ≤ � � � � � � � Bpnp/2E|X|p, p > 2 BpnE|X|p, 1 ≤ p ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The following lemma is instrumental in the proofs for the theoretical guarantees of our algorithm under both the LBS and LBL assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Let (Z1, Z2) be a 2-stage stochastic process and there exists p ≥ 1, such that E[|Z2|p] < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Conditioning on Z1, sample i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Z2(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=', Z2(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 34 Then, E ������ 1 n n � i=1 Z2(i) − E[Z2 | Z1] ����� p� ≤ � � � � � � � B′ p E[|Z2|p] np/2 p > 2 B′ p E[|Z2|p] np−1 1 ≤ p ≤ 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Let p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' For arbitrary fixed Z1 = z1, define ¯Z2(i) := Z2(i)−E[Z2(i) | Z1 = z1], and apply Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2 on the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' mean 0 random variables ¯Z2(i) under the probability distribution π(· | Z1 = z1), we have E ������ 1 n n � i=1 Z2(i) − E[Z2 | Z1] ����� p� = � Ω E ������ 1 n n � i=1 Z2(i) − E[Z2 | Z1] ����� p | Z1 = z1 � π1(dz1) = 1 npE[ ����� n � i=1 ¯Z2(i) ����� p | Z1 = z1]π1(dz1) ≤ Bp np/2E[| ¯Z2(1)|p | Z1 = z1]π1(dz1) ≤ B′ p np/2E[|Z2(1)|p | Z1 = z1]π1(dz1) = B′ p np/2E[|Z2(1)|p|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The second inequality follows from the inequality (a+b)p ≤ 2p−1(|a|p+|b|p) and the monotonicity of a random variable’s Lp norm : E[|X − E[X]|p] ≤ 2p−1(E[|X|p] + |E[X]|p) ≤ 2pE[|X|p] For the case 1 ≤ p ≤ 2, the calculation is identical to the above, except replace the B′ p/np/2 with B′ p/np−1 from Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The following theorem is the Marcinkiewicz-Zygmund law of large numbers from pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 311 [Gut, 2005], which gives us the O � ε−2(1+δ)� sampling complexity for the LBL case for 0 < δ < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 35 Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Suppose that X, X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=', and set Sn = �n k=1 Xk, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' If E|X|p < ∞ and EX = 0 when 1 ≤ p < 2, then E ���� Sn n1/p ���� p = E|Sn|p n n→∞ −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 36 B Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Case 1: d = D When d = D, Algorithm 1 samples one y(D) ∼ πD and outputs RD(y(0:D−1)) := gD(y(0:D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' We first prove our output RD(y(0:D−1)) has a finite expectation for almost surely every fixed y(0:D−1), then its expectation equals γD(y(0:D−1)) follows directly from the algorithm design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' To show the first point, notice that the expectation of |RD(y(0:D−1))| given y(0:D−1) equals the conditional expectation E[|gD(y(0:D))| | y(0:D−1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Since E[|gD|] < ∞ by assumption, we have E[|gD(y(0:D))| | y(0:D−1)] < ∞ almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Therefore Algorithm 1 is unbiased when d = D for almost surely every input y(0:D−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Furthermore, it has computational cost 1, and the output has finite 2D+1-st moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Case 2: 0 ≤ d ≤ D − 1 Now that our base case is proven, we proceed via backwards induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Suppose (a), (b), (c) are satisfied for d+1 where 0 ≤ d ≤ D−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Conditioning on y(0:d−1), we sample y(d) ∼ πd and Nd ∼ Geo(rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Algorithm 1 will call itself independently for 2Nd times, each with input {Depth index: d + 1, Trajectory History: H = y(0:d), Parameters: rd+1, · · · , rD−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' This gives us i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' samples Rd+1(y(0:d))(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=', Rd+1(y(0:d))(2Nd) which are used to compute the following: S2Nd = Rd+1(y(0:d))(1) + Rd+1(y(0:d))(2) + · · · + Rd+1(y(0:d))(2Nd), SO 2Nd−1 = Rd+1(y(0:d))(1) + Rd+1(y(0:d))(3) + · · · + Rd+1(y(0:d))(2Nd − 1), SE 2Nd−1 = Rd+1(y(0:d))(2) + Rd+1(y(0:d))(4) + · · · + Rd+1(y(0:d))(2Nd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 37 Then Algorithm 1 returns as output Rd(y(0:d)) = ∆Nd/prd(Nd), where ∆Nd is the antithetic quantity in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' By the inductive hypothesis on d + 1, we have for almost surely every y(0:d): Eπd+1:D[Rd+1(y(0:d)) | y(0:d)] = γd+1(y(0:d)) and Eπ � |Rd+1(y(0:d))|2d+2� < � D � i=d+1 ˜Ci � ��gD(y(0:D)) ��2D+1 π,2D+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' We will start with showing Rd(y(0:d−1)) has a finite computational cost and a finite 2d+1-th moment, and then show the unbiasedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Finite cost: To show the computational cost, recall that implementing Algorithm 1 with input depth d requires 2Nd calls of Algorithm 1 with input depth d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Since Nd ∼ Geo(rd) with rd > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='5, calling Algorithm 1 with input depth d has an expected cost: rd 2rd − 1 × the expected cost of Algorithm 1 with input depth d + 1, where rd 2rd−1 = E[2Nd] < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' By our inductive hypothesis, the second term in the above product is finite, therefore the expected cost of Algorithm 1 with input depth d is also finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Finite 2d+1-th moment: Next we show Rd+1(y(0:d)) has a finite 2d+1-th moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' For every fixed positive integer n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' doing a Taylor expansions for gd at (y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' γd+1) with 38 respect to the last component gives us: gd � y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' S2n 2n � = gd(y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' γd+1) + ∂d+2gd(y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' γd+1) �S2n 2n − γd+1 � + 1 2∂2 d+2gd(y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' ξ(n)) �S2n 2n − γd+1 �2 gd � y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' SO 2n−1 2n−1 � = gd(y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' γd+1) + ∂d+2gd(y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' γd+1) �SO 2n−1 2n−1 − γd+1 � + 1 2∂2 d+2gd(y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' ξO(n − 1)) �SO 2n−1 2n−1 − γd+1 �2 gd � y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' SE 2n−1 2n−1 � = gd(y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' γd+1) + ∂d+2gd(y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' γd+1) �SE 2n−1 2n−1 − γd+1 � + 1 2∂2 d+2gd(y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' ξE(n − 1)) �SE 2n−1 2n−1 − γd+1 �2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' with ξ(n) between γd+1 and S2n/2n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' ξO(n−1) between γd+1 and SO 2n−1/2n−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' ξE(n − 1) between γd+1 and S2n−1/2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' we have: ∆n = gd � y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' S2n 2n � − 1 2 � gd � y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' SO 2n−1 2n−1 � + gd � y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' SE 2n−1 2n−1 �� = ∂d+2gd(y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' γd+1) �S2n 2n − γd+1 � + 1 2∂2 d+2gd(y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' ξ(n)) �S2n 2n − γd+1 �2 − 1 2 � ∂d+2gd(y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' γd+1) �SO 2n−1 2n−1 − γd+1 � + 1 2∂2 d+2gd(y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' ξO(n − 1)) �SO 2n−1 2n−1 − γd+1 �2 + ∂d+2gd(y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' γd+1) �SE 2n−1 2n−1 − γd+1 � + 1 2∂2 d+2gd(y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' ξE(n − 1)) �SE 2n−1 2n−1 − γd+1 �2 � = 1 2∂2 d+2gd(y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' ξ(n)) �S2n 2n − γd+1 �2 − 1 2 � 1 2∂2 d+2gd(y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' ξO(n − 1)) �SO 2n−1 2n−1 − γd+1 �2 + 1 2∂2 d+2gd(y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' ξE(n − 1)) �SE 2n−1 2n−1 − γd+1 �2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 39 By the LBS assumption which assumes |∂2 d+2gd(y(0:d), z))| < Kd for every (y(0:d), z), and our inductive hypothesis: γd+1 = E[Rd+1(y(0:d)) | y(0:d)] which allows us to use Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='3 with Z1 = y(0:d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Z2 = Rd+1(y(0:d)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' we have: ∥∆n∥π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2d+1 ≤ Kd ����� �S2n 2n − γd+1 �2����� π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2d+1 + Kd ����� �SO 2n−1 2n−1 − γd+1 �2����� π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2d+1 ≤ KdB′ 2d+2 � � Eπ � |Rd+1(y(0:d))|2d+2� 2(2d+1n) � � 1/2d+1 + KdB′ 2d+2 � � Eπ � |Rd+1(y(0:d))|2d+2� 2(2d+1(n−1)) � � 1/2d+1 = KdB′ 2d+2 � � Eπ � |Rd+1(y(0:d))|2d+2� 2(2d+1n) � � 1/2d+1 + 2KdB′ 2d+2 � � Eπ � |Rd+1(y(0:d))|2d+2� 2(2d+1n) � � 1/2d+1 = 3KdB′ 2d+2 � � Eπ � |Rd+1(y(0:d))|2d+2� 2(2d+1n) � � 1/2d+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Therefore, in total: Eπ � |∆n|2d+1� ≤ Dd Eπ � |Rd+1(y(0:d))|2d+2� 2(2d+1n) , with Dd = (3KdB′ 2d+2)(2d+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The result claimed in (c) is obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' We have for (1−rd) = 2−kd 40 for some kd ∈ � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 2d+1 2d+1−1 � : Eπ � |Rd(y(0:d−1))|2d+1� = ∞ � n=0 Eπ � |∆n|2d+1� prd(n)2d+1−1 ≤ DdEπ � |Rd+1(y(0:d))|2d+2� r2d+1−1 d ∞ � n=0 1 22d+1n(1 − rd)(2d+1−1)n ≤ Cd � D � i=d+1 ˜Ci � ��g2 D(y(0:D)) ��2D π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2D ∞ � n=0 � 1 22d+1−kd(2d+1−1) �n = Cd � D � i=d+1 ˜Ci � ��g2 D(y(0:D)) ��2D π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2D � 2(2d+1−kd(2d+1−1)) 2(2d+1−kd(2d+1−1)) − 1 � = � D � i=d ˜Ci � ��g2 D(y(0:D)) ��2D π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' here the second inequality follows from our inductive hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The choice kd ∈ � 1, 2d+1 2d+1−1 � is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' It ensures 2(2d+1)(1 − rd)(2d+1−1) > 1, and in turn ensures the infinite summation of the above geometric series is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Unbiasedness: Now we show the unbiasedness of Rd(y(0:d−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Firstly, since we have just shown Rd(y(0:d−1)) has a finite 2(d + 1)-th moment under π, it directly implies |Rd(y(0:d−1))| has a finite first moment, which further implies E[|Rd(y(0:d−1))| | y(0:d−1)] is finite for π-almost surely y(0:d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' We fix y(0:d−1) from now on, and we will write Eπd:D[·] as a shorthand notation for E[· | y(0:d−1)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Recall that we have output Rd = ∆Nd/prd(Nd), 41 with ∆Nd = gd � y(0:d), S2Nd 2Nd � − 1 2 � gd � y(0:d), SO 2Nd−1 2Nd−1 � + gd � y(0:d), SE 2Nd−1 2Nd−1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Then, Eπd:D[Rd(y(0:d−1))] = Eπd:D � E � ∆Nd prd(Nd) | Nd �� = Eπd:D � ∞ � n=0 ∆n prd(n)prd(n) � = Eπd:D � ∞ � n=0 ∆n � (⋆⋆⋆) = ∞ � n=0 Eπd:D[∆n] = ∞ � n=0 Eπd:D � gd � y(0:d), S2n 2n � − 1 2 � gd � y(0:d), SO 2n−1 2n−1 � + gd � y(0:d), SE 2n−1 2n−1 ��� = Eπd:D � gd � y(0:d), lim n→∞ S2n 2n �� = Eπd:D � gd � y(0:d), γd+1(y(0:d)) �� = γd(y(0:d−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' All the above calculations are straightforward except for (⋆ ⋆ ⋆), which swaps the order of expectation and summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Therefore we complete this proof of unbiasedness by justifying the swap in (⋆ ⋆ ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' To justify the swap, it suffices to show � n E[|∆n|] < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Notice that Rd(y(0:d)) can be equivalently written as �∞ n=1 ∆nI(Nd = n)/prd(n) where Nd independent with {∆i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Calculating Eπd:D[|Rd(y(0:d))|] yields: 42 Eπd:D[|Rd(y(0:d))|] = Eπd:D ������ ∞ � n=1 ∆nI(Nd = n) prd(n) ����� � = Eπd:D � ∞ � n=1 ���� ∆nI(Nd = n) prd(n) ���� � only one term in the summation is non-zero = ∞ � n=1 Eπd:D ����� ∆nI(Nd = n) prd(n) ���� � every term is non-negative = ∞ � n=1 Eπd:D ����� ∆n prd(n) ���� � E [I(Nd = n)] independence between N and {∆i} = ∞ � n=1 Eπd:D [|∆n|] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Since we already know Eπd:D[|Rd(y(0:d−1))|] < ∞, this justifies our swap (⋆⋆⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' C Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='4 The proof strategy of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='4 is very similar to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' We start with a backward induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Case 1: d = D When d = D, Algorithm 1 samples one y(D) ∼ πD and outputs RD(y(0:D−1)) := gD(y(0:D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Again, we first prove our output RD(y(0:D−1)) has a finite expec- tation for almost surely every fixed y(0:D−1), then its expectation equals γD(y(0:D−1)) follows directly from the algorithm design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' To show the first 43 point, notice that the expectation of |RD(y(0:D−1))| equals the conditional expectation E[|gD(y(0:D))| | y(0:D−1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Since E[|gD|] < ∞ by assumption, we have E[|gD(y(0:D))| | y(0:D−1)] < ∞ almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Therefore Algorithm 1 is unbiased when d = D for almost surely every input y(0:D−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Furthermore, it has computational cost 1, and the output has finite � 2 − δ 2D � th moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Case 2: 0 ≤ d ≤ D − 1 Now that our base case is proven, we proceed via backwards induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Let δd := δ/2d for every d ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' , D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Suppose unbiasedness, finite (2 − δd+1)-th moment, and finite expected computational cost are all sat- isfied for d + 1 where 0 ≤ d ≤ D − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Then Algorithm 1 will call itself independently for 2Nd times, each with input {Depth index: d + 1, Trajectory History: H = y(0:d), Parameters: rd+1, · · · , rD−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' This gives us i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' samples Rd+1(y(0:d))(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=', Rd+1(y(0:d))(2Nd) which are used to compute the following: S2Nd = Rd+1(y(0:d))(1) + Rd+1(y(0:d))(2) + · · · + Rd+1(y(0:d))(2Nd), SO 2Nd−1 = Rd+1(y(0:d))(1) + Rd+1(y(0:d))(3) + · · · + Rd+1(y(0:d))(2Nd − 1), SE 2Nd−1 = Rd+1(y(0:d))(2) + Rd+1(y(0:d))(4) + · · · + Rd+1(y(0:d))(2Nd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Then Algorithm 1 returns as output Rd(y(0:d)) = ∆Nd/prd(Nd), where ∆Nd is defined in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' By the inductive hypothesis on d + 1, we have for almost surely every y(0:d): E[Rd+1(y(0:d)) | y(0:d)] = γd+1(y(0:d)) 44 and Eπ � |Rd+1(y(0:d))|2−δd+1� < � D � i=d ˜Ci � ��gD(y(0:D)) ��2−δd+1 π,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' We will start with showing Rd(y(0:d−1)) has a finite computational cost and a finite (2 − δd)-th moment, and then show the unbiasedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Finite cost: To show the computational cost, recall that implementing Algorithm 1 with input depth d requires 2Nd calls of Algorithm 1 with input depth d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' It suffices to check rd > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='5, which reduces to check the the upper bound for kd (defined in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='4) staisfies �2d+2 − 3δ 2d+3 − 3δ � �2d+1 − δ 2d − δ � > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Let t := 2d and the above product becomes: 4t − 3δ 8t − 3δ 2t − δ t − δ = 8t2 + 3δ2 − 10δ 8t2 + 3δ2 − 11δ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Since Nd ∼ Geo(rd) with rd > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='5, calling Algorithm 1 with input depth d has an expected cost: rd 2rd − 1 × the expected cost of Algorithm 1 with input depth d + 1, where rd 2rd−1 = E[2Nd] < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' By our inductive hypothesis, the second term in the above product is finite, therefore the expected cost of Algorithm 1 with input depth d is also finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Finite (2 − δd)-th moment: Next we show Rd has a finite (2−δd)-th moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' By the uniform Ld-Lipschitz 45 property of gd: |∆n| ≤ 1 2 ����gd � y(0:d), S2n 2n � − gd � y(0:d), SO 2n−1 2n−1 ����� + 1 2 ����gd � y(0:d), S2n 2n � − gd � y(0:d), SE 2n−1 2n−1 ����� ≤ Ld 2 ���� SO 2n−1 2n−1 − SE 2n−1 2n−1 ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Therefore, for any fixed 1 ≤ p < 2, applying triangle inequality under the norm ∥·∥π,p, and applying Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='3 with Z1 = y(0:d), Z2 = Rd+1(y(0:d)), we have: ∥∆n∥π,p ≤ Ld 2 ���� SO 2n−1 2n−1 − SE 2n−1 2n−1 ���� π,p ≤ Ld 2 ���� SO 2n−1 2n−1 − γd+1 ���� π,p + Ld 2 ����γd+1 − SE 2n−1 2n−1 ���� π,p ≤ Ld �BpEπ[|Rd+1(y(0:d))|p] 2(n−1)(p−1) �1/p , exponentiate both sides by p yields, Eπ[|∆n|p] ≤ Lp dBpEπ[|Rd+1(y(0:d))|p] 2(n−1)(p−1) ≤ C(d, p)Eπ[|Rd+1(y(0:d))|p] 2(p−1)n , where C(d, p) = Lp dBp21−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Recall that δd = δ/2d, let us choose qd = 2 − (δd + δd+1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Since � 1, �qd − 1 qd � �2 − δd 1 − δd �� = � 1, �2d+2 − 3δ 2d+3 − 3δ � �2d+1 − δ 2d − δ �� , by definition of kd in the Theorem statement we have kd < �qd − 1 qd � �2 − δd 1 − δd � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 46 Now we estimate the (2 − δd)-th moment of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' An important trick in the calculation below is that we are not going to use the above estimate of Eπ[|∆n|p] directly on p = 2 − δd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Instead, we will first use H¨older’s inequality, and then bound Eπ[|∆n|qd](2−δd)/qd via the above estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' It turns out the first way gives us an order of 2−n(1−δd), while the latter is of order 2−n(qd−1)(2−δd)/qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Since the function (x − 1)(2 − δd)/x is increasing with x when x > 1, and equals 1−δd when x = 2−δd, we gain an extra factor 2−Ω(1)n by choosing qd > 2 − δd and use H¨older’s inequality, which is important for establishing our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' The detailed calculation is below: Eπ[|Rd(y(0:d−1))|2−δd] ≤ ∞ � n=0 Eπ[|∆n|2−δd] prd(n)1−δd ≤ ∞ � n=0 � Eπ � |∆n| 2−δd· qd 2−δd �� 2−δd qd prd(n)1−δd H¨older’s inequality ≤ 1 r1−δd d ∞ � n=0 �C(d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' qd)Eπ[|Rd+1(y(0:d))|qd] 2(qd−1)n � 2−δd qd 1 (1 − rd)(1−δd)n estimate of Eπ[|∆n|p] = C′(d) ��Rd+1(y(0:d)) ��2−δd π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='qd ∞ � n=0 � 1 2 (qd−1) qd (2−δd)−kd(1−δd) �n here C′(d) = C(d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' qd)(2−δd)/qd r1−δd d ≤ C′(d) ��Rd+1(y(0:d)) ��2−δd π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2−δd+1 ∞ � n=0 � 1 2 (qd−1) qd (2−δd)−kd(1−δd) �n since qd < 2 − δd+1 ≤ C′(d) � D � i=d+1 ˜Ci � ��gD(y(0:D)) ��2−δd π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2 � � 2 (qd−1) qd (2−δd)−kd(1−δd) 2 (qd−1) qd (2−δd)−kd(1−δd) − 1 � � inductive hypothesis = � D � i=d ˜Ci � ��gD(y(0:D)) ��2−δd π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content='2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 47 and note the RHS is still finite given the assumption of our theorem on gD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Again, as we can see in the proof, the choice of kd and qd is crucial for our calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' It ensures (qd−1) qd (2 − δd) − kd(1 − δd) > 0, and in turn ensures the above summation of the geometric series converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Unbiasedness: The proof of unbiasedness of our estimator in this case is identical to the LBS case, however we still require a justification of the existence of a finite conditional expectation of Rd(y(0:d−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' By what we have just proven, Eπ[|Rd(y(0:d−1))|] ≤ � Eπ � |Rd(y(0:d−1))|2− δ 2d ��1/(2− δ 2d ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' Given Eπ[|Rd(y(0:d−1))|] < ∞, we immediately have Eπd:D[Rd(y(0:d−1))] exists for almost surely every y(0:d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} +page_content=' 48' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfvwi4/content/2301.04095v1.pdf'} diff --git a/b9FJT4oBgHgl3EQfQywo/content/tmp_files/2301.11492v1.pdf.txt b/b9FJT4oBgHgl3EQfQywo/content/tmp_files/2301.11492v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6b72490fb304bf7827e22b3547be877fa84bbad2 --- /dev/null +++ b/b9FJT4oBgHgl3EQfQywo/content/tmp_files/2301.11492v1.pdf.txt @@ -0,0 +1,1357 @@ +arXiv:2301.11492v1 [econ.TH] 27 Jan 2023 +RECOVERING UTILITY +CHRISTOPHER P. CHAMBERS, FEDERICO ECHENIQUE, AND NICOLAS S. LAMBERT +Abstract. We provide sufficient conditions under which a utility function may be +recovered from a finite choice experiment. Identification, as is commonly understood +in decision theory, is not enough. We provide a general recoverability result that +is widely applicable to modern theories of choice under uncertainty. +Key is to +allow for a monetary environment, in which an objective notion of monotonicity is +meaningful. In such environments, we show that subjective expected utility, as well +as variational preferences, and other parametrizations of utilities over uncertain acts +are recoverable. We also consider utility recovery in a statistical model with noise +and random deviations from utility maximization. +(Chambers) Department of Economics, Georgetown University +(Echenique) Department of Economics, UC Berkeley +(Lambert) Department of Economics, University of Southern California +Echenique thanks the National Science Foundation for its support through grant SES 1558757. +Lambert gratefully acknowledges the financial support and hospitality of Microsoft Research New +York and the Yale University Cowles Foundation. +1 + +2 +1. Introduction +Economists are often interested in recovering preferences and utility functions from +data on agents’ choices. If we are able to recover a utility function, then a preference +relation is obviously implied, but the inverse procedure is more delicate. In this paper, +we presume access to data on an agent’s choices, and that these describe the agent’s +preferences (or that preferences have been obtained as the outcome of a statistical +estimation procedure). Our results describe sufficient conditions under which one can +recover, or learn, a utility function from the agents’ choices. +At a high level, the problem is that preferences essentially are choices, because they +encode the choice that would be made from each binary choice problem. When we +write x ≻ y we really mean that x would be chosen from the set {x, y}. Utility func- +tions are much richer objects, and a given choice behavior may be described by many +different utilities. For example, one utility can be used to discuss an agent’s risk pref- +erences: they could have a “constant relative risk aversion” utility, for which a single +parameter describes attitudes towards risk. But the same preferences can be repre- +sented by a utility that does not have such a convenient parametrization. So recover- +ing, or learning, utilities present important challenges that go beyond the problem of +recovering a preference. In the paper, we describe some simple examples that illus- +trate the challenges. Our main results describe when one may (non-parametrically) +recover a utility representation from choice data. +We first consider choice under uncertainty. We adopt the standard (Anscombe- +Aumann) setting of choice under uncertainty, and focus attention on a class of +utility representations that has been extensively studied in the literature. +Spe- +cial cases include subjected expected utility, the max-min expected utility model +of Gilboa and Schmeidler (1989), Choquet expected utility (Schmeidler, 1989), the +variational preferences of Maccheroni, Marinacci, and Rustichini (2006), and many +other popular models. Decision theorists usually place significance on the uniqueness +of their utility representations, arguing that uniqueness provides an identification ar- +gument that allows for utility to be recovered from choice data. We argue, in contrast, +that uniqueness of a utility representation is not enough to recover a utility from finite +choice data. +Counterexamples are not hard to find. Indeed, even when a utility representation +is unique, one may find a convergent sequence of utilities that is consistent with +larger and larger finite datasets, but that does not converge to the utility function +that generated the choices in the data, or to any utility to which it is equivalent. So + +RECOVERING UTILITY +3 +uniqueness is necessary but not sufficient for a utility representation to be empirically +tractable, in the sense of ensuring that a utility is recovered from large, but finite, +choice experiments. +Our main results are positive, and exhibit sufficient conditions for utility recovery. +Key to our results is the availability of an objective direction of improvements in +utility: we focus our attention on models of monotone preferences. Our paper con- +siders choices among monetary acts, meaning state-contingent monetary payoffs. For +such acts, there is a natural notion of monotonicity. Between two acts, if one pays +more in every state of the world, the agent agent should prefer it. As a discipline +on the recovery exercise, this essential notion of monotonicity suffices to ensure that +a sequence of utilities that explains the choices in the data converges to the utility +function that generated the choices. +We proceed by first discussing the continuity of a utility function in its dependence +on the underlying preference relation. If U(⪰, x) is a function of a preference ⪰ and of +choice objects x, then we say that it is a utility function if x �→ U(⪰, x) represents ⪰. +We draw on the existing literature (Theorem 1) to argue that such continuous utilities +exist in very general circumstances. Continuity of this mapping in the preference +ensures that if the choice data allow for preference recovery, they also allow a utility +to be recovered. The drawback, however, of such general utility representation results +is that they do not cover the special theories of utility in which economists generally +take interest. There is no reason to expect that the utility U(⪰, x) coincides with the +standard parametrizations of, for example, subjective expected utility or variational +preferences. +We then go on to our main exercise, which constrains the environment to the +Anscombe-Aumann setting, and considers utility representations that have received +special attention in the theory of choice under uncertainty. +We consider a setup +that is flexible enough to accommodate most theories of choice under uncertainty +that have been studied in the literature. Our main result (Theorem 2) says that, +whenever a choice experiment succeeds in recovering agents’ underlying preferences, +it also serves to recover a utility in the class of utilities of interest. For example, if +an agent has subjective expected utility preferences, and these can be recovered from +a choice experiment, then so can the parameters of the subjective expected utility +representation: the agents’ beliefs and Bernoulli utility index. Or, if the agent has +variational preferences that can be inferred from choice data, then so can the different +components of the variational utility representation. + +4 +CHAMBERS, ECHENIQUE, AND LAMBERT +Actual data on choices may be subject to sampling noise, and agents who randomly +deviate from their preferences. The results we have just mentioned are useful in such +settings, once the randomness in preference estimates is taken into account. As a +complement to our main findings, we proceed with a model that explicitly takes noisy +choice, and randomness, into account. Specifically, we consider choice problems that +are sampled at random, and an agent who may deviate from their preferences. They +make mistakes. In such a setting, we present sufficient conditions for the consistency +of utility function estimates (Theorem 3). +In the last part of the paper we take a step back and revisit the problem of pref- +erence recovery, with the goal of showing how data from a finite choice experiment +can approximate a preference relation, and, in consequence, a utility function. Our +model considers a large, but finite, number of binary choices. We show that when +preferences are monotone, then preference recovery is possible (Theorem 5). In such +environments, utility recovery follows for the models of choice under uncertainty that +we have been interested in (Corollary 1). +Related literature. The literature on revealed preference theory in economics is pri- +marily devoted to tests for consistency with rational choice. The main result in the lit- +erature, Afriat’s theorem (Afriat, 1967a; Diewert, 1973; Varian, 1982), is in the context +of standard demand theory (assuming linear budgets and a finite dataset). Versions of +Afriat’s result have been obtained in a model with infinite data (Reny, 2015), nonlin- +ear budget sets (e.g., Matzkin, 1991; Forges and Minelli, 2009), general choice prob- +lems (e.g., Chavas and Cox, 1993; Nishimura, Ok, and Quah, 2017), and multiperson +equilibrium models (e.g., Brown and Matzkin, 1996; Carvajal, Deb, Fenske, and Quah, +2013). Algorithmic questions related to revealed preference are discussed by Echenique, Golovin, and Wierman +(2011) and Camara (2022). +The monograph by Chambers and Echenique (2016) +presents an overview of results. +The revealed preference literature is primarily concerned with describing the datasets +that are consistent with the theory, not with recovering or learning a preference, or a +utility. In the context of demand theory and choice from linear budgets, Mas-Colell +(1978) introduces sufficient conditions under which a preference relation is recovered, +in the limit, from a sequence of ever richer demand data observations. More recently, +Forges and Minelli (2009) derive the analog of Mas-Colell’s results for nonlinear bud- +get sets. An important strand of literature focuses on non-parametric econometric es- +timation methods applied to demand theory data: Blundell, Browning, and Crawford + +RECOVERING UTILITY +5 +(2003, 2008) propose statistical tests for revealed preference data, and consider coun- +terfactual bounds on demand changes. +The problem of preference and utility recovery has been studied from the perspec- +tive of statistical learning theory. Beigman and Vohra (2006) considers the problem +of learning a demand function within the PAC paradigm, which is closely related to +the exercise we perform in Section 4. A key difference is that we work with data +on pairwise choices, which are common in experimental settings (including in many +recent large-scale online experiments). Zadimoghaddam and Roth (2012) look at the +utility recovery problem, as in Beigman and Vohra (2006), but instead of learning a +demand function they want to understand when a utility can be learned efficiently. +Balcan, Daniely, Mehta, Urner, and Vazirani (2014) follow up on this important work +by providing sample complexity guarantees, while Ugarte (2022) considers the prob- +lem of recovery of preferences under noisy choice data, as in our paper, but within the +demand theory framework. Similarly, the early work of Balcan, Constantin, Iwata, and Wang +(2012) considers a PAC learning question, focusing on important sub-classes of val- +uations in economics. Bei, Chen, Garg, Hoefer, and Sun (2016) pursues the problem +assuming that a seller proposes budgets with the objective of learning an agent’s +utility (they focus on quasilinear utility, and a seller that obtains aggregate demand +data). Zhang and Conitzer (2020) considers this problem under an active-learning +paradigm, and contrasts with the PAC sample complexity. +In all, these works are important precedents for our paper, but they are all within +the demand theory setting. The results do not port to other environments, such as, +for example, binary choice under risk or uncertainty. The closest paper to ours is +Chambers, Echenique, and Lambert (2021), which looks at a host of related ques- +tions to our paper but focusing on preference, not utility, recovery. The work by +Chambers, Echenique, and Lambert considers choices from binary choice problem, +but does not address the question of recovering, or learning, a utility function. As +we explain below in the paper, the problem for utilities is more delicate than the +problem for preferences. In this line of work, Chase and Prasad (2019) obtains im- +portant results on learning a utility but restricted to settings of intertemporal choice. +The work by Basu and Echenique (2020) looks at learnability of utility functions +(within the PAC learning paradigm), but focusing on particular models of choice un- +der uncertainty. Some of our results rely on measures of the richness of a theory, +or of a family of preferences, which is discussed by Basu and Echenique (2020) and +Fudenberg, Gao, and Liang (2021): the former by estimating the VC dimension of + +6 +CHAMBERS, ECHENIQUE, AND LAMBERT +theories of choice under uncertainty, and the latter by proposing and analyzing new +measures of richness that are well-suited for economics, as well as implementing them +one economic datasets. +Finally, it is worth mentioning that preference and utilty recovery is potentially sub- +ject to to strategic manipulations, as emphasized by Dong, Roth, Schutzman, Waggoner, and Wu +(2018) and Echenique and Prasad (2020). This possibility is ignored in our work. +2. The Question +We want to understand when utilities can be recovered from data on an agent’s +choices. Consider an agent with a utility function u. We want know when, given +enough data on the agent’s choices, we can “estimate” or “recover” a utility function +that is guaranteed to be close to u. +In statistical terminology, recovery is analogous to the consistency of an estimator, +and approximation guarantees are analogous to learnability. Imagine a dataset of size +k, obtained from an incentivized experiment with k different choice problems.1 The +observed choice behavior in the data may be described by a preference ⪰k, which +is associated with a utility function uk. The preference ⪰k could be a rationalizing +preference, or a preference estimate. So we choose a utility representation for uk. The +recovery, or consistency, property is that uk → u as k → ∞. +Suppose that the utility u represents preferences ⪰, which summarize the agent’s +full choice behavior. Clearly, unless ⪰k→⪰, the exercise is hopeless. So our first +order of business is to understand when ⪰k→⪰ is enough to ensure that uk → u. +In other words, we want to understand when recovering preferences is sufficient for +recovering utilities. To this end, our main results are in Section 3.4. In recovering +a utility, we are interested in particular parametric representations. In choice over +uncertainty, for example, one may be interested in measures of risk-attitudes, or +uncertainty aversion. It is key then that the utility recovery exercises preserves the +aspects of utility that allow such measures to be have meaning. If, say, preferences +have the “constant relative risk aversion” (CRRA) form, then we want to recover the +Arrow-Pratt measure of risk aversion. +1Such +datasets +are +common +in +experimental +economics, +including +cases +with +very +large +k. +See, +for +example, +von Gaudecker, van Soest, and Wengstrom +(2011), +Chapman, Dean, Ortoleva, Snowberg, and Camerer +(2017), +Chapman, Dean, Ortoleva, Snowberg, and Camerer (2022) and Falk, Becker, Dohmen, Enke, Huffman, and Sunde +(2018). One can also apply our results to roll call data from congress, as in Poole and Rosenthal +(1985) or Clinton, Jackman, and Rivers (2004). Large-scale A/B testing by tech firms may provide +further examples (albeit involving proprietary datasets). + +RECOVERING UTILITY +7 +Our data is presumably obtained in an experimental setting, where an agent’s +behavior may be recorded with errors; o in which the agent may randomly deviate +from their underlying preference ⪰. Despite such errors, with high probability, “on +the sample path,” we should obtain that ⪰k→⪰. In our paper we uncover situations +where this convergence leads to utility recovery. Indeed, the results in Section 3.4 +and 3.5 may be applied to say that, in many popular models in decision theory, when +⪰k→⪰ (with high probability), then the resulting utility representations enable utility +recovery (with high probability). +The next step is to discuss learning and sample complexity. +Here we need to +explicitly account for randomness and errors. We lay out a model of random choice, +with random sampling of choice problems and errors in agents’ choices. The errors +may take a very general form, as long as random choices are more likely to go in the +direction of preferences than against it (if x ≻ y then x is the more likely choice from +the choice problem {x, y}), and that this likelihood ratio remains bounded away from +one. Contrast with the standard theory of discrete choice, where the randomness +usually is taken to be additive, and independent of the particular pair of alternatives +that are being compared. +Here we consider a formal statistical consistency problem, and exhibit situations +where utility recovery is feasible. We use ideas from the literature on PAC learning to +provide formal finite sample-size bounds for each desired approximation guarantee. +See Section 4. +3. The Model +3.1. Basic definitions and notational conventions. Let X be a set. Given a +binary relation R ⊆ X × X, we write x R y when (x, y) ∈ R. A binary relation that +is complete and transitive is called a weak order. If X is a topological space, then +we say that R is continuous if R is closed as a subset of X × X (see, for example, +Bergstrom, Parks, and Rader, 1976). A preference relation is a weak order that is +also continuous. +A preference relation ⪰ is locally strict if, for all x, y ∈ X, x ⪰ y implies that for +each neighborhood U of (x, y), there is (x′, y′) ∈ U with x ≻ y. The notion of local +strictness was first introduced by Border and Segal (1994) as a generalization of the +property of being locally non-satiated from consumer theory. + +8 +CHAMBERS, ECHENIQUE, AND LAMBERT +If ⪰ is a preference on X and u : X → R is a function for which x ⪰ y if and +only if u(x) ≥ u(y) then we say that u is a representation of ⪰, or that u is a utility +function for ⪰. +If A ⊆ Rd is a Borel set, we write ∆(A) for the set of all Borel probability measures +on A. We endow ∆(A) with the weak* topology. If S is a finite set, then we topologize +∆(A)S with the product topology. +For p, q ∈ ∆(A), we say that p is larger than q in the sense of first-order stochastic +dominance if +� +A fdx ≥ +� +A fdy for all monotone increasing, continuous and bounded +functions f on A. +3.2. Topologies on preferences and utilities. The set of preferences over X, +when X is a topological space, is endowed with the topology of closed convergence. +The space of corresponding utility representations is endowed with the compact-open +topology. +These are the standard topologies for preferences and utilities, used in +prior work in mathematical economics. See, for example, Hildenbrand (1970), Kannai +(1970), and Mas-Colell (1974). Here we offer definitions and a brief discussion of our +choice of topology. +Let X be a topological space, and F = {F n}n be a sequence of closed sets in +X ×X (with the product topology). We define Li(F) and Ls(F) to be closed subsets +of X × X as follows: +• (x, y) ∈ Li(F) if and only if, for all neighborhoods V of (x, y), there exists +N ∈ N such that F n ∩ V ̸= ∅ for all n ≥ N. +• (x, y) ∈ Ls(F) if and only if, for all neighborhoods V of (x, y), and all N ∈ N, +there is n ≥ N such that F n ∩ V ̸= ∅. +Observe that Li(F) ⊆ Ls(F). The definition of closed convergence is as follows. +Definition 1. F n converges to F in the topology of closed convergence if Li(F) = +F = Ls(F). +Closed convergence captures the property that agents with similar preferences +should have similar choice behavior—a property that is necessary to be able to learn +the preference from finite data. Specifically, if X ⊆ Rn, and P is the set of all locally +strict and continuous preferences on X, then the topology of closed convergence is +the smallest topology on P for which the sets +{(x, y, ⪰) : x ≻ y} ⊆ X × X × P + +RECOVERING UTILITY +9 +are open.2 In words: suppose that x ≻ y, then for x′ close to x, y′ close to y, and ⪰′ +close to ⪰, we obtain that x′ ≻′ y′. +For utility functions, we adopt the compact-open topology, which we also claim +is a natural choice of topology. The compact-open topology is characterized by the +convergence criterion of uniform convergence on compact sets. The reason it is natural +for utility functions is that a utility usually has two arguments: one is the object +being “consumed” (a lottery, for example) and the other is the ordinal preference that +utility is meant to represent. (The preference argument is usually implicit, but of +course it remains a key aspect of the exercise.) Now an analyst wants the utility to +be “jointly continuous,” or continuous in both of its arguments. For such a purpose, +the natural topology on the set of utilities, when they are viewed solely as functions +of consumption, is indeed the compact-open topology. More formally, consider the +following result, originally due to Mas-Colell (1977).3 +Theorem 1. Let X be a locally compact Polish space, and P the space of all contin- +uous preferences on X endowed with the topology of closed convergence. Then there +exists a continuous function U : P × X → [0, 1] so that x �→ U(⪰, x) represents ⪰. +We may view the map U as a mapping from ⪰ to the space of utility functions. +Then continuity of this induced mapping is equivalent to the joint continuity result +discussed in Theorem 1, as long as we impose the compact-open topology on the space +of utility functions (see Fox (1945)). +3.3. The model. As laid our in Section 2, we want to understand when we may +conclude that uk → u from knowing that ⪰k→⪰. Mas-Colell’s theorem (Theorem 1) +provides general conditions under which there exists one utility representation that +has the requisite convergence property, but he is clear about the practical limita- +tions of his result: “There is probably not a simple constructive (“canonical”) method +to find a U function.” +In contrast, economists are generally interested in specific +parameterizations of utility. +For example, if an agent has subjective expected-utility preferences, economists +want to estimate beliefs and a von-Neumann-Morgenstern index; not some arbitrary +representation of the agent’s preferences. Or, if the data involve intertemporal choices, +and the agent discounts utility exponentially, then an economist will want to estimate +2See Kannai (1970) and Hildenbrand (1970) for a discussion; a proof of this claim is available from +the authors upon request. +3Levin (1983) provides a generalization to incomplete preferences. + +10 +CHAMBERS, ECHENIQUE, AND LAMBERT +their discount factor. Such specific parameterizations of utility are not meaningful in +the context of Theorem 1. +The following (trivial) example shows that there is indeed a problem to be studied. +Convergence of arbitrary utility representations to the correct limit is not guaranteed, +even when recovered utilities form a convergent sequence, and recovered preferences +converge to the correct limit. +Example 1. Consider expected-utility preferences on ∆(K)S, where K is a compact +space, S a finite set of states, and ∆S(K) is the set of Anscombe-Aumann acts. Fix +an affine function v : ∆(K) → R, a prior µ ∈ ∆(S), and consider the preference ⪰ +with representation +� +S v(f(s)) dµ(s). +Now if we set ⪰k=⪰ then ⪰k→⪰ holds trivially. However, it is possible to choose an +expected utility representation +� +S vk(f(s)) dµk(s) that does not converge to a utility +representation (of any kind) for ⪰. In fact one could choose a µk and a “normalization” +for vk, for example ∥vk∥ = 1 (imagine for concreteness that K is finite, and use the +Euclidean norm for vk). Specifically, choose scalars βk with ∥βk + 1 +kv∥ = 1. Then the +utility f �→ +� +S vk(f(s)) dµ(s) represents ⪰k and converges to a constant function. +The punchline is that the limiting utility represents the preference that exhibits +complete indifference among all acts. This is true, no matter what the original pref- +erence ⪰ was. +In the example, we have imposed some discipline on the representation. Given that +the utility converges to a constant, the discipline we have chosen is a particular nor- +malization of the utility representations (their norm is constant). The normalization +just makes the construction of the example slightly more challenging, and reflects +perhaps the most basic care that an analyst could impose on the recovery exercise. +3.4. Anscombe-Aumann acts. We present our first main result in the context +of Anscombe-Aumann acts, the workhorse model of the modern theory of decisions +under uncertainty. Let S be a finite set of states of the world, and fix a closed interval +of the real line [a, b] ⊆ R. An act is a function f : S → ∆([a, b]). We interpret the +elements of ∆([a, b]) as monetary lotteries, so that acts are state-contingent monetary +lotteries. The set of all acts is ∆([a, b])S. When p ∈ ∆([a, b]), we denote the constant +act that is identically equal to p by (p, . . . , p); or sometimes by p for short. +Note that we do not work with abstract, general, Anscombe-Aumann acts, but in +assuming monetary lotteries we impose a particular structure on the objective lotter- +ies in our Anscombe-Aumann framework. The reason is that our theory necessitates + +RECOVERING UTILITY +11 +a certain known and objective direction of preference. Certain preference compar- +isons must be known a priori: monotonicity of preference will do the job, but for +monotonicity to be objective we need the structure of monetary lotteries. +An act f dominates an act g if, for all s ∈ S, f(s) first-order stochastic dominates +g(s). And f strictly dominates g if, for all s ∈ S, f(s) strictly first-order stochastic +dominates g(s). A preference ⪰ over acts is weakly monotone if f ⪰ g whenever f +first-order stochastic dominates g. +Let U be the set of all continuous and monotone weakly increasing functions u : +[a, b] → R with u(a) = 0 and u(b) = 1. A pair (V, u) is a standard representation if V : +∆([a, b])S → R and u ∈ U are continuous functions such that v(p, . . . , p) = +� +[a,b] u dp, +for all constant acts (p, . . . , p). Moreover, we say that a standard representation (V, u) +is aggregative if there is an aggregator H : [0, 1]S → R with V (f) = H(( +� +u df(s))s∈S) +for f ∈ ∆([a, b])S. An aggregative representation with aggregator H is denoted by +(V, u, H). Observe that a standard representation rules out total indifference. +A preference ⪰ on ∆([a, b])S is standard if it is weakly monotone, and there is a +standard representation (V, u) in which V represents ⪰. Roughly, standard prefer- +ences will be those that satisfy the expected utility axioms across constant acts, and +are monotone with respect to the (statewise) first order stochastic dominance rela- +tion. Aggregative preferences will additionally satisfy an analogue of Savage’s P3 or +the Anscombe-Aumann notion of monotonicity. +Example 2. Variational preferences (Maccheroni, Marinacci, and Rustichini, 2006) +are standard and aggregative.4 Let +V (f) = inf{ +� +v(f(s))dπ(s) + c(π) : π ∈ ∆(S)} +where +(1) v : ∆([a, b]) → R is continuous and affine. +(2) c : ∆(S) → [0, ∞] is lower semicontinuous, convex and grounded (meaning +that inf{c(π) : π ∈ ∆(S)} = 0). +Note that V (p, . . . , p) = v(p) + inf{c(π) : π ∈ ∆(S)} = +� +u dp, by the assumption +that c is grounded, and where the existence of u : [a, b] → R so that v(p) = +� +u dp +4Variational preferences are widely used in macroeconomics and finance to capture decision mak- +ers’ concerns for using a misspecified model. +Here it is important to recover the different com- +ponents of a representation, v and c, because they quantify key features of the environment. See +for example Hansen and Sargent (2001); Hansen, Sargent, Turmuhambetova, and Williams (2006); +Hansen and Sargent (2022). + +12 +CHAMBERS, ECHENIQUE, AND LAMBERT +is an instance of the Riesz representation theorem. It is clear that we may choose +u ∈ U. So (V, u) is a standard representation. +Letting H : [0, 1]S → R be defined by H(x) = inf{� +s∈S x(s)π(s) + c(π) : π ∈ +∆(S)}, we see that indeed (V, u, H) is also an aggregative representation of these +preferences. +Some other examples of aggregative preferences include special cases of the varia- +tional model Gilboa and Schmeidler (1989), as well as generalizations of it, Cerreia-Vioglio, Maccheroni, Marinacci, and Montrucchio +(2011); Chandrasekher, Frick, Iijima, and Le Yaouanq (2021), and others which are +not comparable Schmeidler (1989); Chateauneuf, Grabisch, and Rico (2008); Chateauneuf and Faro +(2009).5 +Theorem 2. Let ⪰ be a standard preference with standard representation (V, u), and +{⪰k} a sequence of standard preferences, each with a standard representation (V k, uk). +(1) If ⪰k→⪰, then (V k, uk) → (V, u). +(2) If, in addition, these preferences are aggregative with representations (V k, uk, Hk) +and (V, u, H), then Hk → H. +In terms of interpretation, Theorem 2 suggests that, as preferences converge, risk- +attitudes, or von Neumann morgenstern utility indices also converge in a pointwise +sense. The aggregative part claims that we can study the convergence of risk attitudes +and the convergence of the aggregator controlling for risk separately. So, for example, +in the multiple priors case, two decision makers whose preferences are close will have +similar sets of priors. +3.5. Preferences over lotteries and certainty equivalents. In this section, we +focus on a canonical representation for preferences over lotteries: the certainty equiv- +alent. There are many models of preferences over lotteries, but we have in mind in +particular Cerreia-Vioglio, Dillenberger, and Ortoleva (2015), whereby a preference +representation over lotteries is given by U(p) = infu∈U u−1( +� +udp); a minimum over +a set of certainty equivalents for expected utility maximizers. Key is that for this +representation, and any degenerate lottery δx, U(δx) = x. +5A class of variational preferences that are of particular interest to computer scientists are preferences +with a max-min representation (Gilboa and Schmeidler, 1989). These evaluate acts by +V (f) = inf{ +� +v(f(s))dπ(s) : π ∈ Π}, +with Π ⊆ ∆(S) a closed and convex set. Here c is the indicator function of Π (as defined in convex +analysis). + +RECOVERING UTILITY +13 +Let [a, b] ⊂ R, where a < b, be an interval in the real line and consider ∆([a, b]). +Say that ⪰ on ∆([a, b]) is certainty monotone if when ever p first order stochastically +dominates q, then p ⪰ q, and for all x, y ∈ [a, b] for which x > y, δx ≻ δy. Any cer- +tainty monotone continuous preference ⪰ and any lottery p ∈ ∆([a, b]) then possesses +a unique certainty equivalent x ∈ [0, 1], satisfying δx ∼ p. To this end, we define +ce(⪰, p) to be the certainty equivalent of p for ⪰. It is clear that, fixing ⪰, ce(·, ⪰) is +a continuous utility representation of ⪰. +Proposition 1. Let ⪰ be a certainty monotone preference and let p ∈ ∆([a, b]). Let +{⪰k} be a sequence of certainty monotone preferences and let pk be a sequence in +∆([a, b]). If (⪰k, pk) → (⪰, p), then ce(⪰k, pk) → ce(⪰, p). +To this end, the map carrying each preference to its certainty equivalent represen- +tation is a continuous map in the topology of closed convergence. +4. Utility recovery with noisy choice data +We develop a model of noisy choice data, and consider when utility may be re- +covered from a traditional estimation procedure. Recovery here takes the form of an +explicit consistency result, together with sample complexity bounds in a PAC learning +framework. +The focus is on the Wald representation, analogous to the certainty equivalent +we considered in Section 3.5. When choosing among vectors in x ∈ Rd, the Wald +representation is u(x) ∈ R so that +x ∼ (u(x), . . . , u(x)). +If the choice space is well behaved, a Wald representation exists for any monotone +and continuous preference relation. To this end, we move beyond the Anscombe- +Aumann setting that we considered above, but it should be clear that some versions +of Anscombe-Aumann can be accommodated within the assumptions of this section. +Our main results for the model that explicitly accounts for noisy choice data as- +sumes Wald representations that are either Lipschitz or homogeneous (meaning that +preferences are homothetic). +4.1. Noisy choice data. The primitives of our noisy choice model are collected in +the tuple (X, P, λ, q), where: +• X ⊆ Rd is the ambient choice, or consumption, space. The set X is endowed +with the (relative) topology inherited from Rd. + +14 +CHAMBERS, ECHENIQUE, AND LAMBERT +• P is a class of continuous and locally strict preferences on X. +The class +comes with a set of utility functions U, so that each element of P has a utility +representation in the set U. +• λ is a probability measure on X, assumed to be absolutely continuous with +respect to Lebesgue measure. We also assume that λ ≥ c Leb, where c > 0 is +a constant and Leb denotes Lebesgue measure. +• q : X × X × P → [0, 1] is a random choice function, so q(x, y; ⪰) is the +probability that an agent with preferences ⪰ chooses x over y. Assume that +if x ≻ y, then x is chosen with probability q(x, y; ⪰) > 1/2 and y with +probability q(y, x; ⪰∗) = 1 − q(x, y; ⪰). If x ∼ y then x and y are chosen with +equal probability. +• We shall assume that the error probability q satisfies that +Θ ≡ inf{q(⪰, (x, y)) : x ≻ y and ⪰∈ P} > 1 +2. +The tuple (X, P, λ, q) describes a data-generating process for noisy choice data. Fix +a sample size n and consider an agent with preference ⪰∗∈ P. A sequence of choice +problems {xi, yi}, 1 ≤ i ≤ n are obtained by drawing xi and yi from X, independently, +according to the law λ. Then a choice is made from each problem {xi, yi} according +to q(·, ·; ⪰∗). +Observe that our assumptions on q are mild. We allow errors to depend on the +pair {x, y} under consideration, almost arbitrarily. The only requirement is that one +is more likely to choose according to one’s preference than to go against them, as well +as the more technical assumptions of measurability and a control on how large the +deviation from 1/2-1/2 choice may get. +To keep track of the chosen alternative, we order the elements of each problem so +that (xi, yi) means that xi was chosen from the choice problem {xi, yi}. So a sample +of size n is {(x1, y1), . . . , (xn, yn)}, consisting of 2n iid draws from X × X according +to our stochastic choice model: in the ith draw, the choice problem was {xi, yi} and +xi was chosen. +A utility function un ∈ U is chosen to maximize the number of rationalized choices +in the data. +So un maximizes �n +i=1 1u(xi)≥u(yi). +The space of utility functions is +endowed with a metric, ρ. In this section, all we ask of ρ is that, for any u, u′ ∈ U, +there is x ∈ X with |u(x) − u′(x)| ≥ ρ(u, u′). For example, we could use the sup +norm for the purposes of any of the results in this section. + +RECOVERING UTILITY +15 +4.1.1. Lipschitz utilities. One set of sufficient conditions will need the family of rele- +vant utility representations to satisfy a Lipschitz property with a common Lipschitz +bound. The representations are of the Wald kind, as in Section 3.5. We now add the +requirement of having the Lipschitz property, which allows us to connect differences +in utility functions to quantifiable observable (but noisy) choice behavior. The main +idea is expressed in Lemma 4 of Section 6. +We say that (X, P, λ, q) is a Lipschitz environment if: +(1) X ⊆ Rd is convex, compact, and has nonempty interior. +(2) Each preference ⪰∈ P has a Wald utility representation u⪰ : X → R so that +x ∼ u⪰(x)1. +(3) All utilities in U are Lipschitz, and admit a common Lipschitz constant κ. So, +for any x, x′ ∈ X and u ∈ U, |u(x) − u(x′)| ≤ κ∥x − x′∥. +4.1.2. Homothetic preferences. The second set of sufficient conditions involve homo- +thetic preferences. It turns out, in this case, that the Wald representations have a +homogeneity property, and this allows us to connect differences in utilities to a prob- +ability of detecting such differences. The key insights is contained in Lemma 5 of +Section 6. +We employ the following auxiliary notation. SM +α = {x ∈ Rd : ∥x∥ = M and x ≥ +α1} and DM +α = {θx : x ∈ SM +α and θ ∈ [0, 1]}. +We say that (X, P, λ, q) is a homothetic environment if: +(1) X = DM +α for some (small) α > 0 and (large) M > 0. +(2) P is a class of continuous, monotone, homothetic, and complete preferences +on X ⊆ Rd. +(3) U is a class of Wald representations, so that for each ⪰∈ P there is a utility +function u ∈ U with x ∼ u(x)1. +Remark: if u ∈ U is the Wald representation of ⪰, then u is homogeneous of degree +one because x ∼ u(x)1 iff λx ∼ λu(x)1, so u(λx) = λu(x). +4.1.3. VC dimension. The Vapnik-Chervonenkis (VC) dimension of a set P of prefer- +ences is the largest sample size n for which there exists a utility u ∈ U that perfectly +rationalizes all the choices in the data, no matter what those are. That is so that +n = �n +i=1 1u(xi)≥u(yi) for any dataset (xi, yi)n +i=1 of size n. +VC dimension is a basic ingredient in the standard PAC learning paradigm. It is +a measure of the complexity of a theory used in machine learning, and lies behind stan- +dard results on uniform laws of large numbers (see, for example, Boucheron, Bousquet, and Lugosi + +16 +CHAMBERS, ECHENIQUE, AND LAMBERT +(2005)). Applications of VC to decision theory can be found in Basu and Echenique +(2020) and Chambers, Echenique, and Lambert (2021). +It is worth noting that VC dimension is used in classification tasks. It may not +be obvious, but when it comes to preferences, our exercise may be thought of as +classification. For each pair of alternatives x and y, a preference ⪰ “classifies” the +pair as x ⪰ y or y ≻ x. Then we can think of preference recovery as a problem of +learning a classifier within the class P. +4.2. Consistency and sample complexity. +Theorem 3. Consider a noisy choice environment (X, P, λ, q) that is either a homo- +thetic or a Lipschitz environment. Suppose that u∗ ∈ U is the Wald utility represen- +tation of ⪰∗∈ P. +(1) The estimates un converge to u∗ in probability. +(2) There are constants K and ¯C so that, for any δ ∈ (0, 1) and n, with probability +at least 1 − δ, +ρ(un, u∗) ≤ ¯C +� +K +� +V/n + +� +2 ln(1/δ)/n +�1/D +, +where V is the VC dimension of P, D = d when the environment is Lipschitz +and D = 2d when it is homothetic. +Of course, the second statement in the theorem is only meaningful when the VC +dimension of P is finite. The constants K and ¯C depend on the primitives in the +environment, but not on preferences, utilities, or sample sizes. +5. Recovering preferences and utilities +The discussion in Section 3.4 focused on utility recovery, taking convergence of +preferences as given. Here we take a step back, provide some conditions for pref- +erence recovery that are particularly relevant for the setting of Section 3.4, and +then connect these back to utility recovery in Corollary 1. +First we describe an +experimental setting in which preferences may be elicited: an agent, or subject, +faces a sequence of (incentivized) choice problems, and the choices made produce +data on his preferences. The specific model and description below is borrowed from +Chambers, Echenique, and Lambert (2021), but the setting is completely standard in +choice theory. +Let X = ∆([a, b])S be the set of acts over monetary lotteries, as discussed in +Section 3.4. A choice function is a pair (Σ, c) with Σ ⊆ 2X \ {∅} a collection of + +RECOVERING UTILITY +17 +nonempty subsets of X, and c : Σ → 2X with ∅ ̸= c(A) ⊆ A for all A ∈ Σ. When Σ, +the domain of c, is implied, we refer to c as a choice function. +A choice function (Σ, c) is generated by a preference relation ⪰ over X if +c(A) = {x ∈ A : x ⪰ y for all y ∈ B}, +for all A ∈ Σ. +The notation (Σ, c⪰) means that the choice function (Σ, c⪰) is generated by the +preference relation ⪰ on X. +Our model features an experimenter (a female) and a subject (a male). The subject +chooses among alternatives in a way described by a preference ⪰∗ over X, which we +refer to as data-generating preference. The experimenter seeks to infer ⪰∗ from the +subject’s choices in a finite experiment. +In a finite experiment, the subject is presented with finitely many unordered pairs +of alternatives Bk = {xk, yk} in X. For every pair Bk, the subject is asked to choose +one of the two alternatives: xk or yk. +A sequence of experiments is a collection Σ∞ = {Bi}i∈N of pairs of possible choices +presented to the subject. Let Σk = {B1, . . . , Bk} collect the first k elements of a +sequence of experiments, and B = ∪∞ +k=1Bk be the set of all alternatives that are used +over all the experiments in a sequence. Here Σk is a finite experiment of size k. +We make two assumptions on Σ∞. The first is that B is dense in X. The second +is that, for any x, y ∈ B there is k for which Bk = {x, y}. The first assumption +is obviously needed to obtain any general preference recovery result. +The second +assumption means that the experimenter is able to elicit the subject’s choices over all +pairs used in her experiment.6 +For each k, the subject’s preference ⪰∗ generates a choice function (Σk, c) by letting, +for each Bi ∈ Σk, c(B) be a maximal element of Bi according to ⪰∗. Thus the choice +behavior observed by the experimenter is always consistent with (Σk, c⪰∗). +We introduce two notions of rationalization: weak and strong. A preference ⪰k +weakly rationalizes (Σk, c) if, for all Bi ∈ Σk, c(Bi) ⊆ c⪰k(Bi). +A preference ⪰k +weakly rationalizes a choice sequence (Σ∞, c) if it rationalizes the choice function of +order k (Σk, c), for all k ≥ 1. +A preference ⪰k strongly rationalizes (Σk, c) if, for all Bi ∈ Σk, c(Bi) = c⪰k(Bi). +A preference ⪰k strongly rationalizes a choice sequence (Σ∞, c) if it rationalizes the +choice function of order k (Σk, c), for all k ≥ 1. +6If there is a countable dense A ⊆ X, then one can always construct such a sequence of experiments +via a standard diagonalization argument. + +18 +CHAMBERS, ECHENIQUE, AND LAMBERT +In the history of revealed preference theory in consumer theory, strong rationaliz- +ability came first. It is essentially the notion in Samuelson (1938) and Richter (1966). +Strong rationalizability is the appropriate notion when it is known that all potentially +chosen alternatives are actually chosen, or when we want to impose, as an added dis- +cipline, that the observed choices are uniquely optimal in each choice problem. This +makes sense when studying demand functions, as Samuelson did. Weak rationaliz- +ability was one of the innovations in Afriat (1967b), who was interested in demand +correspondences.7 +5.1. A general “limiting” result. Our next result serves to contrast what can be +achieved with the “limiting” (countably infinite) data with the limit of preferences +recovered from finite choice experiments. +Theorem 4. Suppose that ⪰ and ⪰∗ are two continuous preference relations (com- +plete and transitive). If ⪰ |B×B =⪰∗ |B×B, then ⪰=⪰∗. +Indeed, as the proof makes clear, Theorem 4 would hold more generally for any X +which is a connected topological space, but it may not hold in absence of connect- +edness. There is a sense in which the limiting case with an infinite amount of data +offers no problems for preference recovery. The structure we impose is needed for the +limit of rationalizations drawn from finite data. +5.2. Recovery from finite data in the AA model. Here we adopt the same +structural assumptions as in Section 3.4, namely that X = ∆([a, b])S, endowed with +the weak topology and the first order stochastic dominance relation. However, the +result easily extends to broader environments, as the proof makes clear. +Theorem 5. There is a sequence of finite experiments Σ∞ so that if the subject’s +preference ⪰∗ is continuous and weakly monotone, and for each k ∈ N, ⪰k is a con- +tinuous and weakly monotone preference that strongly rationalizes a choice function +(Σk, c) generated by ⪰∗; then ⪰k→⪰∗. +Corollary 1. Let ⪰∗ and ⪰k be as in the statement of Theorem 5. +If, in addi- +tion, ⪰∗ and ⪰k have standard representations (V, u) and (V k, uk) then (V, u) = +limk→∞(V k, uk). +7As an illustration of the difference between these two notions of rationalizability, note that, in the +setting of consumer theory, one leads to the Strong Axiom of Revealed Preference while the other +to the Generalized Axiom of Revealed Preference. Of course, Afriat’s approach is also distinct in +assuming a finite dataset. See Chambers and Echenique (2016) for a detailed discussion. + +RECOVERING UTILITY +19 +Note that Theorem 5 requires the existence of the data-generating preference ⪰∗. +A “dual” result to Theorem 5 was established in Chambers, Echenique, and Lambert +(2021). There, the focus was on weak rationalization via ⪰k, which is a weaker notion +than the strong rationalization hypothesized here. To achieve a weak rationalization +result, we assumed instead that preferences were strictly monotone. +6. Proofs +In this section, unless we say otherwise, we denote by X the set of acts ∆([a, b])S, +and the elements of X by x, y, z etc. Note that X is compact Polish when ∆([a, b]) +is endowed with the topology of weak convergence of probability measures. Let P be +the set of all complete and continuous binary relations on X. +6.1. Lemmas. The lemmas stated here will be used in the proofs of our results. +Lemma 1. Let X ⊆ Rn. If {x′ +n} is an increasing sequence in X, and {x′′ +n} is a +decreasing sequence, such that sup{x′ +n : n ≥ 1} = x∗ = inf{x′′ +n : n ≥ 1}. Then +lim +n→∞ x′ +n = x∗ = lim +n→∞ x′′ +n. +Proof. This is obviously true for n = 1. For n > 1, convergence and sups and infs are +obtained component-by-component, so the result follows. +□ +Lemma 2. Let X = ∆([a, b]). Let {xn} be a convergent sequence in X, with xn → x∗. +Then there is an increasing sequence {x′ +n} and an a decreasing sequence {x′′ +n} such +that x′ +n ≤ xn ≤ x′′ +n, and limn→∞ x′ +n = x∗ = limn→∞ x′′ +n. +Proof. The set X ordered by first order stochastic dominance is a complete lattice +(see, for example, Lemma 3.1 in Kertz and Rösler (2000)). Suppose that xn → x∗. +Define x′ +n and x′′ +n by x′ +n = inf{xm : n ≤ m} and x′′ +n = sup{xm : n ≤ m}. Clearly, +{x′ +n} is an increasing sequence, {x′′ +n} is decreasing, and x′ +n ≤ xn ≤ x′′ +n. +Let Fx denote the cdf associated with x. Note that Fx′′n(r) = inf{Fxm(r) : n ≤ m} +while Fx′n(r) is the right-continuous modification of sup{Fxm(r) : n ≤ m}. For any +point of continuity r of F, Fxm(r) → Fx∗(r), so +Fx(r) = sup{inf{Fxm(r) : n ≤ m} : n ≥ 1} +by Lemma 1. +Moreover, Fx∗(r) = inf{sup{Fxm(r) : n ≤ m} : n ≥ 1}. Let ε > 0. Then +Fx∗(r − ε) ← sup{Fxm(r − ε) : n ≤ m} ≤ Fx′n(r) ≤ sup{Fxm(r + ε) : n ≤ m} +→ Fx∗(r + ε) + +20 +CHAMBERS, ECHENIQUE, AND LAMBERT +Then Fx′n(r) → Fx∗(r), as r is a point of continuity of Fx∗. +□ +The results we have obtained motivate two definitions that will prove useful. Say +that the set X, together with the collection of finite experiments Σ∞, has the countable +order property if for each x ∈ X and each neighborhood V of x in X there is x′, x′′ ∈ +(∪iBi) ∩ V with x′ ≤ x ≤ x′′. +We say that X has the squeezing property if for +any convergent sequence {xn}n in X, if xn → x∗ then there is an increasing sequence +{x′ +n}n, and an a decreasing sequence {x′′ +n}n, such that x′ +n ≤ xn ≤ x′′ +n, and limn→∞ x′ +n = +x∗ = limn→∞ x′′ +n. +Lemma 3. If X = ∆([a, b])S, then X has the squeezing property, and there is Σ∞ +such that (X, Σ∞) has the countable order property. +Proof. The squeezing property follows from Lemma 2, and the countable order prop- +erty from Theorem 15.11 of Aliprantis and Border (2006): Indeed, let B be the set of +probability distributions p with finite support on Q∩[a, b], where for all q ∈ Q∩[a, b], +p(q) ∈ Q. Then we may choose a sequence of pairs Bi, and let Σ∞ to be Bi with +B = ∪Bi so that the countable order property is satisfied. +□ +6.2. Proof of Theorem 2. Without loss of generality, we may set [a, b] = [0, 1]. First +we show that uk → u in the compact-open topology. To this end, let xk → x. We want +to show that uk(xk) → u(x). Suppose then that this is not the case, and by selecting +a subsequence that uk(xk) → Y > u(x) (without loss). Note that δxk ∼k pk, where +pk is the lottery that pays 1 with probability uk(xk) ∈ [0, 1], and 0 with probability +1−uk(xk). Let p be the lottery that pays 1 with probability Y , and 0 with probability +1 − Y (given that the range of uk is [0, 1], we must have Y ∈ [0, 1]). Now we have +that (δxk, pk) → (δx, p) and δxk ∼k pk implies δx ∼ p. This is a contradiction because +δx is indifferent in ⪰ to the lottery that pays 1 with probability uk(xk) ∈ [0, 1], and 0 +with probability 1 −uk(xk). The latter is strictly first-order stochastically dominated +by the lottery p. +To finish the proof, we show that V k → V . +This is the same as proving that +V k(f k) → V (f) when f k → f. For each k, continuity and weak monotonicity imply +that there is xk ∈ [0, 1] so that +V k(f k) = V k(δxk, . . . , δxk) = uk(xk). +Similarly, there is x with V (f) = V (δx, . . . , δx) = u(x). +Now we argue that xk → x. +Indeed {xk} is a sequence in [0, 1]. +If there is a +subsequence that converges to, say, x′ > x then we may choose x′′ = +x+x′ +2 +and + +RECOVERING UTILITY +21 +eventually +f k ⪰k (δx′′, . . . , δx′′) ≻ (δx, . . . , δx) ∼ f, +using weak monotonicity. This is impossible because (f k, (δxk, . . . , δxk) → (f, (δx′, . . . , δx′)) +and f k ⪰k ((δxk, . . . , δxk) imply that f ⪰ ((δx′, . . . , δx′) ⪰ (δx′′, . . . , δx′′). +Finally, using what we know about the convergence of uk to u, V k(f k) = uk(xk) → +u(x) = V (f). +We now turn to the second statement in the theorem. Observe that Hk is a con- +tinuous function from [0, 1]S onto [0, 1]. Let zk ∈ [0, 1]S be an arbitrary convergent +sequence, and say that zk → z∗. We claim that Hk(zk) → H(z∗). Without loss we +may assume that Hk(zk) → Y , by taking a subsequence if necessary. For each k and +s, choose yk(s) ∈ [0, 1] for which uk(yk(s)) = zk(s). Again, without loss, we may +assume that yk → y∗ by taking a subsequence if necessary, and using the finiteness +of S. +Observe also that u(y∗(s)) = z∗(s) as we have shown that uk → u in the +compact-open topology. +Now, we may also choose ˆzk ∈ [0, 1] so that +uk(ˆzk) = Hk(zk) = Hk((uk(yk(s)))s∈S), +and further may again without loss (by taking a subsequence) assume that ˆzk con- +verges to ˆz∗. Thus u(ˆz∗) = lim uk(ˆzk) = lim Hk(zk) = Y , again using what we have +shown regarding uk → u. Then (δˆzk, . . . , δˆzk) ∼k (yk(s))s∈S so that, by taking limits, +(δˆz∗, . . . , δˆz∗) ∼∗ (y∗(s))s. This implies that Y = u(ˆz∗) = H(u(y∗(s)) = H(z∗). +6.3. Proof of Proposition 1. Take (⪰k, pk) as in the statement of the Proposition, +and observe that for every p ∈ ∆([a, b]), ce(⪰k, pk) ∈ [a, b]. Suppose by means of +contradiction that ce(⪰k, pk) → ce(⪰, p) is false, then there is some ǫ > 0 and a sub- +sequence for which |ce(⪰km, pkm) − ce(⪰, p)| > ǫ, by taking a further subsequence, we +assume without loss that ce(⪰km, pkm) → α ̸= ce(⪰, p). Now, pkm ∼km δce(⪰km,pkm), +and pkm → p and δce(⪰km,pkm) → δα. So by definition of closed convergence, it follows +that p ∼ δα; but this violates certainty monotonicity as α ̸= ce(⪰, p). +7. Proof of Theorem 3 +First some notation. Let µn(⪰) = +1 +n +�n +i=1 1xi⪰yi, and ⪰n∈ P be represented by +un ∈ U. By definition of un, we have that µn(⪰n) ≥ µn(⪰) for all ⪰∈ P. And we use +Vol(A) to denote the volume of a set A in Rd, when this is well defined (see Schneider +(2014)). + +22 +CHAMBERS, ECHENIQUE, AND LAMBERT +Consider the measure µ on X × X defined as +µ(A, ⪰) = +� +A +q(⪰; x, y) dλ(x, y). +In particular +µ(⪰′, ⪰) = +� +X×X +1⪰′(x, y)q(⪰; x, y) dλ(x, y). +is the probability that a choice with error made at a randomly-drawn choice problem +by an agent with preference ⪰ will coincide with ⪰′. +The key identification result shown in Chambers, Echenique, and Lambert (2021) +is that, if ⪰′̸=⪰, then +µ(⪰′, ⪰) < µ(⪰, ⪰). +Lemma 4. Consider a Lipschitz noise choice environment (X, P, λ, q). There is a +constant C with the following property. If ⪰ and ⪰′ are two preferences in P with +representations u and u′ (respectively) in U. Then +Cρ(u, u′)d ≤ µ(⪰, ⪰) − µ(⪰′, ⪰) +Proof. The ball in Rd with center x and radius ε is denoted by Bε(x). First we show +that the map +ε �→ Vol(Bǫ(x) ∩ X) +Vol(Bǫ(x)) +, +defined for x ∈ X, is nonincreasing as a function of ǫ > 0. +Indeed, let ǫ1 < ǫ2, and let y ∈ Bǫ2(x) ∩ X. Then y ∈ X and ∥y − x∥ ≤ ǫ2. By +convexity of X, y1 ≡ x + ǫ1 +ǫ2(y − x) = (1 − ǫ1 +ǫ2)x + ǫ1 +ǫ2y ∈ X, and y1 ∈ Bǫ1(x). Observe +further by properties of Lebesgue measure in Rd that Vol({x+ ǫ1 +ǫ2(y−x) : y ∈ Bǫ2(x)∩ +X}) = +� +ǫ1 +ǫ2 +�d +Vol(Bǫ2(x) ∩ X). Therefore, Vol(Bǫ1(x) ∩ X) ≥ +� +ǫ1 +ǫ2 +�d +Vol(Bǫ2(x) ∩ X). +Since Vol(Bǫ1(x)) = +� +ǫ1 +ǫ2 +�d +Vol(Bǫ2(x)), it follows that +Vol(Bǫ1(x) ∩ X) +Vol(Bǫ1(x)) +≥ Vol(Bǫ2(x) ∩ X) +Vol(Bǫ2(x)) +, +like we wanted to show. +Now observe that there exists ¯ε > 0 large enough that X ⊆ Bε(x) for all ε ≥ ¯ε and +x ∈ X. Hence, for any x ∈ X and ε ∈ (0, ¯ε] +Vol(Bǫ(x) ∩ X) +Vol(Bǫ(x)) +≥ +Vol(X) +Vol(B¯ǫ(x)) ≡ c′ > 0, + +RECOVERING UTILITY +23 +as X has nonempty interior and the volume of a ball in Rd is independent of its +center. +Now we proceed with the proof of the statement in the lemma. Let ∆ = ρ(u, u′) +and fix x ∈ X with (wlog) u(x) − u′(x) = ∆ > 0. Set +ε = ∆ +4κ. +We may assume that ε ≤ 2¯ε as defined above, as otherwise we can use a larger upper +bound on the Lipschitz constants for the functions in U. +Consider the interval +I = [(u′(x) + κε)1, (u(x) − κε)1], +with volume +(u(x) − κε − (u′(x) + κε))d = (∆/2)d. +Consider Bε/2(x). If y ∈ Bε/2(x) then |˜u(y) − ˜u(x)| < κε for any ˜u ∈ U. +Now, if z ∈ I and y ∈ Bε(x) then +u(y) > u(x) − κε = u((x − κε)1) ≥ u(z) +by monotonicity. Similarly, +u′(z) ≥ u′((x + κε)1) = u′(x) + κε > u′(y) +Thus (y, z) ∈≻ \ ⪰′ for any (y, z) ∈ Bε(x) × I, and +µ(⪰, ⪰) − µ(⪰′, ⪰) = +� +1≻\≻′(y, z)[q(⪰; (y, z)) − q(⪰; (z, y))] dλ(y, z) +≥ +� +Bε/2(x)×I +1≻\≻′(y, z)[q(⪰; (y, z)) − q(⪰; (z, y))] dλ(y, z) +≥ λ(Bε(x)/2 × I) inf{q(⪰; (y, z) − q(⪰; (z, y)) : (y, z) ∈ Bε/2(x) × I}. +Where the first identity is shown in Chambers, Echenique, and Lambert (2021). +The second inequality follows because q(⪰; (x, y)) > 1/2 > q(⪰; (y, x)) on (x, y) ∈≻. +The third inequality is because (y, z) ∈≻ \⪰′ ⊆≻ \≻′ on Bε(x) × I. +By the assumptions we have placed on λ, and the calculations above, we know that +λ(Bε(x)/2) ≥ ¯c Vol(B¯ǫ(x) ∩ X) ≥ ¯cc′ Vol(B¯ǫ(x)) = ¯cc′ (ε/2)dπd/2 +Γ(1 + d/2). + +24 +CHAMBERS, ECHENIQUE, AND LAMBERT +So there is a constant C′′ (that only depends on X and ¯c) so that λ(I × Bε/2(x)) is +bounded below by +(∆/2)dC′′(ε/2)dπd/2 +Γ(1 + d/2) += (∆/2)d +C′′∆dπd/2 +(8κ)dΓ(1 + d/2) = C′∆2d. +Here C′ is a constant that only depends on C′′, κ and d. +By the assumption that Θ > 1/2, we get that +µ(⪰, ⪰) − µ(⪰′, ⪰) ≥ C∆2d +for some constant C that depends on C′ and Θ. +□ +Lemma 5. Consider a homothetic noise choice environment (X, P, λ, q). There is a +constant C with the following property. If ⪰ and ⪰′ are two preferences in P with +representations u and u′ (respectively) in U. Then +Cρ(u, u′)2d ≤ µ(⪰, ⪰) − µ(⪰′, ⪰) +Proof. Let x ∈ X be such that +ρ(u, u′) ≤ u(x) − u′(x) = ∆ > 0. +Choose η ∈ (0, 1) so that u(ηx) − u′(x) = ∆/2. Let +I = (u′(x)1, u(ηx)1) +and +Zη = [ηx, x] ∩ DM +α . +Note that I ⊆ X because by homotheticity, ∥x∥ = M and hence x ≥ α1. Then we +must have α1 ≤ u′(x)1 as α1 ̸≤ u′(x)1 would mean that u′(x)1 ≪ α1, contradicting +monotonicity and x ∼′ u′(x)1. +Observe that if y ∈ I and z ∈ Zη then we have that +u(y) < u(u(ηx)1) = u(ηx) ≤ u(z), +as y < u(ηx)1 and ηx ≤ z; while +u′(z) ≤ u′(x) = u′(u′(x)1) < u′(y). +Hence (z, y) ∈ ≻ \ ⪰′. +First we estimate Vol(Zη). Write Z0 for [0, x] ∩ DM +α . Define the function f(z) = +x + (1 − η)(z − x) and note that when z ∈ Z0 then f(z) = ηx + (1 − η)z ∈ [ηx, x] +because z ≥ 0. Note also that f(z) is a convex combination of x and z, so f(z) ∈ DM +α + +RECOVERING UTILITY +25 +as the latter is a convex set. This shows that +Zη = {x} + (1 − η)(Z0 − {x}), +and hence that Vol(Zη) = (1 − η)dVol(Z0). +Now, since Z0 is star shaped we have +Vol(Z0) = 1 +d +� +y∈SM +α +ρ(y, [0, x])d dy ≥ ( α +M )dAM +α , +where AM +α is the surface area of SM +α and ρ(y, [0, x]) = max{θ > 0 : θy ∈ [0, x] is the +radial function of the set [0, x] (see Schneider (2014) page 57). The inequality results +from ρ(y, [0, x]) ≥ α/M as xi ≥ α and yi ≤ M for any y ∈ SM +α . +Now, +1 − η = 1 − ∆/2 + u′(x) +u(x) += ∆/2 +u(x) ≥ ∆/2 +M , +as u(x) ≤ M. Thus we have that +Vol(Zη) ≥ ∆dC′, +with C′ = Vol(Z0)/(2M)d > 0, a constant. +Moreover, we have Vol(I) = (∆/2)d as I ⊆ X. Then we obtain, again using a for- +mula derived in Chambers, Echenique, and Lambert (2021), and that q(⪰; (x, y)) > +1/2 > q(⪰; (y, x)) on (x, y) ∈≻: +µ(⪰, ⪰) − µ(⪰′, ⪰) = +� +1≻\≻′(z, y)[q(⪰; (z, y)) − q(⪰; (y, z))] dλ(z, y) +≥ +� +Zη×I +1≻\≻′(z, y)[q(⪰; (z, y)) − q(⪰; (y, z))] dλ(z, y) +≥ λ(Zλ × I) inf{q(⪰; (z, y) − q(⪰; (y, z)) : (z, y) ∈ Zη × I} +≥ (∆/2)dC′∆dΘ, +where Θ = inf{q(⪰; (z, y) − q(⪰; (y, z)) : (z, y) ∈ Zη × I} > 0. +□ +7.1. Proof of Theorem 3. For the rest of this proof, we denote µ(⪰, ⪰∗) by µ(⪰). +The rest of the proof uses routine ideas from statistical learning theory. By standard +results (see, for example, Theorem 3.1 in Boucheron, Bousquet, and Lugosi (2005)), +there exists an event E with probability at least 1 − δ on which: +sup{|µn(⪰) − µ(⪰)| :⪰∈ P} ≤ E sup{|µn(⪰) − µ(⪰)| :⪰∈ P} + +� +2 ln(1/δ) +n +. + +26 +CHAMBERS, ECHENIQUE, AND LAMBERT +Moreover, again by standard arguments (see Theorem 3.2 in Boucheron, Bousquet, and Lugosi +(2005)), we also have +E sup{|µn(⪰) − µ(⪰)| :⪰∈ P} ≤ 2 E sup{ 1 +n +����� +� +i +σi1˜xi⪰yi +����� :⪰∈ P}, +where +Rn(P) = E sup{ 1 +n +����� +� +i +σi1˜xi⪰yi +����� :⪰∈ P} +is the Rademacher average of P. +Now, by the Vapnik-Chervonenkis inequality (see Theorem 3.4 in Boucheron, Bousquet, and Lugosi +(2005)), we have that +E sup{|µn(⪰) − µ(⪰)| :⪰∈ P} ≤ K +� +V +n , +where V is the VC dimension of P, and K is a universal constant. +So on the event E, we have we have that +sup{|µn(⪰) − µ(⪰)| :⪰∈ P} ≤ K +� +V/n + +� +2 ln(1/δ) +n +. +We now combine these statements with Lemmas 4 and 5. In particular, we let +D = d or D = 2d depending on which of the lemmas we invoke. Let u∗ ∈ U represent +⪰∗ and un ∈ U represent ⪰n. Let ∆ = ρ(u∗, un), a magnitude that depends on the +sample. Then, on the event E, by Lemma 4 or 5, we have that +C∆D ≤ µ(⪰∗) − µ(⪰n) += µ(⪰∗) − µn(⪰∗) + µn(⪰∗) − µn(⪰n) + µn(⪰n) − µ(⪰n) +≤ 2K +� +V +n + 2 +� +2 ln(1/δ) +n +, +where we have used that µn(⪰∗) − µn(⪰n) < 0 by definition of ⪰n. This proves the +second statement in the theorem. +To prove the first statement in the theorem, by Lemmas 4 and 5 again, and using +that µn(⪰n) ≥ µn(⪰∗), we have that, for any η > 0, +Pr(ρ(u∗, un) > η) ≤ Pr(µ(⪰∗) − µ(⪰n) > CηD) +≤ Pr(µ(⪰∗) − µn(⪰∗) > CηD/2) + Pr(µn(⪰n) − µ(⪰n) > CηD/2) +≤ 2 Pr(sup{|µ(⪰′) − µn(⪰′)| :⪰′∈ P} > CηD/2) → 0 + +RECOVERING UTILITY +27 +as n → ∞ by the uniform convergence in probability result shown in Chambers, Echenique, and Lambert +(2021). +7.2. Proof of Theorem 5. By standard results (see Hildenbrand (1970)), since X +is locally compact Polish, the topology of closed convergence is compact metric. +We will show that for any subsequence of ⪰k, there is a subsubsequence converging +to ⪰∗, which will establish that ⪰k→⪰∗. +So choose a convergent subsubsequence of the given subsequence. +To simplify +notation and with a slight abuse of notation, let us also refer to this subsubsequence +as ⪰k. Call its limit ⪰; ⪰ is complete as the set of complete relations is closed in +the closed convergence topology. It is therefore sufficient to establish that ≻∗⊆≻ and +⪰∗⊆⪰. +First we show that x ≻∗ y implies that x ≻ y. So let x ≻∗ y. Let U and V +be neighborhoods of x and y, respectively, such that x′ ≻∗ y′ for all x′ ∈ U and +y′ ∈ V . Such neighborhoods exist by the continuity of ⪰∗. We prove first that if +(x′, y′) ∈ U × V , then there exists N such that x′ ≻n y′ for all n ≥ N. Recall that +B = ∪{B′ : B′ ∈ Σ∞}. By hypothesis, there exist x′′ ∈ U ∩ B and y′′ ∈ V ∩ B such +that x′′ ≤ x′ and y′ ≤ y′′. Each ⪰n is a strong rationalization of the finite experiment +of order n, so if {˜x, ˜y} ∈ Σn then ˜x ≻n ˜y implies that ˜x ≻m ˜y for all m ≥ n. Since +x′′, y′′ ∈ B, there is N is such that {x′′, y′′} ∈ ΣN. +Thus x′′ ≻∗ y′′ implies that +x′′ ≻n y′′ for all n ≥ N. So, for n ≥ N, x′ ≻n y′, as ⪰n is weakly monotone. +Now we establish that x ≻ y. +Let {(xn, yn)} be an arbitrary sequence with +(xn, yn) → (x, y). By hypothesis, there is an increasing sequence {x′ +n}, and a decreas- +ing sequence {y′ +n}, such that x′ +n ≤ xn and yn ≤ y′ +n while (x, y) = limn→∞(x′ +n, y′ +n). +Let N be large enough that x′ +N ∈ U and y′ +N ∈ V . Let N′ ≥ N be such that +x′ +N ≻n y′ +N for all n ≥ N′ (we established the existence of such N′ above). Then, for +any n ≥ N′ we have that +xn ≥ x′ +n ≥ x′ +N ≻n y′ +N ≥ y′ +n ≥ yn. +By the weak monotonicity of ⪰n, then, xn ≻n yn. +The sequence {(xn, yn)} was +arbitrary, so (y, x) /∈⪰= limn→∞ ⪰n. Thus ¬(y ⪰ x). Completeness of ⪰ implies +that x ≻ y. +In second place we show that if x ⪰∗ y then x ⪰ y, thus completing the proof. So +let x ⪰∗ y. We recursively construct sequences xnk, ynk such that xnk ⪰nk ynk and +xnk → x, ynk → y. + +28 +CHAMBERS, ECHENIQUE, AND LAMBERT +So, for any k ≥ 1, choose x′ ∈ Nx(1/k) ∩ B with x′ ≥ x, and y′ ∈ Ny(1/k) ∩ B +with y′ ≤ y; so that x′ ⪰∗ x ⪰∗ y ⪰∗ y′, as ⪰∗ is weakly monotone. Recall that ⪰n +strongly rationalizes c⪰∗ for Σn. So x′ ⪰∗ y′ and x′, y′ ∈ B imply that x′ ⪰n y′ for all +n large enough. Let nk > nk−1 (where we can take n0 = 0) such that x′ ⪰nk y′; and +let xnk = x′ and ynk = y′. +Then we have (xnk, ynk) → (x, y) and xnk ⪰nk ynk. Thus x ⪰ y. +7.3. Proof of Theorem 4. First, it is straightforward to show that x ≻ y implies +x ⪰′ y. Because otherwise there are x, y for which x ≻ y and y ≻′ x. Take an open +neighborhood U about (x, y) and a pair (z, w) ∈ U ∩ (B × B) for which z ≻ w and +w ≻′ z, a contradiction. Symmetrically, we also have x ≻′ y implies x ⪰ y. +Now, without loss, suppose that there is a pair x, y for which x ≻ y and x ∼′ y. +By connectedness and continuity, V = {z : x ≻ z ≻ y} is nonempty. Indeed if we +assume, towards a contradiction that V = ∅, then {z : x ≻ z} and {z : z ≻ y} +are nonempty open sets. Further, for any z ∈ X, either x ≻ z or z ≻ y (because +if ¬(x ≻ z) then by completeness z ⪰ x, which implies that z ≻ y). Conclude that +{z : x ≻ z} ∪ {z : z ≻ y} = X and each of the sets are nonempty and open (by +continuity of the preference ⪰); these sets are disjoint, violating connectedness of X. +So we conclude that V is nonempty. By continuity of the preference ⪰, V os open. +We claim that there is a pair (w, z) ∈ (V × V ) ∩ (B × B) for which w ≻ z. For +otherwise, for all (w, z) ∈ V × V ∩ (B × B), w ∼ z. Conclude then by continuity +that for all (w, z) ∈ V × V , w ∼ z. Observe that this implies that, for any w ∈ V , +the set {z : w ≻ z ≻ y} = ∅, as if w ≻ z ≻ y, we also have that x ⪰ w ≻ z, from +which we conclude x ≻ z, so that z ∈ V and hence z ∼ w, a contradiction. Observe +that {z : w ≻ z ≻ y} = ∅ contradicts the continuity of ⪰ and the connectedness of +X (same argument as nonemptyness of V ; see our discussion above). +We have shown that there is (w, z) ∈ (V × V ) ∩ (B × B) for which w ≻ z, so that +x ≻ w ≻ z ≻ y. Further, we have hypothesized that x ∼′ y. By the first paragraph, +we know that x ⪰′ w ⪰′ z ⪰′ y. If, by means of contradiction, we have w ≻′ z, then +x ≻′ y, a contradiction. So w ∼′ z and w ≻ z, a contradiction to ⪰B×B=⪰′ +B×B. + +RECOVERING UTILITY +29 +References +Afriat, S. N. 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Conitzer (2020): “Learning the Valuations of a k-demand +Agent,” in International Conference on Machine Learning. + diff --git a/b9FJT4oBgHgl3EQfQywo/content/tmp_files/load_file.txt b/b9FJT4oBgHgl3EQfQywo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..95909e511c163043712f0d84cb54840ff84ba2db --- /dev/null +++ b/b9FJT4oBgHgl3EQfQywo/content/tmp_files/load_file.txt @@ -0,0 +1,1004 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf,len=1003 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='11492v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='TH] 27 Jan 2023 RECOVERING UTILITY CHRISTOPHER P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' CHAMBERS, FEDERICO ECHENIQUE, AND NICOLAS S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' LAMBERT Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We provide sufficient conditions under which a utility function may be recovered from a finite choice experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Identification, as is commonly understood in decision theory, is not enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We provide a general recoverability result that is widely applicable to modern theories of choice under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Key is to allow for a monetary environment, in which an objective notion of monotonicity is meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In such environments, we show that subjective expected utility, as well as variational preferences, and other parametrizations of utilities over uncertain acts are recoverable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We also consider utility recovery in a statistical model with noise and random deviations from utility maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (Chambers) Department of Economics, Georgetown University (Echenique) Department of Economics, UC Berkeley (Lambert) Department of Economics, University of Southern California Echenique thanks the National Science Foundation for its support through grant SES 1558757.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Lambert gratefully acknowledges the financial support and hospitality of Microsoft Research New York and the Yale University Cowles Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Introduction Economists are often interested in recovering preferences and utility functions from data on agents’ choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' If we are able to recover a utility function, then a preference relation is obviously implied, but the inverse procedure is more delicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In this paper, we presume access to data on an agent’s choices, and that these describe the agent’s preferences (or that preferences have been obtained as the outcome of a statistical estimation procedure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Our results describe sufficient conditions under which one can recover, or learn, a utility function from the agents’ choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' At a high level, the problem is that preferences essentially are choices, because they encode the choice that would be made from each binary choice problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' When we write x ≻ y we really mean that x would be chosen from the set {x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Utility func- tions are much richer objects, and a given choice behavior may be described by many different utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' For example, one utility can be used to discuss an agent’s risk pref- erences: they could have a “constant relative risk aversion” utility, for which a single parameter describes attitudes towards risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' But the same preferences can be repre- sented by a utility that does not have such a convenient parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' So recover- ing, or learning, utilities present important challenges that go beyond the problem of recovering a preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In the paper, we describe some simple examples that illus- trate the challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Our main results describe when one may (non-parametrically) recover a utility representation from choice data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We first consider choice under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We adopt the standard (Anscombe- Aumann) setting of choice under uncertainty, and focus attention on a class of utility representations that has been extensively studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Spe- cial cases include subjected expected utility, the max-min expected utility model of Gilboa and Schmeidler (1989), Choquet expected utility (Schmeidler, 1989), the variational preferences of Maccheroni, Marinacci, and Rustichini (2006), and many other popular models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Decision theorists usually place significance on the uniqueness of their utility representations, arguing that uniqueness provides an identification ar- gument that allows for utility to be recovered from choice data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We argue, in contrast, that uniqueness of a utility representation is not enough to recover a utility from finite choice data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Counterexamples are not hard to find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Indeed, even when a utility representation is unique, one may find a convergent sequence of utilities that is consistent with larger and larger finite datasets, but that does not converge to the utility function that generated the choices in the data, or to any utility to which it is equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' So RECOVERING UTILITY 3 uniqueness is necessary but not sufficient for a utility representation to be empirically tractable, in the sense of ensuring that a utility is recovered from large, but finite, choice experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Our main results are positive, and exhibit sufficient conditions for utility recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Key to our results is the availability of an objective direction of improvements in utility: we focus our attention on models of monotone preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Our paper con- siders choices among monetary acts, meaning state-contingent monetary payoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' For such acts, there is a natural notion of monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Between two acts, if one pays more in every state of the world, the agent agent should prefer it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' As a discipline on the recovery exercise, this essential notion of monotonicity suffices to ensure that a sequence of utilities that explains the choices in the data converges to the utility function that generated the choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We proceed by first discussing the continuity of a utility function in its dependence on the underlying preference relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' If U(⪰, x) is a function of a preference ⪰ and of choice objects x, then we say that it is a utility function if x �→ U(⪰, x) represents ⪰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We draw on the existing literature (Theorem 1) to argue that such continuous utilities exist in very general circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Continuity of this mapping in the preference ensures that if the choice data allow for preference recovery, they also allow a utility to be recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The drawback, however, of such general utility representation results is that they do not cover the special theories of utility in which economists generally take interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' There is no reason to expect that the utility U(⪰, x) coincides with the standard parametrizations of, for example, subjective expected utility or variational preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We then go on to our main exercise, which constrains the environment to the Anscombe-Aumann setting, and considers utility representations that have received special attention in the theory of choice under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We consider a setup that is flexible enough to accommodate most theories of choice under uncertainty that have been studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Our main result (Theorem 2) says that, whenever a choice experiment succeeds in recovering agents’ underlying preferences, it also serves to recover a utility in the class of utilities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' For example, if an agent has subjective expected utility preferences, and these can be recovered from a choice experiment, then so can the parameters of the subjective expected utility representation: the agents’ beliefs and Bernoulli utility index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Or, if the agent has variational preferences that can be inferred from choice data, then so can the different components of the variational utility representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 4 CHAMBERS, ECHENIQUE, AND LAMBERT Actual data on choices may be subject to sampling noise, and agents who randomly deviate from their preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The results we have just mentioned are useful in such settings, once the randomness in preference estimates is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' As a complement to our main findings, we proceed with a model that explicitly takes noisy choice, and randomness, into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Specifically, we consider choice problems that are sampled at random, and an agent who may deviate from their preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' They make mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In such a setting, we present sufficient conditions for the consistency of utility function estimates (Theorem 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In the last part of the paper we take a step back and revisit the problem of pref- erence recovery, with the goal of showing how data from a finite choice experiment can approximate a preference relation, and, in consequence, a utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Our model considers a large, but finite, number of binary choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We show that when preferences are monotone, then preference recovery is possible (Theorem 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In such environments, utility recovery follows for the models of choice under uncertainty that we have been interested in (Corollary 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Related literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The literature on revealed preference theory in economics is pri- marily devoted to tests for consistency with rational choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The main result in the lit- erature, Afriat’s theorem (Afriat, 1967a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Diewert, 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Varian, 1982), is in the context of standard demand theory (assuming linear budgets and a finite dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Versions of Afriat’s result have been obtained in a model with infinite data (Reny, 2015), nonlin- ear budget sets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=', Matzkin, 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Forges and Minelli, 2009), general choice prob- lems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=', Chavas and Cox, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Nishimura, Ok, and Quah, 2017), and multiperson equilibrium models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=', Brown and Matzkin, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Carvajal, Deb, Fenske, and Quah, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Algorithmic questions related to revealed preference are discussed by Echenique, Golovin, and Wierman (2011) and Camara (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The monograph by Chambers and Echenique (2016) presents an overview of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The revealed preference literature is primarily concerned with describing the datasets that are consistent with the theory, not with recovering or learning a preference, or a utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In the context of demand theory and choice from linear budgets, Mas-Colell (1978) introduces sufficient conditions under which a preference relation is recovered, in the limit, from a sequence of ever richer demand data observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' More recently, Forges and Minelli (2009) derive the analog of Mas-Colell’s results for nonlinear bud- get sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' An important strand of literature focuses on non-parametric econometric es- timation methods applied to demand theory data: Blundell, Browning, and Crawford RECOVERING UTILITY 5 (2003, 2008) propose statistical tests for revealed preference data, and consider coun- terfactual bounds on demand changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The problem of preference and utility recovery has been studied from the perspec- tive of statistical learning theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Beigman and Vohra (2006) considers the problem of learning a demand function within the PAC paradigm, which is closely related to the exercise we perform in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' A key difference is that we work with data on pairwise choices, which are common in experimental settings (including in many recent large-scale online experiments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Zadimoghaddam and Roth (2012) look at the utility recovery problem, as in Beigman and Vohra (2006), but instead of learning a demand function they want to understand when a utility can be learned efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Balcan, Daniely, Mehta, Urner, and Vazirani (2014) follow up on this important work by providing sample complexity guarantees, while Ugarte (2022) considers the prob- lem of recovery of preferences under noisy choice data, as in our paper, but within the demand theory framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Similarly, the early work of Balcan, Constantin, Iwata, and Wang (2012) considers a PAC learning question, focusing on important sub-classes of val- uations in economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Bei, Chen, Garg, Hoefer, and Sun (2016) pursues the problem assuming that a seller proposes budgets with the objective of learning an agent’s utility (they focus on quasilinear utility, and a seller that obtains aggregate demand data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Zhang and Conitzer (2020) considers this problem under an active-learning paradigm, and contrasts with the PAC sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In all, these works are important precedents for our paper, but they are all within the demand theory setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The results do not port to other environments, such as, for example, binary choice under risk or uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The closest paper to ours is Chambers, Echenique, and Lambert (2021), which looks at a host of related ques- tions to our paper but focusing on preference, not utility, recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The work by Chambers, Echenique, and Lambert considers choices from binary choice problem, but does not address the question of recovering, or learning, a utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' As we explain below in the paper, the problem for utilities is more delicate than the problem for preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In this line of work, Chase and Prasad (2019) obtains im- portant results on learning a utility but restricted to settings of intertemporal choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The work by Basu and Echenique (2020) looks at learnability of utility functions (within the PAC learning paradigm), but focusing on particular models of choice un- der uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Some of our results rely on measures of the richness of a theory, or of a family of preferences, which is discussed by Basu and Echenique (2020) and Fudenberg, Gao, and Liang (2021): the former by estimating the VC dimension of 6 CHAMBERS, ECHENIQUE, AND LAMBERT theories of choice under uncertainty, and the latter by proposing and analyzing new measures of richness that are well-suited for economics, as well as implementing them one economic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Finally, it is worth mentioning that preference and utilty recovery is potentially sub- ject to to strategic manipulations, as emphasized by Dong, Roth, Schutzman, Waggoner, and Wu (2018) and Echenique and Prasad (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' This possibility is ignored in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The Question We want to understand when utilities can be recovered from data on an agent’s choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Consider an agent with a utility function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We want know when, given enough data on the agent’s choices, we can “estimate” or “recover” a utility function that is guaranteed to be close to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In statistical terminology, recovery is analogous to the consistency of an estimator, and approximation guarantees are analogous to learnability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Imagine a dataset of size k, obtained from an incentivized experiment with k different choice problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='1 The observed choice behavior in the data may be described by a preference ⪰k, which is associated with a utility function uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The preference ⪰k could be a rationalizing preference, or a preference estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' So we choose a utility representation for uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The recovery, or consistency, property is that uk → u as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Suppose that the utility u represents preferences ⪰, which summarize the agent’s full choice behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Clearly, unless ⪰k→⪰, the exercise is hopeless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' So our first order of business is to understand when ⪰k→⪰ is enough to ensure that uk → u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In other words, we want to understand when recovering preferences is sufficient for recovering utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' To this end, our main results are in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In recovering a utility, we are interested in particular parametric representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In choice over uncertainty, for example, one may be interested in measures of risk-attitudes, or uncertainty aversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' It is key then that the utility recovery exercises preserves the aspects of utility that allow such measures to be have meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' If, say, preferences have the “constant relative risk aversion” (CRRA) form, then we want to recover the Arrow-Pratt measure of risk aversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 1Such datasets are common in experimental economics, including cases with very large k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' See, for example, von Gaudecker, van Soest, and Wengstrom (2011), Chapman, Dean, Ortoleva, Snowberg, and Camerer (2017), Chapman, Dean, Ortoleva, Snowberg, and Camerer (2022) and Falk, Becker, Dohmen, Enke, Huffman, and Sunde (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' One can also apply our results to roll call data from congress, as in Poole and Rosenthal (1985) or Clinton, Jackman, and Rivers (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Large-scale A/B testing by tech firms may provide further examples (albeit involving proprietary datasets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' RECOVERING UTILITY 7 Our data is presumably obtained in an experimental setting, where an agent’s behavior may be recorded with errors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' o in which the agent may randomly deviate from their underlying preference ⪰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Despite such errors, with high probability, “on the sample path,” we should obtain that ⪰k→⪰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In our paper we uncover situations where this convergence leads to utility recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Indeed, the results in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='5 may be applied to say that, in many popular models in decision theory, when ⪰k→⪰ (with high probability), then the resulting utility representations enable utility recovery (with high probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The next step is to discuss learning and sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Here we need to explicitly account for randomness and errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We lay out a model of random choice, with random sampling of choice problems and errors in agents’ choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The errors may take a very general form, as long as random choices are more likely to go in the direction of preferences than against it (if x ≻ y then x is the more likely choice from the choice problem {x, y}), and that this likelihood ratio remains bounded away from one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Contrast with the standard theory of discrete choice, where the randomness usually is taken to be additive, and independent of the particular pair of alternatives that are being compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Here we consider a formal statistical consistency problem, and exhibit situations where utility recovery is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We use ideas from the literature on PAC learning to provide formal finite sample-size bounds for each desired approximation guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The Model 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Basic definitions and notational conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let X be a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Given a binary relation R ⊆ X × X, we write x R y when (x, y) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' A binary relation that is complete and transitive is called a weak order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' If X is a topological space, then we say that R is continuous if R is closed as a subset of X × X (see, for example, Bergstrom, Parks, and Rader, 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' A preference relation is a weak order that is also continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' A preference relation ⪰ is locally strict if, for all x, y ∈ X, x ⪰ y implies that for each neighborhood U of (x, y), there is (x′, y′) ∈ U with x ≻ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The notion of local strictness was first introduced by Border and Segal (1994) as a generalization of the property of being locally non-satiated from consumer theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 8 CHAMBERS, ECHENIQUE, AND LAMBERT If ⪰ is a preference on X and u : X → R is a function for which x ⪰ y if and only if u(x) ≥ u(y) then we say that u is a representation of ⪰, or that u is a utility function for ⪰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' If A ⊆ Rd is a Borel set, we write ∆(A) for the set of all Borel probability measures on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We endow ∆(A) with the weak* topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' If S is a finite set, then we topologize ∆(A)S with the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' For p, q ∈ ∆(A), we say that p is larger than q in the sense of first-order stochastic dominance if � A fdx ≥ � A fdy for all monotone increasing, continuous and bounded functions f on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Topologies on preferences and utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The set of preferences over X, when X is a topological space, is endowed with the topology of closed convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The space of corresponding utility representations is endowed with the compact-open topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' These are the standard topologies for preferences and utilities, used in prior work in mathematical economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' See, for example, Hildenbrand (1970), Kannai (1970), and Mas-Colell (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Here we offer definitions and a brief discussion of our choice of topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let X be a topological space, and F = {F n}n be a sequence of closed sets in X ×X (with the product topology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We define Li(F) and Ls(F) to be closed subsets of X × X as follows: (x, y) ∈ Li(F) if and only if, for all neighborhoods V of (x, y), there exists N ∈ N such that F n ∩ V ̸= ∅ for all n ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (x, y) ∈ Ls(F) if and only if, for all neighborhoods V of (x, y), and all N ∈ N, there is n ≥ N such that F n ∩ V ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Observe that Li(F) ⊆ Ls(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The definition of closed convergence is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' F n converges to F in the topology of closed convergence if Li(F) = F = Ls(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Closed convergence captures the property that agents with similar preferences should have similar choice behavior—a property that is necessary to be able to learn the preference from finite data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Specifically, if X ⊆ Rn, and P is the set of all locally strict and continuous preferences on X, then the topology of closed convergence is the smallest topology on P for which the sets {(x, y, ⪰) : x ≻ y} ⊆ X × X × P RECOVERING UTILITY 9 are open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='2 In words: suppose that x ≻ y, then for x′ close to x, y′ close to y, and ⪰′ close to ⪰, we obtain that x′ ≻′ y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' For utility functions, we adopt the compact-open topology, which we also claim is a natural choice of topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The compact-open topology is characterized by the convergence criterion of uniform convergence on compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The reason it is natural for utility functions is that a utility usually has two arguments: one is the object being “consumed” (a lottery, for example) and the other is the ordinal preference that utility is meant to represent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (The preference argument is usually implicit, but of course it remains a key aspect of the exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=') Now an analyst wants the utility to be “jointly continuous,” or continuous in both of its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' For such a purpose, the natural topology on the set of utilities, when they are viewed solely as functions of consumption, is indeed the compact-open topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' More formally, consider the following result, originally due to Mas-Colell (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='3 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let X be a locally compact Polish space, and P the space of all contin- uous preferences on X endowed with the topology of closed convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Then there exists a continuous function U : P × X → [0, 1] so that x �→ U(⪰, x) represents ⪰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We may view the map U as a mapping from ⪰ to the space of utility functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Then continuity of this induced mapping is equivalent to the joint continuity result discussed in Theorem 1, as long as we impose the compact-open topology on the space of utility functions (see Fox (1945)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' As laid our in Section 2, we want to understand when we may conclude that uk → u from knowing that ⪰k→⪰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Mas-Colell’s theorem (Theorem 1) provides general conditions under which there exists one utility representation that has the requisite convergence property, but he is clear about the practical limita- tions of his result: “There is probably not a simple constructive (“canonical”) method to find a U function.” In contrast, economists are generally interested in specific parameterizations of utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' For example, if an agent has subjective expected-utility preferences, economists want to estimate beliefs and a von-Neumann-Morgenstern index;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' not some arbitrary representation of the agent’s preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Or, if the data involve intertemporal choices, and the agent discounts utility exponentially, then an economist will want to estimate 2See Kannai (1970) and Hildenbrand (1970) for a discussion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' a proof of this claim is available from the authors upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 3Levin (1983) provides a generalization to incomplete preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 10 CHAMBERS, ECHENIQUE, AND LAMBERT their discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Such specific parameterizations of utility are not meaningful in the context of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The following (trivial) example shows that there is indeed a problem to be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Convergence of arbitrary utility representations to the correct limit is not guaranteed, even when recovered utilities form a convergent sequence, and recovered preferences converge to the correct limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Consider expected-utility preferences on ∆(K)S, where K is a compact space, S a finite set of states, and ∆S(K) is the set of Anscombe-Aumann acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Fix an affine function v : ∆(K) → R, a prior µ ∈ ∆(S), and consider the preference ⪰ with representation � S v(f(s)) dµ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Now if we set ⪰k=⪰ then ⪰k→⪰ holds trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' However, it is possible to choose an expected utility representation � S vk(f(s)) dµk(s) that does not converge to a utility representation (of any kind) for ⪰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In fact one could choose a µk and a “normalization” for vk, for example ∥vk∥ = 1 (imagine for concreteness that K is finite, and use the Euclidean norm for vk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Specifically, choose scalars βk with ∥βk + 1 kv∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Then the utility f �→ � S vk(f(s)) dµ(s) represents ⪰k and converges to a constant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The punchline is that the limiting utility represents the preference that exhibits complete indifference among all acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' This is true, no matter what the original pref- erence ⪰ was.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In the example, we have imposed some discipline on the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Given that the utility converges to a constant, the discipline we have chosen is a particular nor- malization of the utility representations (their norm is constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The normalization just makes the construction of the example slightly more challenging, and reflects perhaps the most basic care that an analyst could impose on the recovery exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Anscombe-Aumann acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We present our first main result in the context of Anscombe-Aumann acts, the workhorse model of the modern theory of decisions under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let S be a finite set of states of the world, and fix a closed interval of the real line [a, b] ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' An act is a function f : S → ∆([a, b]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We interpret the elements of ∆([a, b]) as monetary lotteries, so that acts are state-contingent monetary lotteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The set of all acts is ∆([a, b])S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' When p ∈ ∆([a, b]), we denote the constant act that is identically equal to p by (p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' , p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' or sometimes by p for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Note that we do not work with abstract, general, Anscombe-Aumann acts, but in assuming monetary lotteries we impose a particular structure on the objective lotter- ies in our Anscombe-Aumann framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The reason is that our theory necessitates RECOVERING UTILITY 11 a certain known and objective direction of preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Certain preference compar- isons must be known a priori: monotonicity of preference will do the job, but for monotonicity to be objective we need the structure of monetary lotteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' An act f dominates an act g if, for all s ∈ S, f(s) first-order stochastic dominates g(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' And f strictly dominates g if, for all s ∈ S, f(s) strictly first-order stochastic dominates g(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' A preference ⪰ over acts is weakly monotone if f ⪰ g whenever f first-order stochastic dominates g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let U be the set of all continuous and monotone weakly increasing functions u : [a, b] → R with u(a) = 0 and u(b) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' A pair (V, u) is a standard representation if V : ∆([a, b])S → R and u ∈ U are continuous functions such that v(p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' , p) = � [a,b] u dp, for all constant acts (p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' , p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Moreover, we say that a standard representation (V, u) is aggregative if there is an aggregator H : [0, 1]S → R with V (f) = H(( � u df(s))s∈S) for f ∈ ∆([a, b])S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' An aggregative representation with aggregator H is denoted by (V, u, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Observe that a standard representation rules out total indifference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' A preference ⪰ on ∆([a, b])S is standard if it is weakly monotone, and there is a standard representation (V, u) in which V represents ⪰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Roughly, standard prefer- ences will be those that satisfy the expected utility axioms across constant acts, and are monotone with respect to the (statewise) first order stochastic dominance rela- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Aggregative preferences will additionally satisfy an analogue of Savage’s P3 or the Anscombe-Aumann notion of monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Variational preferences (Maccheroni, Marinacci, and Rustichini, 2006) are standard and aggregative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='4 Let V (f) = inf{ � v(f(s))dπ(s) + c(π) : π ∈ ∆(S)} where (1) v : ∆([a, b]) → R is continuous and affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (2) c : ∆(S) → [0, ∞] is lower semicontinuous, convex and grounded (meaning that inf{c(π) : π ∈ ∆(S)} = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Note that V (p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' , p) = v(p) + inf{c(π) : π ∈ ∆(S)} = � u dp, by the assumption that c is grounded, and where the existence of u : [a, b] → R so that v(p) = � u dp 4Variational preferences are widely used in macroeconomics and finance to capture decision mak- ers’ concerns for using a misspecified model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Here it is important to recover the different com- ponents of a representation, v and c, because they quantify key features of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' See for example Hansen and Sargent (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Hansen, Sargent, Turmuhambetova, and Williams (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Hansen and Sargent (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 12 CHAMBERS, ECHENIQUE, AND LAMBERT is an instance of the Riesz representation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' It is clear that we may choose u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' So (V, u) is a standard representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Letting H : [0, 1]S → R be defined by H(x) = inf{� s∈S x(s)π(s) + c(π) : π ∈ ∆(S)}, we see that indeed (V, u, H) is also an aggregative representation of these preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Some other examples of aggregative preferences include special cases of the varia- tional model Gilboa and Schmeidler (1989), as well as generalizations of it, Cerreia-Vioglio, Maccheroni, Marinacci, and Montrucchio (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Chandrasekher, Frick, Iijima, and Le Yaouanq (2021), and others which are not comparable Schmeidler (1989);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Chateauneuf, Grabisch, and Rico (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Chateauneuf and Faro (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='5 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let ⪰ be a standard preference with standard representation (V, u), and {⪰k} a sequence of standard preferences, each with a standard representation (V k, uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (1) If ⪰k→⪰, then (V k, uk) → (V, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (2) If, in addition, these preferences are aggregative with representations (V k, uk, Hk) and (V, u, H), then Hk → H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In terms of interpretation, Theorem 2 suggests that, as preferences converge, risk- attitudes, or von Neumann morgenstern utility indices also converge in a pointwise sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The aggregative part claims that we can study the convergence of risk attitudes and the convergence of the aggregator controlling for risk separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' So, for example, in the multiple priors case, two decision makers whose preferences are close will have similar sets of priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Preferences over lotteries and certainty equivalents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In this section, we focus on a canonical representation for preferences over lotteries: the certainty equiv- alent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' There are many models of preferences over lotteries, but we have in mind in particular Cerreia-Vioglio, Dillenberger, and Ortoleva (2015), whereby a preference representation over lotteries is given by U(p) = infu∈U u−1( � udp);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' a minimum over a set of certainty equivalents for expected utility maximizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Key is that for this representation, and any degenerate lottery δx, U(δx) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 5A class of variational preferences that are of particular interest to computer scientists are preferences with a max-min representation (Gilboa and Schmeidler, 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' These evaluate acts by V (f) = inf{ � v(f(s))dπ(s) : π ∈ Π}, with Π ⊆ ∆(S) a closed and convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Here c is the indicator function of Π (as defined in convex analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' RECOVERING UTILITY 13 Let [a, b] ⊂ R, where a < b, be an interval in the real line and consider ∆([a, b]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Say that ⪰ on ∆([a, b]) is certainty monotone if when ever p first order stochastically dominates q, then p ⪰ q, and for all x, y ∈ [a, b] for which x > y, δx ≻ δy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Any cer- tainty monotone continuous preference ⪰ and any lottery p ∈ ∆([a, b]) then possesses a unique certainty equivalent x ∈ [0, 1], satisfying δx ∼ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' To this end, we define ce(⪰, p) to be the certainty equivalent of p for ⪰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' It is clear that, fixing ⪰, ce(·, ⪰) is a continuous utility representation of ⪰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let ⪰ be a certainty monotone preference and let p ∈ ∆([a, b]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let {⪰k} be a sequence of certainty monotone preferences and let pk be a sequence in ∆([a, b]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' If (⪰k, pk) → (⪰, p), then ce(⪰k, pk) → ce(⪰, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' To this end, the map carrying each preference to its certainty equivalent represen- tation is a continuous map in the topology of closed convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Utility recovery with noisy choice data We develop a model of noisy choice data, and consider when utility may be re- covered from a traditional estimation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Recovery here takes the form of an explicit consistency result, together with sample complexity bounds in a PAC learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The focus is on the Wald representation, analogous to the certainty equivalent we considered in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' When choosing among vectors in x ∈ Rd, the Wald representation is u(x) ∈ R so that x ∼ (u(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' , u(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' If the choice space is well behaved, a Wald representation exists for any monotone and continuous preference relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' To this end, we move beyond the Anscombe- Aumann setting that we considered above, but it should be clear that some versions of Anscombe-Aumann can be accommodated within the assumptions of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Our main results for the model that explicitly accounts for noisy choice data as- sumes Wald representations that are either Lipschitz or homogeneous (meaning that preferences are homothetic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Noisy choice data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The primitives of our noisy choice model are collected in the tuple (X, P, λ, q), where: X ⊆ Rd is the ambient choice, or consumption, space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The set X is endowed with the (relative) topology inherited from Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 14 CHAMBERS, ECHENIQUE, AND LAMBERT P is a class of continuous and locally strict preferences on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The class comes with a set of utility functions U, so that each element of P has a utility representation in the set U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' λ is a probability measure on X, assumed to be absolutely continuous with respect to Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We also assume that λ ≥ c Leb, where c > 0 is a constant and Leb denotes Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' q : X × X × P → [0, 1] is a random choice function, so q(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' ⪰) is the probability that an agent with preferences ⪰ chooses x over y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Assume that if x ≻ y, then x is chosen with probability q(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' ⪰) > 1/2 and y with probability q(y, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' ⪰∗) = 1 − q(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' ⪰).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' If x ∼ y then x and y are chosen with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We shall assume that the error probability q satisfies that Θ ≡ inf{q(⪰, (x, y)) : x ≻ y and ⪰∈ P} > 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The tuple (X, P, λ, q) describes a data-generating process for noisy choice data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Fix a sample size n and consider an agent with preference ⪰∗∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' A sequence of choice problems {xi, yi}, 1 ≤ i ≤ n are obtained by drawing xi and yi from X, independently, according to the law λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Then a choice is made from each problem {xi, yi} according to q(·, ·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' ⪰∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Observe that our assumptions on q are mild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We allow errors to depend on the pair {x, y} under consideration, almost arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The only requirement is that one is more likely to choose according to one’s preference than to go against them, as well as the more technical assumptions of measurability and a control on how large the deviation from 1/2-1/2 choice may get.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' To keep track of the chosen alternative, we order the elements of each problem so that (xi, yi) means that xi was chosen from the choice problem {xi, yi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' So a sample of size n is {(x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' , (xn, yn)}, consisting of 2n iid draws from X × X according to our stochastic choice model: in the ith draw, the choice problem was {xi, yi} and xi was chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' A utility function un ∈ U is chosen to maximize the number of rationalized choices in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' So un maximizes �n i=1 1u(xi)≥u(yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The space of utility functions is endowed with a metric, ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In this section, all we ask of ρ is that, for any u, u′ ∈ U, there is x ∈ X with |u(x) − u′(x)| ≥ ρ(u, u′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' For example, we could use the sup norm for the purposes of any of the results in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' RECOVERING UTILITY 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Lipschitz utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' One set of sufficient conditions will need the family of rele- vant utility representations to satisfy a Lipschitz property with a common Lipschitz bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The representations are of the Wald kind, as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We now add the requirement of having the Lipschitz property, which allows us to connect differences in utility functions to quantifiable observable (but noisy) choice behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The main idea is expressed in Lemma 4 of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We say that (X, P, λ, q) is a Lipschitz environment if: (1) X ⊆ Rd is convex, compact, and has nonempty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (2) Each preference ⪰∈ P has a Wald utility representation u⪰ : X → R so that x ∼ u⪰(x)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (3) All utilities in U are Lipschitz, and admit a common Lipschitz constant κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' So, for any x, x′ ∈ X and u ∈ U, |u(x) − u(x′)| ≤ κ∥x − x′∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Homothetic preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The second set of sufficient conditions involve homo- thetic preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' It turns out, in this case, that the Wald representations have a homogeneity property, and this allows us to connect differences in utilities to a prob- ability of detecting such differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The key insights is contained in Lemma 5 of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We employ the following auxiliary notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' SM α = {x ∈ Rd : ∥x∥ = M and x ≥ α1} and DM α = {θx : x ∈ SM α and θ ∈ [0, 1]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We say that (X, P, λ, q) is a homothetic environment if: (1) X = DM α for some (small) α > 0 and (large) M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (2) P is a class of continuous, monotone, homothetic, and complete preferences on X ⊆ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (3) U is a class of Wald representations, so that for each ⪰∈ P there is a utility function u ∈ U with x ∼ u(x)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Remark: if u ∈ U is the Wald representation of ⪰, then u is homogeneous of degree one because x ∼ u(x)1 iff λx ∼ λu(x)1, so u(λx) = λu(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' VC dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The Vapnik-Chervonenkis (VC) dimension of a set P of prefer- ences is the largest sample size n for which there exists a utility u ∈ U that perfectly rationalizes all the choices in the data, no matter what those are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' That is so that n = �n i=1 1u(xi)≥u(yi) for any dataset (xi, yi)n i=1 of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' VC dimension is a basic ingredient in the standard PAC learning paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' It is a measure of the complexity of a theory used in machine learning, and lies behind stan- dard results on uniform laws of large numbers (see, for example, Boucheron, Bousquet, and Lugosi 16 CHAMBERS, ECHENIQUE, AND LAMBERT (2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Applications of VC to decision theory can be found in Basu and Echenique (2020) and Chambers, Echenique, and Lambert (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' It is worth noting that VC dimension is used in classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' It may not be obvious, but when it comes to preferences, our exercise may be thought of as classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' For each pair of alternatives x and y, a preference ⪰ “classifies” the pair as x ⪰ y or y ≻ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Then we can think of preference recovery as a problem of learning a classifier within the class P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Consistency and sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Consider a noisy choice environment (X, P, λ, q) that is either a homo- thetic or a Lipschitz environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Suppose that u∗ ∈ U is the Wald utility represen- tation of ⪰∗∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (1) The estimates un converge to u∗ in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (2) There are constants K and ¯C so that, for any δ ∈ (0, 1) and n, with probability at least 1 − δ, ρ(un, u∗) ≤ ¯C � K � V/n + � 2 ln(1/δ)/n �1/D , where V is the VC dimension of P, D = d when the environment is Lipschitz and D = 2d when it is homothetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Of course, the second statement in the theorem is only meaningful when the VC dimension of P is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The constants K and ¯C depend on the primitives in the environment, but not on preferences, utilities, or sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Recovering preferences and utilities The discussion in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='4 focused on utility recovery, taking convergence of preferences as given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Here we take a step back, provide some conditions for pref- erence recovery that are particularly relevant for the setting of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='4, and then connect these back to utility recovery in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' First we describe an experimental setting in which preferences may be elicited: an agent, or subject, faces a sequence of (incentivized) choice problems, and the choices made produce data on his preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The specific model and description below is borrowed from Chambers, Echenique, and Lambert (2021), but the setting is completely standard in choice theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let X = ∆([a, b])S be the set of acts over monetary lotteries, as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' A choice function is a pair (Σ, c) with Σ ⊆ 2X \\ {∅} a collection of RECOVERING UTILITY 17 nonempty subsets of X, and c : Σ → 2X with ∅ ̸= c(A) ⊆ A for all A ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' When Σ, the domain of c, is implied, we refer to c as a choice function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' A choice function (Σ, c) is generated by a preference relation ⪰ over X if c(A) = {x ∈ A : x ⪰ y for all y ∈ B}, for all A ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The notation (Σ, c⪰) means that the choice function (Σ, c⪰) is generated by the preference relation ⪰ on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Our model features an experimenter (a female) and a subject (a male).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The subject chooses among alternatives in a way described by a preference ⪰∗ over X, which we refer to as data-generating preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The experimenter seeks to infer ⪰∗ from the subject’s choices in a finite experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In a finite experiment, the subject is presented with finitely many unordered pairs of alternatives Bk = {xk, yk} in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' For every pair Bk, the subject is asked to choose one of the two alternatives: xk or yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' A sequence of experiments is a collection Σ∞ = {Bi}i∈N of pairs of possible choices presented to the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let Σk = {B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' , Bk} collect the first k elements of a sequence of experiments, and B = ∪∞ k=1Bk be the set of all alternatives that are used over all the experiments in a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Here Σk is a finite experiment of size k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We make two assumptions on Σ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The first is that B is dense in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The second is that, for any x, y ∈ B there is k for which Bk = {x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The first assumption is obviously needed to obtain any general preference recovery result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The second assumption means that the experimenter is able to elicit the subject’s choices over all pairs used in her experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='6 For each k, the subject’s preference ⪰∗ generates a choice function (Σk, c) by letting, for each Bi ∈ Σk, c(B) be a maximal element of Bi according to ⪰∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Thus the choice behavior observed by the experimenter is always consistent with (Σk, c⪰∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We introduce two notions of rationalization: weak and strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' A preference ⪰k weakly rationalizes (Σk, c) if, for all Bi ∈ Σk, c(Bi) ⊆ c⪰k(Bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' A preference ⪰k weakly rationalizes a choice sequence (Σ∞, c) if it rationalizes the choice function of order k (Σk, c), for all k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' A preference ⪰k strongly rationalizes (Σk, c) if, for all Bi ∈ Σk, c(Bi) = c⪰k(Bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' A preference ⪰k strongly rationalizes a choice sequence (Σ∞, c) if it rationalizes the choice function of order k (Σk, c), for all k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 6If there is a countable dense A ⊆ X, then one can always construct such a sequence of experiments via a standard diagonalization argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 18 CHAMBERS, ECHENIQUE, AND LAMBERT In the history of revealed preference theory in consumer theory, strong rationaliz- ability came first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' It is essentially the notion in Samuelson (1938) and Richter (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Strong rationalizability is the appropriate notion when it is known that all potentially chosen alternatives are actually chosen, or when we want to impose, as an added dis- cipline, that the observed choices are uniquely optimal in each choice problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' This makes sense when studying demand functions, as Samuelson did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Weak rationaliz- ability was one of the innovations in Afriat (1967b), who was interested in demand correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' A general “limiting” result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Our next result serves to contrast what can be achieved with the “limiting” (countably infinite) data with the limit of preferences recovered from finite choice experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Suppose that ⪰ and ⪰∗ are two continuous preference relations (com- plete and transitive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' If ⪰ |B×B =⪰∗ |B×B, then ⪰=⪰∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Indeed, as the proof makes clear, Theorem 4 would hold more generally for any X which is a connected topological space, but it may not hold in absence of connect- edness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' There is a sense in which the limiting case with an infinite amount of data offers no problems for preference recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The structure we impose is needed for the limit of rationalizations drawn from finite data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Recovery from finite data in the AA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Here we adopt the same structural assumptions as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='4, namely that X = ∆([a, b])S, endowed with the weak topology and the first order stochastic dominance relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' However, the result easily extends to broader environments, as the proof makes clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' There is a sequence of finite experiments Σ∞ so that if the subject’s preference ⪰∗ is continuous and weakly monotone, and for each k ∈ N, ⪰k is a con- tinuous and weakly monotone preference that strongly rationalizes a choice function (Σk, c) generated by ⪰∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' then ⪰k→⪰∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let ⪰∗ and ⪰k be as in the statement of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' If, in addi- tion, ⪰∗ and ⪰k have standard representations (V, u) and (V k, uk) then (V, u) = limk→∞(V k, uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 7As an illustration of the difference between these two notions of rationalizability, note that, in the setting of consumer theory, one leads to the Strong Axiom of Revealed Preference while the other to the Generalized Axiom of Revealed Preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Of course, Afriat’s approach is also distinct in assuming a finite dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' See Chambers and Echenique (2016) for a detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' RECOVERING UTILITY 19 Note that Theorem 5 requires the existence of the data-generating preference ⪰∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' A “dual” result to Theorem 5 was established in Chambers, Echenique, and Lambert (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' There, the focus was on weak rationalization via ⪰k, which is a weaker notion than the strong rationalization hypothesized here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' To achieve a weak rationalization result, we assumed instead that preferences were strictly monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Proofs In this section, unless we say otherwise, we denote by X the set of acts ∆([a, b])S, and the elements of X by x, y, z etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Note that X is compact Polish when ∆([a, b]) is endowed with the topology of weak convergence of probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let P be the set of all complete and continuous binary relations on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The lemmas stated here will be used in the proofs of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let X ⊆ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' If {x′ n} is an increasing sequence in X, and {x′′ n} is a decreasing sequence, such that sup{x′ n : n ≥ 1} = x∗ = inf{x′′ n : n ≥ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Then lim n→∞ x′ n = x∗ = lim n→∞ x′′ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' This is obviously true for n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' For n > 1, convergence and sups and infs are obtained component-by-component, so the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let X = ∆([a, b]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let {xn} be a convergent sequence in X, with xn → x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Then there is an increasing sequence {x′ n} and an a decreasing sequence {x′′ n} such that x′ n ≤ xn ≤ x′′ n, and limn→∞ x′ n = x∗ = limn→∞ x′′ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The set X ordered by first order stochastic dominance is a complete lattice (see, for example, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='1 in Kertz and Rösler (2000)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Suppose that xn → x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Define x′ n and x′′ n by x′ n = inf{xm : n ≤ m} and x′′ n = sup{xm : n ≤ m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Clearly, {x′ n} is an increasing sequence, {x′′ n} is decreasing, and x′ n ≤ xn ≤ x′′ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let Fx denote the cdf associated with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Note that Fx′′n(r) = inf{Fxm(r) : n ≤ m} while Fx′n(r) is the right-continuous modification of sup{Fxm(r) : n ≤ m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' For any point of continuity r of F, Fxm(r) → Fx∗(r), so Fx(r) = sup{inf{Fxm(r) : n ≤ m} : n ≥ 1} by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Moreover, Fx∗(r) = inf{sup{Fxm(r) : n ≤ m} : n ≥ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Then Fx∗(r − ε) ← sup{Fxm(r − ε) : n ≤ m} ≤ Fx′n(r) ≤ sup{Fxm(r + ε) : n ≤ m} → Fx∗(r + ε) 20 CHAMBERS, ECHENIQUE, AND LAMBERT Then Fx′n(r) → Fx∗(r), as r is a point of continuity of Fx∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' □ The results we have obtained motivate two definitions that will prove useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Say that the set X, together with the collection of finite experiments Σ∞, has the countable order property if for each x ∈ X and each neighborhood V of x in X there is x′, x′′ ∈ (∪iBi) ∩ V with x′ ≤ x ≤ x′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We say that X has the squeezing property if for any convergent sequence {xn}n in X, if xn → x∗ then there is an increasing sequence {x′ n}n, and an a decreasing sequence {x′′ n}n, such that x′ n ≤ xn ≤ x′′ n, and limn→∞ x′ n = x∗ = limn→∞ x′′ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' If X = ∆([a, b])S, then X has the squeezing property, and there is Σ∞ such that (X, Σ∞) has the countable order property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The squeezing property follows from Lemma 2, and the countable order prop- erty from Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='11 of Aliprantis and Border (2006): Indeed, let B be the set of probability distributions p with finite support on Q∩[a, b], where for all q ∈ Q∩[a, b], p(q) ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Then we may choose a sequence of pairs Bi, and let Σ∞ to be Bi with B = ∪Bi so that the countable order property is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Without loss of generality, we may set [a, b] = [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' First we show that uk → u in the compact-open topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' To this end, let xk → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We want to show that uk(xk) → u(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Suppose then that this is not the case, and by selecting a subsequence that uk(xk) → Y > u(x) (without loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Note that δxk ∼k pk, where pk is the lottery that pays 1 with probability uk(xk) ∈ [0, 1], and 0 with probability 1−uk(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let p be the lottery that pays 1 with probability Y , and 0 with probability 1 − Y (given that the range of uk is [0, 1], we must have Y ∈ [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Now we have that (δxk, pk) → (δx, p) and δxk ∼k pk implies δx ∼ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' This is a contradiction because δx is indifferent in ⪰ to the lottery that pays 1 with probability uk(xk) ∈ [0, 1], and 0 with probability 1 −uk(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The latter is strictly first-order stochastically dominated by the lottery p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' To finish the proof, we show that V k → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' This is the same as proving that V k(f k) → V (f) when f k → f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' For each k, continuity and weak monotonicity imply that there is xk ∈ [0, 1] so that V k(f k) = V k(δxk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' , δxk) = uk(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Similarly, there is x with V (f) = V (δx, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' , δx) = u(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Now we argue that xk → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Indeed {xk} is a sequence in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' If there is a subsequence that converges to, say, x′ > x then we may choose x′′ = x+x′ 2 and RECOVERING UTILITY 21 eventually f k ⪰k (δx′′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' , δx′′) ≻ (δx, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' , δx) ∼ f, using weak monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' This is impossible because (f k, (δxk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' , δxk) → (f, (δx′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' , δx′)) and f k ⪰k ((δxk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' , δxk) imply that f ⪰ ((δx′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' , δx′) ⪰ (δx′′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' , δx′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Finally, using what we know about the convergence of uk to u, V k(f k) = uk(xk) → u(x) = V (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We now turn to the second statement in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Observe that Hk is a con- tinuous function from [0, 1]S onto [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let zk ∈ [0, 1]S be an arbitrary convergent sequence, and say that zk → z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We claim that Hk(zk) → H(z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Without loss we may assume that Hk(zk) → Y , by taking a subsequence if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' For each k and s, choose yk(s) ∈ [0, 1] for which uk(yk(s)) = zk(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Again, without loss, we may assume that yk → y∗ by taking a subsequence if necessary, and using the finiteness of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Observe also that u(y∗(s)) = z∗(s) as we have shown that uk → u in the compact-open topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Now, we may also choose ˆzk ∈ [0, 1] so that uk(ˆzk) = Hk(zk) = Hk((uk(yk(s)))s∈S), and further may again without loss (by taking a subsequence) assume that ˆzk con- verges to ˆz∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Thus u(ˆz∗) = lim uk(ˆzk) = lim Hk(zk) = Y , again using what we have shown regarding uk → u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Then (δˆzk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' , δˆzk) ∼k (yk(s))s∈S so that, by taking limits, (δˆz∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' , δˆz∗) ∼∗ (y∗(s))s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' This implies that Y = u(ˆz∗) = H(u(y∗(s)) = H(z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Take (⪰k, pk) as in the statement of the Proposition, and observe that for every p ∈ ∆([a, b]), ce(⪰k, pk) ∈ [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Suppose by means of contradiction that ce(⪰k, pk) → ce(⪰, p) is false, then there is some ǫ > 0 and a sub- sequence for which |ce(⪰km, pkm) − ce(⪰, p)| > ǫ, by taking a further subsequence, we assume without loss that ce(⪰km, pkm) → α ̸= ce(⪰, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Now, pkm ∼km δce(⪰km,pkm), and pkm → p and δce(⪰km,pkm) → δα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' So by definition of closed convergence, it follows that p ∼ δα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' but this violates certainty monotonicity as α ̸= ce(⪰, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Proof of Theorem 3 First some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let µn(⪰) = 1 n �n i=1 1xi⪰yi, and ⪰n∈ P be represented by un ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' By definition of un, we have that µn(⪰n) ≥ µn(⪰) for all ⪰∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' And we use Vol(A) to denote the volume of a set A in Rd, when this is well defined (see Schneider (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 22 CHAMBERS, ECHENIQUE, AND LAMBERT Consider the measure µ on X × X defined as µ(A, ⪰) = � A q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' x, y) dλ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In particular µ(⪰′, ⪰) = � X×X 1⪰′(x, y)q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' x, y) dλ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' is the probability that a choice with error made at a randomly-drawn choice problem by an agent with preference ⪰ will coincide with ⪰′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The key identification result shown in Chambers, Echenique, and Lambert (2021) is that, if ⪰′̸=⪰, then µ(⪰′, ⪰) < µ(⪰, ⪰).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Consider a Lipschitz noise choice environment (X, P, λ, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' There is a constant C with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' If ⪰ and ⪰′ are two preferences in P with representations u and u′ (respectively) in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Then Cρ(u, u′)d ≤ µ(⪰, ⪰) − µ(⪰′, ⪰) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The ball in Rd with center x and radius ε is denoted by Bε(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' First we show that the map ε �→ Vol(Bǫ(x) ∩ X) Vol(Bǫ(x)) , defined for x ∈ X, is nonincreasing as a function of ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Indeed, let ǫ1 < ǫ2, and let y ∈ Bǫ2(x) ∩ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Then y ∈ X and ∥y − x∥ ≤ ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' By convexity of X, y1 ≡ x + ǫ1 ǫ2(y − x) = (1 − ǫ1 ǫ2)x + ǫ1 ǫ2y ∈ X, and y1 ∈ Bǫ1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Observe further by properties of Lebesgue measure in Rd that Vol({x+ ǫ1 ǫ2(y−x) : y ∈ Bǫ2(x)∩ X}) = � ǫ1 ǫ2 �d Vol(Bǫ2(x) ∩ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Therefore, Vol(Bǫ1(x) ∩ X) ≥ � ǫ1 ǫ2 �d Vol(Bǫ2(x) ∩ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Since Vol(Bǫ1(x)) = � ǫ1 ǫ2 �d Vol(Bǫ2(x)), it follows that Vol(Bǫ1(x) ∩ X) Vol(Bǫ1(x)) ≥ Vol(Bǫ2(x) ∩ X) Vol(Bǫ2(x)) , like we wanted to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Now observe that there exists ¯ε > 0 large enough that X ⊆ Bε(x) for all ε ≥ ¯ε and x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Hence, for any x ∈ X and ε ∈ (0, ¯ε] Vol(Bǫ(x) ∩ X) Vol(Bǫ(x)) ≥ Vol(X) Vol(B¯ǫ(x)) ≡ c′ > 0, RECOVERING UTILITY 23 as X has nonempty interior and the volume of a ball in Rd is independent of its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Now we proceed with the proof of the statement in the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let ∆ = ρ(u, u′) and fix x ∈ X with (wlog) u(x) − u′(x) = ∆ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Set ε = ∆ 4κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We may assume that ε ≤ 2¯ε as defined above, as otherwise we can use a larger upper bound on the Lipschitz constants for the functions in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Consider the interval I = [(u′(x) + κε)1, (u(x) − κε)1], with volume (u(x) − κε − (u′(x) + κε))d = (∆/2)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Consider Bε/2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' If y ∈ Bε/2(x) then |˜u(y) − ˜u(x)| < κε for any ˜u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Now, if z ∈ I and y ∈ Bε(x) then u(y) > u(x) − κε = u((x − κε)1) ≥ u(z) by monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Similarly, u′(z) ≥ u′((x + κε)1) = u′(x) + κε > u′(y) Thus (y, z) ∈≻ \\ ⪰′ for any (y, z) ∈ Bε(x) × I, and µ(⪰, ⪰) − µ(⪰′, ⪰) = � 1≻\\≻′(y, z)[q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (y, z)) − q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (z, y))] dλ(y, z) ≥ � Bε/2(x)×I 1≻\\≻′(y, z)[q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (y, z)) − q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (z, y))] dλ(y, z) ≥ λ(Bε(x)/2 × I) inf{q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (y, z) − q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (z, y)) : (y, z) ∈ Bε/2(x) × I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Where the first identity is shown in Chambers, Echenique, and Lambert (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The second inequality follows because q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (x, y)) > 1/2 > q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (y, x)) on (x, y) ∈≻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The third inequality is because (y, z) ∈≻ \\⪰′ ⊆≻ \\≻′ on Bε(x) × I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' By the assumptions we have placed on λ, and the calculations above, we know that λ(Bε(x)/2) ≥ ¯c Vol(B¯ǫ(x) ∩ X) ≥ ¯cc′ Vol(B¯ǫ(x)) = ¯cc′ (ε/2)dπd/2 Γ(1 + d/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 24 CHAMBERS, ECHENIQUE, AND LAMBERT So there is a constant C′′ (that only depends on X and ¯c) so that λ(I × Bε/2(x)) is bounded below by (∆/2)dC′′(ε/2)dπd/2 Γ(1 + d/2) = (∆/2)d C′′∆dπd/2 (8κ)dΓ(1 + d/2) = C′∆2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Here C′ is a constant that only depends on C′′, κ and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' By the assumption that Θ > 1/2, we get that µ(⪰, ⪰) − µ(⪰′, ⪰) ≥ C∆2d for some constant C that depends on C′ and Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Consider a homothetic noise choice environment (X, P, λ, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' There is a constant C with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' If ⪰ and ⪰′ are two preferences in P with representations u and u′ (respectively) in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Then Cρ(u, u′)2d ≤ µ(⪰, ⪰) − µ(⪰′, ⪰) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let x ∈ X be such that ρ(u, u′) ≤ u(x) − u′(x) = ∆ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Choose η ∈ (0, 1) so that u(ηx) − u′(x) = ∆/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let I = (u′(x)1, u(ηx)1) and Zη = [ηx, x] ∩ DM α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Note that I ⊆ X because by homotheticity, ∥x∥ = M and hence x ≥ α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Then we must have α1 ≤ u′(x)1 as α1 ̸≤ u′(x)1 would mean that u′(x)1 ≪ α1, contradicting monotonicity and x ∼′ u′(x)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Observe that if y ∈ I and z ∈ Zη then we have that u(y) < u(u(ηx)1) = u(ηx) ≤ u(z), as y < u(ηx)1 and ηx ≤ z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' while u′(z) ≤ u′(x) = u′(u′(x)1) < u′(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Hence (z, y) ∈ ≻ \\ ⪰′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' First we estimate Vol(Zη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Write Z0 for [0, x] ∩ DM α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Define the function f(z) = x + (1 − η)(z − x) and note that when z ∈ Z0 then f(z) = ηx + (1 − η)z ∈ [ηx, x] because z ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Note also that f(z) is a convex combination of x and z, so f(z) ∈ DM α RECOVERING UTILITY 25 as the latter is a convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' This shows that Zη = {x} + (1 − η)(Z0 − {x}), and hence that Vol(Zη) = (1 − η)dVol(Z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Now, since Z0 is star shaped we have Vol(Z0) = 1 d � y∈SM α ρ(y, [0, x])d dy ≥ ( α M )dAM α , where AM α is the surface area of SM α and ρ(y, [0, x]) = max{θ > 0 : θy ∈ [0, x] is the radial function of the set [0, x] (see Schneider (2014) page 57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The inequality results from ρ(y, [0, x]) ≥ α/M as xi ≥ α and yi ≤ M for any y ∈ SM α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Now, 1 − η = 1 − ∆/2 + u′(x) u(x) = ∆/2 u(x) ≥ ∆/2 M , as u(x) ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Thus we have that Vol(Zη) ≥ ∆dC′, with C′ = Vol(Z0)/(2M)d > 0, a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Moreover, we have Vol(I) = (∆/2)d as I ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Then we obtain, again using a for- mula derived in Chambers, Echenique, and Lambert (2021), and that q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (x, y)) > 1/2 > q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (y, x)) on (x, y) ∈≻: µ(⪰, ⪰) − µ(⪰′, ⪰) = � 1≻\\≻′(z, y)[q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (z, y)) − q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (y, z))] dλ(z, y) ≥ � Zη×I 1≻\\≻′(z, y)[q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (z, y)) − q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (y, z))] dλ(z, y) ≥ λ(Zλ × I) inf{q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (z, y) − q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (y, z)) : (z, y) ∈ Zη × I} ≥ (∆/2)dC′∆dΘ, where Θ = inf{q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (z, y) − q(⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' (y, z)) : (z, y) ∈ Zη × I} > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' For the rest of this proof, we denote µ(⪰, ⪰∗) by µ(⪰).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The rest of the proof uses routine ideas from statistical learning theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' By standard results (see, for example, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='1 in Boucheron, Bousquet, and Lugosi (2005)), there exists an event E with probability at least 1 − δ on which: sup{|µn(⪰) − µ(⪰)| :⪰∈ P} ≤ E sup{|µn(⪰) − µ(⪰)| :⪰∈ P} + � 2 ln(1/δ) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 26 CHAMBERS, ECHENIQUE, AND LAMBERT Moreover, again by standard arguments (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='2 in Boucheron, Bousquet, and Lugosi (2005)), we also have E sup{|µn(⪰) − µ(⪰)| :⪰∈ P} ≤ 2 E sup{ 1 n ����� � i σi1˜xi⪰yi ����� :⪰∈ P}, where Rn(P) = E sup{ 1 n ����� � i σi1˜xi⪰yi ����� :⪰∈ P} is the Rademacher average of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Now, by the Vapnik-Chervonenkis inequality (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='4 in Boucheron, Bousquet, and Lugosi (2005)), we have that E sup{|µn(⪰) − µ(⪰)| :⪰∈ P} ≤ K � V n , where V is the VC dimension of P, and K is a universal constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' So on the event E, we have we have that sup{|µn(⪰) − µ(⪰)| :⪰∈ P} ≤ K � V/n + � 2 ln(1/δ) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We now combine these statements with Lemmas 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In particular, we let D = d or D = 2d depending on which of the lemmas we invoke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let u∗ ∈ U represent ⪰∗ and un ∈ U represent ⪰n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let ∆ = ρ(u∗, un), a magnitude that depends on the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Then, on the event E, by Lemma 4 or 5, we have that C∆D ≤ µ(⪰∗) − µ(⪰n) = µ(⪰∗) − µn(⪰∗) + µn(⪰∗) − µn(⪰n) + µn(⪰n) − µ(⪰n) ≤ 2K � V n + 2 � 2 ln(1/δ) n , where we have used that µn(⪰∗) − µn(⪰n) < 0 by definition of ⪰n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' This proves the second statement in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' To prove the first statement in the theorem, by Lemmas 4 and 5 again, and using that µn(⪰n) ≥ µn(⪰∗), we have that, for any η > 0, Pr(ρ(u∗, un) > η) ≤ Pr(µ(⪰∗) − µ(⪰n) > CηD) ≤ Pr(µ(⪰∗) − µn(⪰∗) > CηD/2) + Pr(µn(⪰n) − µ(⪰n) > CηD/2) ≤ 2 Pr(sup{|µ(⪰′) − µn(⪰′)| :⪰′∈ P} > CηD/2) → 0 RECOVERING UTILITY 27 as n → ∞ by the uniform convergence in probability result shown in Chambers, Echenique, and Lambert (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' By standard results (see Hildenbrand (1970)), since X is locally compact Polish, the topology of closed convergence is compact metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We will show that for any subsequence of ⪰k, there is a subsubsequence converging to ⪰∗, which will establish that ⪰k→⪰∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' So choose a convergent subsubsequence of the given subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' To simplify notation and with a slight abuse of notation, let us also refer to this subsubsequence as ⪰k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Call its limit ⪰;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' ⪰ is complete as the set of complete relations is closed in the closed convergence topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' It is therefore sufficient to establish that ≻∗⊆≻ and ⪰∗⊆⪰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' First we show that x ≻∗ y implies that x ≻ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' So let x ≻∗ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let U and V be neighborhoods of x and y, respectively, such that x′ ≻∗ y′ for all x′ ∈ U and y′ ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Such neighborhoods exist by the continuity of ⪰∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We prove first that if (x′, y′) ∈ U × V , then there exists N such that x′ ≻n y′ for all n ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Recall that B = ∪{B′ : B′ ∈ Σ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' By hypothesis, there exist x′′ ∈ U ∩ B and y′′ ∈ V ∩ B such that x′′ ≤ x′ and y′ ≤ y′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Each ⪰n is a strong rationalization of the finite experiment of order n, so if {˜x, ˜y} ∈ Σn then ˜x ≻n ˜y implies that ˜x ≻m ˜y for all m ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Since x′′, y′′ ∈ B, there is N is such that {x′′, y′′} ∈ ΣN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Thus x′′ ≻∗ y′′ implies that x′′ ≻n y′′ for all n ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' So, for n ≥ N, x′ ≻n y′, as ⪰n is weakly monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Now we establish that x ≻ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let {(xn, yn)} be an arbitrary sequence with (xn, yn) → (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' By hypothesis, there is an increasing sequence {x′ n}, and a decreas- ing sequence {y′ n}, such that x′ n ≤ xn and yn ≤ y′ n while (x, y) = limn→∞(x′ n, y′ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let N be large enough that x′ N ∈ U and y′ N ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let N′ ≥ N be such that x′ N ≻n y′ N for all n ≥ N′ (we established the existence of such N′ above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Then, for any n ≥ N′ we have that xn ≥ x′ n ≥ x′ N ≻n y′ N ≥ y′ n ≥ yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' By the weak monotonicity of ⪰n, then, xn ≻n yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' The sequence {(xn, yn)} was arbitrary, so (y, x) /∈⪰= limn→∞ ⪰n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Thus ¬(y ⪰ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Completeness of ⪰ implies that x ≻ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' In second place we show that if x ⪰∗ y then x ⪰ y, thus completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' So let x ⪰∗ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We recursively construct sequences xnk, ynk such that xnk ⪰nk ynk and xnk → x, ynk → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 28 CHAMBERS, ECHENIQUE, AND LAMBERT So, for any k ≥ 1, choose x′ ∈ Nx(1/k) ∩ B with x′ ≥ x, and y′ ∈ Ny(1/k) ∩ B with y′ ≤ y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' so that x′ ⪰∗ x ⪰∗ y ⪰∗ y′, as ⪰∗ is weakly monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Recall that ⪰n strongly rationalizes c⪰∗ for Σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' So x′ ⪰∗ y′ and x′, y′ ∈ B imply that x′ ⪰n y′ for all n large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Let nk > nk−1 (where we can take n0 = 0) such that x′ ⪰nk y′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' and let xnk = x′ and ynk = y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Then we have (xnk, ynk) → (x, y) and xnk ⪰nk ynk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Thus x ⪰ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' First, it is straightforward to show that x ≻ y implies x ⪰′ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Because otherwise there are x, y for which x ≻ y and y ≻′ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Take an open neighborhood U about (x, y) and a pair (z, w) ∈ U ∩ (B × B) for which z ≻ w and w ≻′ z, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Symmetrically, we also have x ≻′ y implies x ⪰ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Now, without loss, suppose that there is a pair x, y for which x ≻ y and x ∼′ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' By connectedness and continuity, V = {z : x ≻ z ≻ y} is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Indeed if we assume, towards a contradiction that V = ∅, then {z : x ≻ z} and {z : z ≻ y} are nonempty open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Further, for any z ∈ X, either x ≻ z or z ≻ y (because if ¬(x ≻ z) then by completeness z ⪰ x, which implies that z ≻ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Conclude that {z : x ≻ z} ∪ {z : z ≻ y} = X and each of the sets are nonempty and open (by continuity of the preference ⪰);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' these sets are disjoint, violating connectedness of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' So we conclude that V is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' By continuity of the preference ⪰, V os open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We claim that there is a pair (w, z) ∈ (V × V ) ∩ (B × B) for which w ≻ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' For otherwise, for all (w, z) ∈ V × V ∩ (B × B), w ∼ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Conclude then by continuity that for all (w, z) ∈ V × V , w ∼ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Observe that this implies that, for any w ∈ V , the set {z : w ≻ z ≻ y} = ∅, as if w ≻ z ≻ y, we also have that x ⪰ w ≻ z, from which we conclude x ≻ z, so that z ∈ V and hence z ∼ w, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Observe that {z : w ≻ z ≻ y} = ∅ contradicts the continuity of ⪰ and the connectedness of X (same argument as nonemptyness of V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' see our discussion above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' We have shown that there is (w, z) ∈ (V × V ) ∩ (B × B) for which w ≻ z, so that x ≻ w ≻ z ≻ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Further, we have hypothesized that x ∼′ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' By the first paragraph, we know that x ⪰′ w ⪰′ z ⪰′ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' If, by means of contradiction, we have w ≻′ z, then x ≻′ y, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' So w ∼′ z and w ≻ z, a contradiction to ⪰B×B=⪰′ B×B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' RECOVERING UTILITY 29 References Afriat, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Roth (2012): “Efficiently learning from revealed preference,” in International Workshop on Internet and Network Economics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' 114–127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' RECOVERING UTILITY 33 Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=', and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} +page_content=' Conitzer (2020): “Learning the Valuations of a k-demand Agent,” in International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FJT4oBgHgl3EQfQywo/content/2301.11492v1.pdf'} diff --git a/bNE5T4oBgHgl3EQfew_U/content/tmp_files/2301.05622v1.pdf.txt b/bNE5T4oBgHgl3EQfew_U/content/tmp_files/2301.05622v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..95640a041c3e0935d3c998d8b297086f3f196887 --- /dev/null +++ b/bNE5T4oBgHgl3EQfew_U/content/tmp_files/2301.05622v1.pdf.txt @@ -0,0 +1,951 @@ +Search for large topological gaps in atomic spin chains +on proximitized superconducting heavy metal layers +Philip Beck1,†, Bendeg´uz Ny´ari2,†, Lucas Schneider1, Levente R´ozsa3, Andr´as L´aszl´offy4, Kriszti´an +Palot´as2,4,5, L´aszl´o Szunyogh2,6, Bal´azs Ujfalussy4, Jens Wiebe1 and Roland Wiesendanger1 +1Department of Physics, University of Hamburg, Jungiusstrasse 9A, 20355 Hamburg, Germany. +2Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and +Economics, M˝uegyetem rkp. 3., H-1111 Budapest, Hungary. +3Department of Physics, University of Konstanz, D-78457 Konstanz, Germany. +4Wigner Research Centre for Physics, Institute for Solid State Physics and Optics, H-1525 Bu- +dapest, Hungary. +5ELKH-SZTE Reaction Kinetics and Surface Chemistry Research Group, University of Szeged, +H-6720 Szeged, Hungary. +6ELKH-BME Condensed Matter Research Group, Budapest University of Technology and Eco- +nomics, M˝uegyetem rkp. 3., H-1111 Budapest, Hungary. +† These authors contributed equally to this work. +Abstract +One-dimensional systems comprising s-wave superconductivity with meticulously tuned mag- +netism and spin-orbit coupling can realize topologically gapped superconductors hosting Ma- +jorana edge modes whose stability is determined by the gap’s size. The ongoing quest for +larger topological gaps evolved into a material science issue. However, for atomic spin chains +on superconductor surfaces, the effect of the substrate’s spin-orbit coupling on the system’s +topological gap size is largely unexplored. Here, we introduce an atomic layer of the heavy +metal Au on Nb(110) which combines strong spin-orbit coupling and a large superconducting +gap with a high crystallographic quality enabling the assembly of defect-free Fe chains using +a scanning tunneling microscope tip. Scanning tunneling spectroscopy experiments and den- +sity functional theory calculations reveal ferromagnetic coupling and ungapped YSR bands +in the chain despite of the heavy substrate. By artificially imposing a spin spiral state our cal- +culations indicate a minigap opening and zero-energy edge state formation. The presented +methodology paves the way towards a material screening of heavy metal layers on elemental +superconductors for ideal systems hosting Majorana edge modes protected by large topolog- +ical gaps. +1 +arXiv:2301.05622v1 [cond-mat.supr-con] 13 Jan 2023 + +Main +Inducing spin-orbit coupling (SOC) in nanostructures has recently been of great interest in a +variety of disciplines related to surface science1 due to its close ties with the existence of non- +collinear magnetic states2–4, spin-split surface states5,6, topological surface states7,8 and topologi- +cal superconductivity9–11 which can be accompanied by Majorana bound states (MBS)12–18. Since +the latter is a promising candidate as a building block of topological quantum computation19, sys- +tems which may potentially host MBS have attracted a lot of interest. MBS may be realized in +chains of magnetic atoms, also called atomic spin chains, on s-wave superconductors14,18,20,21, ar- +tificially fabricated by atom manipulation with the tip of a scanning tunneling microscope (STM)22. +Although first experimental realizations, e.g. Fe chains on Re(0001), show signatures of MBS in +scanning tunneling spectroscopy (STS)18, the system suffers from the small energy gap ∆s of the +superconducting rhenium substrate, making a clear allocation of in-gap features difficult20. More +recent results of such atomic spin chains on Nb(110)23–25 circumvent this, but presumably at the +price of lower SOC, which manifests itself in the dominance of collinear magnetic ground states +due to weak Dzyaloshinskii–Moriya interaction (DMI) terms26, and the hybridization of the spa- +tially extended precursors of MBS in experimentally accessible chain lengths24. +Aiming at maintaining the largest ∆s = 1.50 meV of all elemental superconductors combined +with the possibility of atom manipulation which the Nb substrate offer, there are two apparent ap- +proaches to induce a higher SOC in the system. On the one hand, one may think of using atoms +with larger atomic SOC, e.g. rare earth metals, for the formation of the atomic spin chain. On +the other hand, one can try to couple the chain to a heavy metal substrate with potentially larger +SOC, which is grown on Nb(110) and therefore becomes superconducting by proximity. First at- +tempts combining both approaches using Gd atoms on bismuth thin films grown on Nb(110) show +hybridization of the Yu–Shiba–Rusinov (YSR) states of small ensembles of 3-4 Gd atoms27. How- +ever, the assembly of longer chains and a formation of bands from the hybridizing YSR states, +which are both prerequisites for the emergence of topological superconductivity and MBSs, were +not possible for that system. Moreover, it is even unclear, whether a larger SOC in the constituents, +i.e. chain, substrate, or both of them, will lead to a larger SOC in the YSR bands of the hybrid +chain system which is finally relevant for the emergence of topological superconductivity with a +large topological gap and well-defined MBSs isolated at the ends of the chain. +Here, we investigate these questions pursuing the second approach by constructing Fe chains +on ultrathin Au films grown on Nb(110) used as a substrate. Au is well known to exhibit large +SOC5,28 and the proximity to Au has been demonstrated to enhance SOC-induced effects in light +elements, including the scattering rate29, the Rashba spin-splitting30,31, and also the magnetocrys- +talline anisotropy energy32. Moreover, twisted spin textures were predicted to occur around single +Fe atoms33 and in Mn chains34 on Au(111). Since previous LEED studies hint at the possibility +to grow pseudomorphic thin films of Au on Nb(110)35, it is a natural candidate to be used as a +proximitized superconducting heavy metal layer on Nb(110). Combining experimental STS and +density functional theory (DFT) calculations by solving the fully relativistic Dirac–Bogoliubov–de +Gennes (DBdG) equations of the single Fe adatom, dimers and chains, we study (i) whether there +2 + +is indeed a topological gap opening in the YSR bands due to the large SOC in the Au substrate, +and (ii) the effect of a different spin structure in the chain on the topological gap width and the +localization of MBSs. +Monolayer Au on Nb(110) +We first describe the growth and the superconducting properties of the heavy metal layer. We +aimed at the preparation of ultrathin Au films, maintaining the surface structure of Nb(110) as it +offers multiple distinct building directions for artificial chains, which enables some tuning of the +hybridization of YSR states26,36 and, therefore, of the in-gap band structure of such chains23,24. An +overview STM image of the Au/Nb(110) sample after the low-temperature deposition of Fe atoms +is shown in Figure 1a (preparation details in the Methods section). The Nb(110) surface is almost +completely covered by one monolayer (ML) of Au (see sketch in bottom panel of Figure 1a), with +only a few remaining holes. Pseudomorphic growth, as predicted by LEED studies35, is confirmed +by manipulated atom STM images37 (Supplementary Note 1 and Supplementary Figure 1) of the +first ML. Partially, the second ML Au has started to grow at step edges and in the form of a few +free-standing islands. In this ultrathin limit, we find that the energy gap of the superconducting Nb +is fully preserved in the ML Au due to the proximity effect, as demonstrated by the deconvoluted +dI/dV spectrum (see Methods and Supplementary Note 2 for the deconvolution procedure) taken +on the bare ML Au on Nb(110) far from any Fe atom in Figure 1c (gray curve). The energy gap on +the ML Au is of equal size as that of bare Nb(110) at our measurement temperature and the in-gap +dI/dV signal is zero (see Supplementary Figure 2). This behavior is crucial for our experiments, +but might be altered for thicker films as indicated by recent theoretical38,39 and experimental40,41 +studies. +YSR states of single Fe atoms +We continue with the investigation of the YSR states induced by single Fe atoms, i.e. the building +blocks of chains. According to our DFT calculations the Fe adatom has a magnetic moment of +3.57µB with a preferred orientation perpendicular to the Au surface (Supplementary Note 5). A +close-up STM image of the statistically distributed single Fe adatoms is shown in Figure 1b, where +they appear as shallower protrusions on the first ML and as brighter spheres on the second ML. +We restrict ourselves to the first ML since only this layer has large enough terraces to construct +artificial chains. The Fe adatoms are adsorbed on the fourfold coordinated hollow sites on this ML +Au (Supplementary Note 1 and Supplementary Figure 1). Fe adatoms which are adsorbed far from +other adatoms or defects show similar dI/dV spectra as shown in the top panel of Figure 1c (red +curve). This reproducibility is required for a well-defined band formation in bottom-up fabricated +nanostructures made from such individual adatoms. On the adatoms we find two pairs of YSR +3 + +states induced in the gap of the Au ML which are marked by black arrows and greek letters42–44. +One of them is energetically located close to the Au ML gap edge (±α, ±1.23 meV) while the +other one is located close to the Fermi energy EF (±β, ±0.27 meV). We use constant-contour +maps (see Methods) to resolve the spatial distribution of both YSR states43,44, as shown in Fig- +ure 1d. The ±α state has a spatial distribution resembling that of a dyz orbital with two lobes +pointing along the [110] direction. On the other hand, a spatial distribution resembling that of a +dxz orbital extended along the [001] direction is observed for the ±β state. Note that the shapes of +the +β and the −β state are somewhat different since the former has additional faint lobes along +the [110] direction probably indicating contributions from a dx2−y2-like YSR state. The spatial +distributions are explainable by the orbital origin of YSR states26,43,44 and the point group C2v for +the system. However, note that the energetic order of the states is interchanged with respect to the +case of bare Nb(110)26, and that there are no obvious indications for the other two possible, dxy- +and dz2-like, YSR states, which might indicate an overlap of these peaks in energy with the dxz-, +dyz-, or dx2−y2-like YSR states, or that they are hidden in the coherence peaks of the substrate. +In order to further clarify these experimental results we calculated the local density of states +(LDOS) for a single Fe atom on the Au/Nb(110) film as shown in the top panel of Figure 2a +and Figure 2b (see Methods for the calculation details). There are three very close-by, almost +overlapping YSR states in the vicinity of the substrate’s coherence peaks (see also Supplementary +Figure 5a). They correspond to the dz2, dxy and dyz orbitals (see the +1.32 meV map in Figure 2b) +and the spatial distributions of the LDOS around these peaks are very sensitive to the exact en- +ergy (Supplementary Figure 5b). Due to their energetic location and orbital symmetries, we assign +them to the experimental ±α YSR state (c.f. Figure 1c and d) which appear as a single peak in the +dI/dV spectrum due to the finite-temperature smearing. Additionally, there are two most intense +peaks in the calculations which stem from two energetically close-by YSR states near EF where +the one closest to EF corresponds to the dx2−y2 orbital (see the +0.38 meV map in Figure 2b), and +the less intense one further apart from EF to the dxz orbital (see the −0.58 meV map in Figure 2b). +While the latter resembles the experimentally observed −β state, the former has more similarities +to the +β state (c.f. Figure 1d). These theoretical results indicate that the different spatial experi- +mental distributions of the +β and −β states in Figure 1d can be explained by supposing that they +correspond to two different YSR states of dx2−y2 and dxz orbital character which overlap within the +experimental energy resolution, rather than to a single YSR state. The two YSR peaks also overlap +in the theoretical calculations if a larger imaginary part is chosen for the energy, corresponding to +a higher effective temperature. Finally, note that the electron-hole asymmetries in the intensities of +the calculated peaks appear to be inverted compared to the experiment. With the exception of the +dxz-like YSR state, each pair of peaks has a larger electron contribution above EF (Supplementary +Figure 5a). The dxz-like YSR state has a higher electron contribution below EF which implies that +this state has the strongest coupling to the substrate. +4 + +YSR states of Fe dimers +Before we continue with the investigation of Fe dimers, we consider some intuitive ideas about the +most promising orientations of chains built from individual Fe atoms towards the goal of topologi- +cally gapped YSR bands. As found in previous works23–25,45, enabling a sufficient hybridization of +a YSR state which is already close to EF, while, at the same time, minimizing the hybridizations of +all the other YSR states far from EF may lead to a single YSR band overlapping with EF. Together +with SOC, this can be a sufficient condition for the opening of a topologically non-trivial gap in the +lowest-energy band. Starting from the experimentally detected shapes and energies of the α and β +YSR states (Figure 1d) we thus regard chains along the [001] direction as most promising. For this +orientation, we expect weak and strong hybridizations, respectively, for the α and β YSR states +which are far and close to EF. While a manipulation of close-packed dimers along [001] turned out +to be impossible, we were able to tune the system into the above conditions using dimers with a +distance of 2a along [001] (see STM image in Figure 1e). A dI/dV spectrum measured above the +center of the dimer as well as constant-contour maps of the spatial distributions of the three evident +states are displayed in the bottom panel (orange curve) of Figure 1c and Figure 1e, respectively. In +this configuration, the ±α YSR states of the two atoms do not overlap significantly such that they +do not split into hybridized states, but only slightly shift in energy. In contrast, the ±β YSR states +of the two atoms strongly overlap, and split into an energetically higher one with a clear nodal line +in the center between both impurities (±βa) and another energetically lower one with an increased +intensity in the center (±βs). +These experimental conclusions are corroborated by our calculations (bottom panel of Figure 2a +and Figure 2c). Apparently, all five pairs of single-atom YSR states are split, as expected from +previous experimental and theoretical studies26,46. Although based on the orbital decomposition it +is possible to separate all of the ten pairs (Supplementary Figure 6), the splitting of the three YSR +states contributing to the α YSR state is particularly small, in accordance with the experiment, +which makes it hard to resolve them in the total LDOS. In Figure 2c we plot the LDOS maps of +the six most relevant peaks in Figure 2a. We find that the very weakly splitted dyz YSR states at ++1.32 meV and +1.30 meV which are assigned to the experimental α YSR state appear with an +almost identical shape as the single atom dyz YSR state (c.f. Figure 2b). In contrast, the dx2−y2 and +dxz YSR states strongly split into states with larger (at +0.79 meV and −0.53 meV) and smaller +(at +0.04 meV and −0.66 meV) intensities in the center between both impurities and are thus +associated with the experimental ±βs and ±βa YSR states, respectively. We thus conclude, that +while the α YSR states hybridize only very weakly, the β YSR states hybridize strongly and split +into states which resemble anti-symmetric and symmetric linear combinations of the single atom +YSR states47–49, which can be seen as a prerequisite for band formation from the hybridizing β +YSR states. +5 + +Gapless YSR band in ferromagnetic Fe chains on Au monolayer +Having identified a promising orientation and interatomic spacing from the investigation of the +single atom and the dimer above, we move on to study artificial chains with the same interatomic +separation, called 2a − [001] chains in the following. A sketch illustrating this geometry and an +STM image of a nine Fe atoms long Fe9 2a − [001] chain are shown in the top panels of Figure 3a +and b. Spin-polarized measurements of a Fe19 2a−[001] chain indicate that the atoms in this chain +configuration prefer ferromagnetic alignment (Supplementary Note 3). This is also supported by +our DFT calculations (Supplementary Note 5). We found that the DMI is around 10% of the +Heisenberg exchange interaction in the dimer. Although this is not particularly weak, the SOC in +the Au layer additionally induces a very strong out-of-plane on-site anisotropy, which prevents the +formation of spin-spirals and stabilizes a normal-to-plane ferromagnetic spin structure. +A dI/dV line profile (see Methods) was measured in the center of such a chain along its main +axis and is plotted in Figure 3a (bottom panel) alongside the acquired stabilization height profile +(middle panel). The first apparent characteristic of this measurement is the modulation of every +feature with the interatomic spacing of 2a in these chains, which is also visible in the height pro- +file. It should be emphasized that this is not a feature of the chains’ in-gap band structure but is +just due to the lattice-periodic part of the wave function. However, we find additional states with +different well-defined numbers of maxima at increasing energy and also very close to EF as indi- +cated by the labels nβ (nβ − 1) for the numbers of maxima (nodes). Note that all these states have +particle-hole partners occurring on the other side of EF with the same energetic distance to EF and +equal numbers of maxima and nodes. However, they mostly have much smaller intensities such +that they are barely visible. These pairs of states can thus be assigned to confined Bogoliubov- +de-Gennes (BdG) quasiparticles residing in a YSR band induced by the finite magnetic chain in +the superconductor23. To determine the orbital origin of these states, we show dI/dV maps (see +Methods) of the Fe9 2a − [001] chain in Figure 3b. We find that the confined BdG states identified +before in Figure 3a are localized inside the spatial extent of the chain deduced from the STM im- +age (dashed red elliptical circumference). We assign those states to a band formed by the strong +hybridization of the ±β YSR states of the single adatoms as they are expected to be largely local- +ized along the longitudinal axis of the chain. Additionally, there is a state at a similar energy as +the single adatom and dimer ±α YSR states around ±1.09 meV. This state has exactly as many +maxima as there are atoms in the chain, namely 9, which are spatially localized along both sides of +the chain with a similar distance to the chain axis as the lobes of the single adatom and dimer’s ±α +YSR states (c.f. Figure 1d and e). Therefore, we assign this state to the very weakly hybridizing +±α YSR states of the single atom. The state is not observed in the dI/dV line profile of Figure 3a +due to its nodal line along the longitudinal chain axis. +In order to measure the dispersion of the confined BdG states from the β YSR band, we collect +similar dI/dV line profiles as the one in Figure 3a of defect-free chains for lengths ranging from +N = 7 to N = 14 atoms (Fe7 − Fe14, see Supplementary Note 4 and Supplementary Figure 4). +It can be observed that the confined BdG quasiparticle states shift in energy as a function of the +length L = N · d = N · 2a of the chain, as expected from the length-dependent interference +6 + +condition +q = |q| = ±2πn +L +(1) +where n is an integer and |q| is the length of the BdG quasiparticle scattering vector23. For par- +ticular chain lengths, the confined BdG quasiparticle states can be located very close to EF (c.f. +Fe8 and Fe10 in Supplementary Figure 4). We perform one-dimensional fast Fourier transforms +(1D-FFT) of the columns of the dI/dV line profiles at fixed energy E averaging all data sets taken +for chains of multiple lengths, and thereby obtain the dispersion of the scattering vectors E(q) +(Figure 3c). This dispersion is closely linked to the β YSR band structure. We find that this band +has an approximately parabolic dispersion ranging from −0.9 meV at q/2 = 0 to +0.5 meV at +q/2 = π/d. Note that, as already discussed for the dI/dV line profiles above, the particle-hole +partner of this band has a much lower intensity. It is only visible around the Brillouin zone center +(q/2 = 0) in our measurements. Most importantly, the β YSR band smoothly crosses EF without +any indications of a minigap opening. +An overall similar behaviour is found using our ab-initio framework. We performed calculations +for 2a − [001] chains of lengths ranging from 9 to 19 Fe atoms with ferromagnetic spin alignment +(Supplementary Note 7 and Supplementary Figure 7). Exemplarily, the calculated LDOS along +a Fe9 chain is shown in Figure 4a and can be directly compared to the measured line profile in +Figure 3a. The band formation of the YSR states can clearly be observed in a wide range of the +substrate gap in the form of LDOS lines with a well-defined number of maxima along the chain +as indicated in the figure. In Figure 4b we present the corresponding spatial distributions of the +LDOS of the Fe9 chain in the form of two-dimensional maps for a selection of confined BdG states +with the indicated dominant orbital characters and numbers of maxima (see the Methods section +for calculation details). The states closest to the substrate’s coherence peaks with nyz = 2 (and +admixed nyz = 4) and nz2 = 3 maxima have dyz and dz2 orbital characters, respectively. They +reside in a very narrow band formed by the weakly interacting α YSR states of the Fe atoms (Fig- +ure 2), which explains the low dispersion of this band. On the contrary, wide bands are formed by +the strong hybridization of the β YSR states, i.e. a dx2−y2 YSR band (between -0.2 meV and +1.1 +meV) having high intensities on both sides along the longitudinal axis of the chain and a dxz YSR +band (between -0.8 meV and 0 meV) characterized by high intensities between the atoms of the +chain (Figure 4a,b). In order to deduce the dispersions of these YSR bands from the theoretical +calculations we applied the same 1D-FFT method as in the experiment (see also Ref. 23), and +averaged over chains containing 9, 11, 13, 14, 17 and 19 Fe atoms (Supplementary Figure 7). The +result is plotted in Figure 4c and can be compared to the experimental dispersion in Figure 3c. +The most characteristic, broad bands are the dx2−y2 and dxz YSR bands between -0.2 meV and +1.1 meV and between -0.8 and 0 meV, respectively. While the energy range of the dxz YSR band +agrees reasonably well with that of the experimental β band, the dx2−y2 YSR band is probably not +detected significantly in the experimental data (Figure 3c). The latter might be explained by the +small intensity of the experimental +β state (Figure 1d) which is not reproduced by the calcula- +tions (Figure 2a). Most importantly, the theoretical study confirms the lack of a detectable minigap +at EF in the YSR bands. +7 + +Minigap and end states in spin spiral Fe chains on Au monolayer +At first sight, the missing minigap seems surprising. For similar ferromagnetic chains on the lighter +substrates Nb(110) and Ta(110) there are already clear indications for the openings of topological +minigaps 23,50. It is widely accepted that topological minigaps hosting MBSs can open in the +quasiparticle spectrum of one-dimensional helical spin systems being proximity-coupled to a con- +ventional s-wave superconductor 14,15,18,51. For ferromagnetic chains, this phenomenon has been +attributed to a Rashba-type SOC induced by the substrate52, which is equivalent to a spin spiral +structure without SOC in a single-band tight-binding model53. As outlined in the introduction +above, the heavier material Au is well known to exhibit large SOC5,28. However, as our experi- +ments and calculations show, it obviously does not induce a spin-spiral state in the Fe chain, and +likewise does not induce a SOC of sufficient strength in the YSR bands of the ferromagnetic chain +to open a detectable minigap. In order to trace whether we can still force the system into a state with +a large topological gap ∆ind just by artificially imposing a suitable non-collinear spin state onto the +chain, we performed calculations for the same chains as before, but now imposing a helical spin +spiral state (Figure 5, Supplementary Note 8, and Supplementary Figure 8). The configuration of +the spin spiral was such that the first Fe site had its spin pointing along the positive z direction and +then each spin is rotated by 90◦ around the chain axis when moving along the chain (Figure 5a). +Indeed, there are two significant features which emerge in the LDOS of the spin-spiral chain with +19 iron atoms (Figure 5b), which were absent in the LDOS of the ferromagnetic chain (Figure 4a). +First, a minigap at EF opens up between −0.22 meV and +0.22 meV. Second, inside this minigap +a single state can be observed at EF with a pronounced intensity localized at the ends of the chain. +This state has an electron-hole ratio of 1 and is robust against the variation of the chain length from +9 to 19 atoms as illustrated in Figure 5b and Supplementary Figure 8. The strongly different spa- +tial LDOS distribution of the zero-energy state compared to that of some exemplary higher-energy +states is further illustrated in Figure 5c. The former is localized over a few atoms at the two ends +of the chain, while the latter states outside the minigap are extended along the whole chain. It +should be mentioned that all these states, both the zero-energy one as well as those outside of the +minigap show the same orbital character, indicating that the minigap emerges from the dx2−y2 YSR +states of the ferromagnetic chain. The induced minigap of 2∆ind = 0.44 meV width and the narrow +spectral weight around EF stemming from the zero-energy end states are also clearly visible in the +dispersion of the scattering wave vectors deduced from the averaged 1D-FFTs of the LDOSs of +chains of different lengths (Figure 5d). Thus, the calculations show evidence for the formation of a +topological, most probably p-wave-like, minigap which hosts a MBS, if the Fe chain on Au(111) is +forced into a helical spin spiral state, indicating that the absence of the non-collinear ground state +is the limiting factor of this experimental system. +8 + +Conclusions and outlook +In summary, our combined experimental and theoretical investigation shows that in contrast to +what might be suggested by simplified tight-binding models52,53, a strong substrate SOC alone +generally is not a sufficient condition for the opening of a topological minigap in a ferromagnetic +chain in contact to an s-wave superconductor, since the SOC has to exist in the lowest-energy YSR +band. In fact, first-principles calculations of the magnetic interaction parameters in ultrathin film +systems have demonstrated that also the connection between the formation of a spin-spiral state +and SOC is considerably more complicated. In particular, the DMI preferring a non-collinear spin +alignment is typically weak when a 3d transition metal is deposited on a Au surface compared +to other 5d substrates54–56, which may be tentatively attributed to the fully occupied 5d band of +Au having a reduced effect on the DMI. Proximity to a Au layer is known to give rise to strong +Heisenberg exchange interactions and anisotropy32 in the magnetic layer instead, both of which +prefer a collinear spin alignment and the latter being induced by the SOC. Our results indicate that, +similarly to the competition between DMI and anisotropy terms in the formation of non-collinear +spin structures, the role of SOC may be more complex for inducing topological superconductivity +in the YSR bands of ferromagnetic spin chains. +Our study proves that it is experimentally possible to grow proximitized ultrathin heavy metal lay- +ers on a superconductor with a large Tc that can be used as a substrate for the deposition of transi- +tion metal atoms and to construct defect-free one-dimensional structures with an excellent quality, +enabling the tailoring of YSR bands. Further, we presented an ab-initio method that accurately +reproduces the main LDOS features observed in the experiments. Our work thus demonstrates the +theoretical feasibility of an ab-initio screening of other combinations of transition metal chains +on heavy metal thin films on bulk superconductors in order to find the optimal conditions for the +opening of a large topological minigap. +Methods +STM and STS measurements +The experiments were performed in a custom home-built ultra-high vacuum system, equipped with +an STM setup, which was operated at a temperature of 320 mK57. STM images were obtained by +applying a bias voltage Vbias to the sample upon which the tip-sample distance is controlled by a +feedback loop such that a constant current I is achieved. dI/dV spectra were obtained in open +feedback mode after stabilizing the tip at Vstab = 6 mV and Istab = 1 nA using a standard lock-in +technique with an AC voltage Vmod = 20 µV (rms value) of frequency fmod = 4142 Hz added +to the ramped Vbias. If other stabilization parameters were used for a particular measurement, it is +indicated in the respective figure caption. dI/dV maps were obtained by measuring dI/dV spectra +9 + +on a predefined spatial grid, which was positioned over the structure of interest, and selecting a slice +at a given voltage. Typical measurement parameters are the same as for individual dI/dV spectra. +dI/dV line profiles are measured similarly to dI/dV maps, with the exception that the spatial +grid is one-dimensional. Constant-contour maps were obtained by repeated scanning of individual +lines of STM images. First, each line is measured as it would be the case in a regular STM image. +The z-signal of this sweep is saved. Then, the feedback is turned off, the bias voltage Vbias is set +to a predefined value of interest, and for the next sweep on the same scan line the dI/dV signal is +measured while restoring the previously recorded z-signal. +A mechanically sharpened and in-situ flashed (50 W) bulk Nb tip was used for all measurements. +While the usage of a superconducting tip is a crucial factor for obtaining a very good energy +resolution, it has the downside that the dI/dV spectra are convolutions of the tip and sample DOSs. +However, we can determine the superconducting gaps of the tip and the sample, and deconvolute +the dI/dV spectra. This process is described in Supplementary Note 2 and is performed for every +spectrum in the main manuscript. +Sample preparation +A Nb(110) single crystal with a purity of 99.99 % was transferred into the ultra-high vacuum +chamber. The sample was cleaned by cycles of Ar ion sputtering and flashes up to 2400 °C, which +results in a clean surface with only few oxygen impurities remaining58. We established flashing +parameters which clean the surface of oxygen, and checked the results by STM. Once this cleaning +procedure was reproducible with the given parameters, we evaporated Au from an e-beam evapora- +tor (EFM3 by FOCUS GmbH) equipped with a Au rod (99.99 % purity). Following this procedure, +we achieved flat and spatially extended films (Figure 1a). +Fe was evaporated onto the surface from a carefully outgassed Fe rod using a second e-beam evap- +orator while keeping the sample temperature below T = 10 K to avoid clustering and diffusion and +thus achieve a random distribution of single Fe adatoms (Figure 1b). From Supplementary Note 1, +we conclude that the Au film grows pseudomorphically and that the Fe atoms are adsorbed in the +fourfold coordinated hollow sites in the center of four Au atoms. This is further supported by the +similarity of the spatial distributions of the YSR states of the Fe/Au/Nb(110) system compared to +the Mn/Nb(110) system26. +STM tip-induced atom manipulation22,37 is used to position individual Fe atoms and construct +artificial structures, such as dimers and chains. The structures built in this study have sufficient +interatomic spacing to unambiguously identify the positions of the individual atoms forming the +structure using STM images. We restrict the investigations here to Fe atoms positioned on the first +ML of Au. Fe atoms on the first and second ML can easily be distinguished by their apparent +height. In the top part of Figure 1b one can see that an Fe atom on the second ML appears as a +bright spherical protrusion, while an Fe atom on the first ML is more shallow and has a relatively +irregular shape. Thus, we can be sure that all of the experiments were carried out on the first ML. +10 + +First-principles calculations +The calculations were performed in terms of the Screened Korringa-Kohn-Rostoker method (SKKR), +based on a fully relativistic Green’s function formalism by solving the Dirac equation for the nor- +mal state32 and the Dirac-Bogoliubov-de Gennes (DBdG) equation for the superconducting state +within multiple scattering theory (MST)46,59. The impurities are included within an embedding +scheme60, being an efficient method to address the electronic and magnetic properties or the in-gap +spectra of real-space atomic structures without introducing a supercell. The system consists of +seven atomic layers of Nb, a single atomic layer of Au and four atomic layers of vacuum between +semi-infinite bulk Nb and semi-infinite vacuum. The Fe adatoms are placed in the hollow position +in the vacuum above the Au layer and relaxed towards the surface by 21%, while the top Au layer +is also relaxed inwards by 2%. The relaxations are obtained from total-energy minimization in a +VASP61–63 calculation for a single Fe adatom and are used for the dimer and all the chains. For +the potentials we employ the atomic sphere approximation (ASA), the normal state is calculated +self-consistently in the local density approximation (LDA) as parametrized by Vosko et al.64 The +partial waves within MST are treated with an angular momentum cutoff of ℓmax = 2. In the self- +consistent normal state calculations we used a Brillouin zone (BZ) integration with 253 k points in +the irreducible wedge of the BZ and a semicircular energy contour on the upper complex half plane +with 16 points for energy integration. In order to take into account charge relaxation around the +magnetic sites, the single impurity and the 2a − [001] dimer are calculated by including a neigh- +borhood containing 48 and 84 atomic sites, respectively, corresponding to a spherical radius of +r = 1.66 a. The atomic chains are calculated with a somewhat smaller neighborhood correspond- +ing to 2 atomic shells or a spherical radius of r = 1.01 a around the Fe atoms. This way our largest +atomic cluster in the calculation with 19 Fe chain atoms contained 339 atomic sites. After having +obtained the self-consistent potentials in the normal state, the superconducting state is simulated +within single-shot calculations by solving the DBdG equation with the experimental band gap used +as the pairing potential in the Nb layers46. In the case of the single impurity and the dimer, the BZ +integration for the host Green’s function is performed by using the same k mesh as for the normal +state, but in order to achieve convergence for the chains we had to increase the number of k points +up to 1891 in the irreducible wedge of the BZ. A sufficient energy resolution of the LDOS in the +superconducting gap is acquired by considering 301 energy points between ±1.95 meV with an +imaginary part of 13.6 µeV related to the smearing of the resulting LDOS. Both the electron and +the hole components of the LDOS are calculated, but in this paper we present only the electron part +leading to the asymmetry of the spectrum as also seen in the experiments. Due to the ASA used +in our method we obtain the LDOS for each atomic site of the cluster averaged inside the atomic +spheres. The orbital resolution of the YSR states can be determined based on the orbital-resolved +LDOS of the Fe atoms. Since the canonical d orbitals hybridize due to the symmetry of the clus- +ter and due to SOC, we assign the labels based on the orbital which has the largest contribution +to the given peak in the LDOS. In addition, in order to mimic the constant-contour maps in the +experiments, we evaluate the spatial distribution of the LDOS. These LDOS maps are taken from +the first vacuum layer above the surface in which the magnetic atoms are embedded, reflecting the +orbital characteristics obtained from the resolution of the LDOS of the Fe atoms. In order to better +11 + +reproduce the experimental constant-contour maps taken from the vacuum region, the LDOS of +the magnetic sites are replaced by the average LDOS over the two vacuum sites (empty spheres) +closest to them in the layer above. 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Ab-initio simulations of materials using VASP: Density-functional theory and be- +yond 29, 2044–2078 (2008). +64. Vosko, S. H., Wilk, L. & Nusair, M. Accurate spin-dependent electron liquid correlation +energies for local spin density calculations: a critical analysis. Canadian Journal of Physics +58, 1200–1211 (1980). +65. Gouraud, H. Continuous shading of curved surfaces. IEEE Transactions on Computers 100, +623–629 (1971). +Acknowledgements +P.B., R.W., and J.W. gratefully acknowledge funding by the Deutsche Forschungs- +gemeinschaft (DFG, German Research Foundation) – SFB-925 – project 170620586. L.S., R.W., and J.W. +gratefully acknowledge funding by the Cluster of Excellence ’Advanced Imaging of Matter’ (EXC 2056 - +project ID 390715994) of the DFG. R.W. gratefully acknowledges funding of the European Union via the +ERC Advanced Grant ADMIRE (project no. 786020). B.Ny., L.R., A.L., K.P., L.Sz. and B.U. acknowledge +financial support by the National Research, Development, and Innovation Office (NRDI) of Hungary under +Project Nos. FK124100 and K131938. B.Ny. and L.Sz. acknowledge support by the Ministry for Innova- +tion and Technology and the NRDI Office within the Quantum Information National Laboratory of Hungary. +B.Ny. and K.P. acknowledge the support by the ´UNKP-21-3 and the ´UNKP-21-5 New National Excellence +16 + +Program of the Ministry for Innovation and Technology from the source of the National Research, Devel- +opment and Innovation Fund. K.P. acknowledges the J´anos Bolyai Research Scholarship of the Hungarian +Academy of Sciences. +Author contributions +P.B., L.S. and J.W. conceived the experiments. P.B. and L.S. performed the mea- +surements and P.B. analyzed the experimental data together with J.W.. L.R. performed the VASP calcu- +lations. B.N. performed the SKKR calculations and discussed the data with L.R., A.L., K.P., L.Sz. and +B.U., and A.L. also contributed to the spin model calculations. P.B. and B.N. prepared the figures and wrote +the first version of the manuscript. L.S., L.R., A.L., K.P., L.Sz., B.U., J.W. and R.W. contributed to the +discussions and the finalization of the manuscript. +Competing Interests +The authors declare no competing interests. +Correspondence +Correspondence and requests for materials should be addressed to J. Wiebe (email: +jwiebe@physnet.uni-hamburg.de). +17 + +Figure 1 | Measured YSR states of Fe atoms and dimers on monolayer Au on Nb(110). +a, STM image of an ultrathin film of Au on Nb(110) with an approximate coverage of 1 ML Au +and additionally deposited Fe atoms. An extracted line profile along the red line is displayed in +the bottom panel (red curve). Rectangles show the surface composition underneath the line profile. +The white scale bar has a length of 20 nm (Vbias = 50 mV and I = 100 pA). b, STM image of +randomly distributed Fe atoms on 1 ML Au (bottom) and 2 ML Au (top). The white scale bar +has a length of 4 nm (Vbias = 6 mV and I = 3 nA). c, Deconvoluted dI/dV spectra measured on +the Au/Nb(110) substrate (gray), a single Fe atom (red, top panel), and the center of a dimer of +Fe atoms spaced by 2a along the [001] direction (orange, bottom panel). Black arrows and Greek +letters label YSR states. Gray ticks mark the position of the superconducting energy gap of the +sample ∆s = 1.50 meV as determined in Supplementary Note 2 (Vstab = 6 mV, Istab = 1 nA and +Vmod = 20 µV). d and e, STM images and constant-contour maps of a single Fe atom (d) and +a Fe dimer (e) spaced by 2a along the [001] direction. Constant-contour maps were obtained for +every energy for which we identified a peak in the corresponding spectrum of c as indicated by the +corresponding Greek letters. White arrows indicate crystallographic directions, red dashed circles +depict the positions of the Fe atoms as determined from the topographies, and white scale bars +represent a length of 1 nm (Vbias = 6 mV and I = 1 nA). +18 + +- ++△s +c +d +1.23 meV ++ 0.27 meV +HI +Au/Nb(110) +1.5 +single Fe atom +Fe atom +(n'que) ^p/lp +2a-[0011 dimer +z (pm) 37 +topogr. +-β +^p/p +- 1.23 meV +- 0.27 meV +0.5E +0- +0 +0.13 +decon. ++β ++α +[001], x +LO +0.0 +AU +-α +-β +Au +Nb(110) +-βs +Nb(110) +e ++ 1.13 meV +0.53.mev ++ 0.35 meV +0 +distance (nm) +28 +2a-[001] dimer +(arb. +0 +z (pm)72 +540 +> +0.4 +topogr. +1p//p ++βs +20- ++α +ed- ↑ +I+βa +z (pm) +10], +- 1.13 meV +- 0.53 meV +- 0.35 meV ++α +[001], +0.0 +-2 -△s -1 +0 +1 +△s 2 +E- E- (meV) +5 +20- +-B +-βsFigure 2 | Calculated YSR states of Fe atoms and dimers on monolayer Au on Nb(110). +a, Electron component of the LDOS of the single Fe atom (top panel) and the ferromagnetic 2a − +[001] dimer (bottom panel). Gray dashed vertical lines indicate the superconducting gap of the +substrate ∆s. b, Spatial distributions of the three YSR peaks of the Fe atom with the highest +intensities visible in the top panel of a. c, Spatial distributions of the six YSR peaks of the FM +2a − [001] dimer with the highest intensities visible in the bottom panel of a. The energies are +indicated in the bottom of the panels of b and c. Red circles indicate the positions of Fe atoms and +the white scale bars correspond to a distance of a. +19 + +a +Fe atom +2a-[001] dimer +b ++△ +LO +LDOS (arb.u.) +HI +3500 ++1.32 meV +-0.58 meV ++0.38 meV +3000 ++ +(n +600个 +(arb. +400 +110] +LDOS ( +[0011. x +200 +0 +c +2000 ++1.32 meV +-0.53 meV ++0.79 meV +1500 +LDOS (arb. u.) +400个 +200 ++1.30 meV +-0.66 meV ++0.04 meV +0 +[0011. +1 +0 ++△ +E- E (meV)Figure 3 | Measured dispersion of BdG quasiparticles in Fe chains on Au monolayer. +a, Deconvoluted dI/dV line profile (bottom panel) and corresponding topographic line profile +(middle panel) measured along the longitudinal axis of a Fe9 2a − [001] chain, as illustrated in the +top panel. Black arrows mark the energies in the bottom panel at which nβ maxima as indicated +are observed along the chain. The subscript of this label refers to the orbital origin of this state +(Vstab = 6 mV, Istab = 1 nA, Vmod = 20 µV). b, The top panel shows an STM image of a Fe9 +2a−[001] chain and the lower panels are dI/dV maps of this chain, obtained at energies indicated +in the top right corner of each panel. The maps are labeled by nβ in a similar fashion as the states +in a. The red lines mark the spatial extent of the chain in the STM image. The white scale bar +represents a length of 1 nm (Vstab = −6 mV, Istab = 1 nA, Vmod = 20 µV). c, Averaged energy- +wise 1D-FFT obtained from dI/dV line profiles of FeN 2a − [001] chains with lengths N ranging +from seven to fourteen atoms. Prior to the 1D-FFT, the spectra were deconvoluted. All dI/dV line +profiles were obtained with the following parameters: Vstab = 6 mV, Istab = 1 nA, Vmod = 20 µV. +20 + +a +b +76 +topogr +c +0.7 +V/FFTI (arb.u.) +1.6 +(pm) +50 +1.5 +(ud) ++0.42 meV +0 +1.0 +N +HI +nβ= ++0.02meV +2 +0.5 +三(n'que) ^p/p +nβ = +nβ=3 +0.02meV +0.0 +F +2 +4 +山 -0.5 +0.47meV +33 +0 +decon. +-1.0 +LO +-0.77 mev +TB +1 +-1.5 +LO +1.09 meV [110] +-1.0 +-0.5 +0.0 +0:5 +1.0 +-2 +q/2 (元/d) +0.0 +2.5 +5.0 +[001] +x (nm)Figure 4 | Calculated dispersion of BdG quasiparticles in ferromagnetic Fe chains on Au +monolayer. +a, Electron component of the LDOS extracted along the longitudinal axis of a ferromagnetic Fe9 +2a − [001] chain, calculated on the Fe and the vacuum sites in between. b, Spatial distributions of +the LDOS evaluated in the first vacuum layer above the chain. Energies are indicated in the top +right corners of each panel, while the numbers of maxima and the orbital origins of the states are +indicated in the top left corners. They correspond to the states marked by the same arrows and +numbers in a. The nyz state has two dominant Fourier components nyz = 2 and nyz = 4, where +the former is dominating. The white scale bar has a length of 2a. c, Dispersion of scattering wave +vectors extracted from the calculated LDOSs of ferromagnetically coupled FeN 2a − [001] chains +and averaged for lengths N of 9, 11, 13, 14, 17 and 19 atoms (Supplementary Figure 7). The green +dashed lines in panels a and c indicate the energy gap of the superconducting substrate. +21 + +b +a +0 LDOS(arb. u.) 800 +LO +HI +c +LDOS (arb.u.) +2 +VIFFTI (arb.u.) +20 +Feg +1.30 meV +1.5 +1.5 +1.22 meV +1.0 +1.0 +nz² = 3 +nx2.y2 = 5 +0.43 meV +0.5 +0.5 +> nx2.y2 = 5 +0.21 meV +0.0 +0.0 +山 -0.5 +nx2.y2 = 3 +0.00 meV +山-0.5 +-1.0 +Nx2-y2 = +-0.13 meV +-1.0 +-1.5 + 4 +-0.51 meV +-1.5 +110], +0.0 +2.5 +5.0 +-1.0 +-0.5 +0.0 +0:5 +1.0 +[001], x +x (nm) +q/2 (π/d)Figure 5 | Minigap and MBS enforced by helical spin spirals in Fe chains on Au monolayer. +a, Illustration of the helical spin spiral state with a rotation angle of 90◦ of a chain containing 19 +Fe atoms. b, Electron component of the LDOS extracted along the longitudinal axis of a Fe19 +2a − [001] chain in the helical spin spiral state shown in panel a. c, The spatial distributions of the +LDOS evaluated in the first vacuum layer above the Fe19 2a − [001] chain at the energies indicated +in the top right corner of each panel. d, Dispersion of scattering wave vectors averaged from the +calculated LDOSs of the FeN 2a − [001] chains including N = 9, 11, 13, 14, 17 and 19 Fe atoms +(Supplementary Figure 8). The green and red dashed lines in b and d indicate the substrate gap +and the minigap, respectively. +22 + +b +d +a +2 +V/IFFTI (arb.u.) +20 +0 LDOS (arb. u.) 700 +1.5 +(meV) +1.0 +0 +(meV) +0.5 +E-1 +0.0 +0 +5 +10 +Distance (nm) +山-0.5 +c +0.42 meV +-1.0 +0.31 meV +0.00 meV +-1.5 +-0.26 meV +110], +LO LDOS (arb.u.)HI +-1.0 +-0.5 +0.0 +0:5 +1.0 +[001], +q/2 (元/d) \ No newline at end of file diff --git a/bNE5T4oBgHgl3EQfew_U/content/tmp_files/load_file.txt b/bNE5T4oBgHgl3EQfew_U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8d9dba5534a1ad0e8d4284d3569b3e51917e7423 --- /dev/null +++ b/bNE5T4oBgHgl3EQfew_U/content/tmp_files/load_file.txt @@ -0,0 +1,946 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf,len=945 +page_content='Search for large topological gaps in atomic spin chains on proximitized superconducting heavy metal layers Philip Beck1,†, Bendeg´uz Ny´ari2,†, Lucas Schneider1, Levente R´ozsa3, Andr´as L´aszl´offy4, Kriszti´an Palot´as2,4,5, L´aszl´o Szunyogh2,6, Bal´azs Ujfalussy4, Jens Wiebe1 and Roland Wiesendanger1 1Department of Physics, University of Hamburg, Jungiusstrasse 9A, 20355 Hamburg, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 2Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics, M˝uegyetem rkp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', H-1111 Budapest, Hungary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 3Department of Physics, University of Konstanz, D-78457 Konstanz, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 4Wigner Research Centre for Physics, Institute for Solid State Physics and Optics, H-1525 Bu- dapest, Hungary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 5ELKH-SZTE Reaction Kinetics and Surface Chemistry Research Group, University of Szeged, H-6720 Szeged, Hungary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 6ELKH-BME Condensed Matter Research Group, Budapest University of Technology and Eco- nomics, M˝uegyetem rkp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', H-1111 Budapest, Hungary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' † These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Abstract One-dimensional systems comprising s-wave superconductivity with meticulously tuned mag- netism and spin-orbit coupling can realize topologically gapped superconductors hosting Ma- jorana edge modes whose stability is determined by the gap’s size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The ongoing quest for larger topological gaps evolved into a material science issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' However, for atomic spin chains on superconductor surfaces, the effect of the substrate’s spin-orbit coupling on the system’s topological gap size is largely unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Here, we introduce an atomic layer of the heavy metal Au on Nb(110) which combines strong spin-orbit coupling and a large superconducting gap with a high crystallographic quality enabling the assembly of defect-free Fe chains using a scanning tunneling microscope tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Scanning tunneling spectroscopy experiments and den- sity functional theory calculations reveal ferromagnetic coupling and ungapped YSR bands in the chain despite of the heavy substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' By artificially imposing a spin spiral state our cal- culations indicate a minigap opening and zero-energy edge state formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The presented methodology paves the way towards a material screening of heavy metal layers on elemental superconductors for ideal systems hosting Majorana edge modes protected by large topolog- ical gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='05622v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='supr-con] 13 Jan 2023 Main Inducing spin-orbit coupling (SOC) in nanostructures has recently been of great interest in a variety of disciplines related to surface science1 due to its close ties with the existence of non- collinear magnetic states2–4, spin-split surface states5,6, topological surface states7,8 and topologi- cal superconductivity9–11 which can be accompanied by Majorana bound states (MBS)12–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Since the latter is a promising candidate as a building block of topological quantum computation19, sys- tems which may potentially host MBS have attracted a lot of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' MBS may be realized in chains of magnetic atoms, also called atomic spin chains, on s-wave superconductors14,18,20,21, ar- tificially fabricated by atom manipulation with the tip of a scanning tunneling microscope (STM)22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Although first experimental realizations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Fe chains on Re(0001), show signatures of MBS in scanning tunneling spectroscopy (STS)18, the system suffers from the small energy gap ∆s of the superconducting rhenium substrate, making a clear allocation of in-gap features difficult20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' More recent results of such atomic spin chains on Nb(110)23–25 circumvent this, but presumably at the price of lower SOC, which manifests itself in the dominance of collinear magnetic ground states due to weak Dzyaloshinskii–Moriya interaction (DMI) terms26, and the hybridization of the spa- tially extended precursors of MBS in experimentally accessible chain lengths24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Aiming at maintaining the largest ∆s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='50 meV of all elemental superconductors combined with the possibility of atom manipulation which the Nb substrate offer, there are two apparent ap- proaches to induce a higher SOC in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' On the one hand, one may think of using atoms with larger atomic SOC, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' rare earth metals, for the formation of the atomic spin chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' On the other hand, one can try to couple the chain to a heavy metal substrate with potentially larger SOC, which is grown on Nb(110) and therefore becomes superconducting by proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' First at- tempts combining both approaches using Gd atoms on bismuth thin films grown on Nb(110) show hybridization of the Yu–Shiba–Rusinov (YSR) states of small ensembles of 3-4 Gd atoms27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' How- ever, the assembly of longer chains and a formation of bands from the hybridizing YSR states, which are both prerequisites for the emergence of topological superconductivity and MBSs, were not possible for that system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Moreover, it is even unclear, whether a larger SOC in the constituents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' chain, substrate, or both of them, will lead to a larger SOC in the YSR bands of the hybrid chain system which is finally relevant for the emergence of topological superconductivity with a large topological gap and well-defined MBSs isolated at the ends of the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Here, we investigate these questions pursuing the second approach by constructing Fe chains on ultrathin Au films grown on Nb(110) used as a substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Au is well known to exhibit large SOC5,28 and the proximity to Au has been demonstrated to enhance SOC-induced effects in light elements, including the scattering rate29, the Rashba spin-splitting30,31, and also the magnetocrys- talline anisotropy energy32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Moreover, twisted spin textures were predicted to occur around single Fe atoms33 and in Mn chains34 on Au(111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Since previous LEED studies hint at the possibility to grow pseudomorphic thin films of Au on Nb(110)35, it is a natural candidate to be used as a proximitized superconducting heavy metal layer on Nb(110).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Combining experimental STS and density functional theory (DFT) calculations by solving the fully relativistic Dirac–Bogoliubov–de Gennes (DBdG) equations of the single Fe adatom, dimers and chains, we study (i) whether there 2 is indeed a topological gap opening in the YSR bands due to the large SOC in the Au substrate, and (ii) the effect of a different spin structure in the chain on the topological gap width and the localization of MBSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Monolayer Au on Nb(110) We first describe the growth and the superconducting properties of the heavy metal layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' We aimed at the preparation of ultrathin Au films, maintaining the surface structure of Nb(110) as it offers multiple distinct building directions for artificial chains, which enables some tuning of the hybridization of YSR states26,36 and, therefore, of the in-gap band structure of such chains23,24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' An overview STM image of the Au/Nb(110) sample after the low-temperature deposition of Fe atoms is shown in Figure 1a (preparation details in the Methods section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The Nb(110) surface is almost completely covered by one monolayer (ML) of Au (see sketch in bottom panel of Figure 1a), with only a few remaining holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Pseudomorphic growth, as predicted by LEED studies35, is confirmed by manipulated atom STM images37 (Supplementary Note 1 and Supplementary Figure 1) of the first ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Partially, the second ML Au has started to grow at step edges and in the form of a few free-standing islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' In this ultrathin limit, we find that the energy gap of the superconducting Nb is fully preserved in the ML Au due to the proximity effect, as demonstrated by the deconvoluted dI/dV spectrum (see Methods and Supplementary Note 2 for the deconvolution procedure) taken on the bare ML Au on Nb(110) far from any Fe atom in Figure 1c (gray curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The energy gap on the ML Au is of equal size as that of bare Nb(110) at our measurement temperature and the in-gap dI/dV signal is zero (see Supplementary Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' This behavior is crucial for our experiments, but might be altered for thicker films as indicated by recent theoretical38,39 and experimental40,41 studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' YSR states of single Fe atoms We continue with the investigation of the YSR states induced by single Fe atoms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' the building blocks of chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' According to our DFT calculations the Fe adatom has a magnetic moment of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='57µB with a preferred orientation perpendicular to the Au surface (Supplementary Note 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' A close-up STM image of the statistically distributed single Fe adatoms is shown in Figure 1b, where they appear as shallower protrusions on the first ML and as brighter spheres on the second ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' We restrict ourselves to the first ML since only this layer has large enough terraces to construct artificial chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The Fe adatoms are adsorbed on the fourfold coordinated hollow sites on this ML Au (Supplementary Note 1 and Supplementary Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Fe adatoms which are adsorbed far from other adatoms or defects show similar dI/dV spectra as shown in the top panel of Figure 1c (red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' This reproducibility is required for a well-defined band formation in bottom-up fabricated nanostructures made from such individual adatoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' On the adatoms we find two pairs of YSR 3 states induced in the gap of the Au ML which are marked by black arrows and greek letters42–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' One of them is energetically located close to the Au ML gap edge (±α, ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='23 meV) while the other one is located close to the Fermi energy EF (±β, ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='27 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' We use constant-contour maps (see Methods) to resolve the spatial distribution of both YSR states43,44, as shown in Fig- ure 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The ±α state has a spatial distribution resembling that of a dyz orbital with two lobes pointing along the [110] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' On the other hand, a spatial distribution resembling that of a dxz orbital extended along the [001] direction is observed for the ±β state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Note that the shapes of the +β and the −β state are somewhat different since the former has additional faint lobes along the [110] direction probably indicating contributions from a dx2−y2-like YSR state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The spatial distributions are explainable by the orbital origin of YSR states26,43,44 and the point group C2v for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' However, note that the energetic order of the states is interchanged with respect to the case of bare Nb(110)26, and that there are no obvious indications for the other two possible, dxy- and dz2-like, YSR states, which might indicate an overlap of these peaks in energy with the dxz-, dyz-, or dx2−y2-like YSR states, or that they are hidden in the coherence peaks of the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' In order to further clarify these experimental results we calculated the local density of states (LDOS) for a single Fe atom on the Au/Nb(110) film as shown in the top panel of Figure 2a and Figure 2b (see Methods for the calculation details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' There are three very close-by, almost overlapping YSR states in the vicinity of the substrate’s coherence peaks (see also Supplementary Figure 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' They correspond to the dz2, dxy and dyz orbitals (see the +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='32 meV map in Figure 2b) and the spatial distributions of the LDOS around these peaks are very sensitive to the exact en- ergy (Supplementary Figure 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Due to their energetic location and orbital symmetries, we assign them to the experimental ±α YSR state (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Figure 1c and d) which appear as a single peak in the dI/dV spectrum due to the finite-temperature smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Additionally, there are two most intense peaks in the calculations which stem from two energetically close-by YSR states near EF where the one closest to EF corresponds to the dx2−y2 orbital (see the +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='38 meV map in Figure 2b), and the less intense one further apart from EF to the dxz orbital (see the −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='58 meV map in Figure 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' While the latter resembles the experimentally observed −β state, the former has more similarities to the +β state (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Figure 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' These theoretical results indicate that the different spatial experi- mental distributions of the +β and −β states in Figure 1d can be explained by supposing that they correspond to two different YSR states of dx2−y2 and dxz orbital character which overlap within the experimental energy resolution, rather than to a single YSR state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The two YSR peaks also overlap in the theoretical calculations if a larger imaginary part is chosen for the energy, corresponding to a higher effective temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Finally, note that the electron-hole asymmetries in the intensities of the calculated peaks appear to be inverted compared to the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' With the exception of the dxz-like YSR state, each pair of peaks has a larger electron contribution above EF (Supplementary Figure 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The dxz-like YSR state has a higher electron contribution below EF which implies that this state has the strongest coupling to the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 4 YSR states of Fe dimers Before we continue with the investigation of Fe dimers, we consider some intuitive ideas about the most promising orientations of chains built from individual Fe atoms towards the goal of topologi- cally gapped YSR bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' As found in previous works23–25,45, enabling a sufficient hybridization of a YSR state which is already close to EF, while, at the same time, minimizing the hybridizations of all the other YSR states far from EF may lead to a single YSR band overlapping with EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Together with SOC, this can be a sufficient condition for the opening of a topologically non-trivial gap in the lowest-energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Starting from the experimentally detected shapes and energies of the α and β YSR states (Figure 1d) we thus regard chains along the [001] direction as most promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' For this orientation, we expect weak and strong hybridizations, respectively, for the α and β YSR states which are far and close to EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' While a manipulation of close-packed dimers along [001] turned out to be impossible, we were able to tune the system into the above conditions using dimers with a distance of 2a along [001] (see STM image in Figure 1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' A dI/dV spectrum measured above the center of the dimer as well as constant-contour maps of the spatial distributions of the three evident states are displayed in the bottom panel (orange curve) of Figure 1c and Figure 1e, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' In this configuration, the ±α YSR states of the two atoms do not overlap significantly such that they do not split into hybridized states, but only slightly shift in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' In contrast, the ±β YSR states of the two atoms strongly overlap, and split into an energetically higher one with a clear nodal line in the center between both impurities (±βa) and another energetically lower one with an increased intensity in the center (±βs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' These experimental conclusions are corroborated by our calculations (bottom panel of Figure 2a and Figure 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Apparently, all five pairs of single-atom YSR states are split, as expected from previous experimental and theoretical studies26,46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Although based on the orbital decomposition it is possible to separate all of the ten pairs (Supplementary Figure 6), the splitting of the three YSR states contributing to the α YSR state is particularly small, in accordance with the experiment, which makes it hard to resolve them in the total LDOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' In Figure 2c we plot the LDOS maps of the six most relevant peaks in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' We find that the very weakly splitted dyz YSR states at +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='32 meV and +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='30 meV which are assigned to the experimental α YSR state appear with an almost identical shape as the single atom dyz YSR state (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Figure 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' In contrast, the dx2−y2 and dxz YSR states strongly split into states with larger (at +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='79 meV and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='53 meV) and smaller (at +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='04 meV and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='66 meV) intensities in the center between both impurities and are thus associated with the experimental ±βs and ±βa YSR states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' We thus conclude, that while the α YSR states hybridize only very weakly, the β YSR states hybridize strongly and split into states which resemble anti-symmetric and symmetric linear combinations of the single atom YSR states47–49, which can be seen as a prerequisite for band formation from the hybridizing β YSR states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 5 Gapless YSR band in ferromagnetic Fe chains on Au monolayer Having identified a promising orientation and interatomic spacing from the investigation of the single atom and the dimer above, we move on to study artificial chains with the same interatomic separation, called 2a − [001] chains in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' A sketch illustrating this geometry and an STM image of a nine Fe atoms long Fe9 2a − [001] chain are shown in the top panels of Figure 3a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Spin-polarized measurements of a Fe19 2a−[001] chain indicate that the atoms in this chain configuration prefer ferromagnetic alignment (Supplementary Note 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' This is also supported by our DFT calculations (Supplementary Note 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' We found that the DMI is around 10% of the Heisenberg exchange interaction in the dimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Although this is not particularly weak, the SOC in the Au layer additionally induces a very strong out-of-plane on-site anisotropy, which prevents the formation of spin-spirals and stabilizes a normal-to-plane ferromagnetic spin structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' A dI/dV line profile (see Methods) was measured in the center of such a chain along its main axis and is plotted in Figure 3a (bottom panel) alongside the acquired stabilization height profile (middle panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The first apparent characteristic of this measurement is the modulation of every feature with the interatomic spacing of 2a in these chains, which is also visible in the height pro- file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' It should be emphasized that this is not a feature of the chains’ in-gap band structure but is just due to the lattice-periodic part of the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' However, we find additional states with different well-defined numbers of maxima at increasing energy and also very close to EF as indi- cated by the labels nβ (nβ − 1) for the numbers of maxima (nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Note that all these states have particle-hole partners occurring on the other side of EF with the same energetic distance to EF and equal numbers of maxima and nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' However, they mostly have much smaller intensities such that they are barely visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' These pairs of states can thus be assigned to confined Bogoliubov- de-Gennes (BdG) quasiparticles residing in a YSR band induced by the finite magnetic chain in the superconductor23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' To determine the orbital origin of these states, we show dI/dV maps (see Methods) of the Fe9 2a − [001] chain in Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' We find that the confined BdG states identified before in Figure 3a are localized inside the spatial extent of the chain deduced from the STM im- age (dashed red elliptical circumference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' We assign those states to a band formed by the strong hybridization of the ±β YSR states of the single adatoms as they are expected to be largely local- ized along the longitudinal axis of the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Additionally, there is a state at a similar energy as the single adatom and dimer ±α YSR states around ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='09 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' This state has exactly as many maxima as there are atoms in the chain, namely 9, which are spatially localized along both sides of the chain with a similar distance to the chain axis as the lobes of the single adatom and dimer’s ±α YSR states (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Figure 1d and e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Therefore, we assign this state to the very weakly hybridizing ±α YSR states of the single atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The state is not observed in the dI/dV line profile of Figure 3a due to its nodal line along the longitudinal chain axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' In order to measure the dispersion of the confined BdG states from the β YSR band, we collect similar dI/dV line profiles as the one in Figure 3a of defect-free chains for lengths ranging from N = 7 to N = 14 atoms (Fe7 − Fe14, see Supplementary Note 4 and Supplementary Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' It can be observed that the confined BdG quasiparticle states shift in energy as a function of the length L = N · d = N · 2a of the chain, as expected from the length-dependent interference 6 condition q = |q| = ±2πn L (1) where n is an integer and |q| is the length of the BdG quasiparticle scattering vector23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' For par- ticular chain lengths, the confined BdG quasiparticle states can be located very close to EF (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Fe8 and Fe10 in Supplementary Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' We perform one-dimensional fast Fourier transforms (1D-FFT) of the columns of the dI/dV line profiles at fixed energy E averaging all data sets taken for chains of multiple lengths, and thereby obtain the dispersion of the scattering vectors E(q) (Figure 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' This dispersion is closely linked to the β YSR band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' We find that this band has an approximately parabolic dispersion ranging from −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='9 meV at q/2 = 0 to +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='5 meV at q/2 = π/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Note that, as already discussed for the dI/dV line profiles above, the particle-hole partner of this band has a much lower intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' It is only visible around the Brillouin zone center (q/2 = 0) in our measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Most importantly, the β YSR band smoothly crosses EF without any indications of a minigap opening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' An overall similar behaviour is found using our ab-initio framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' We performed calculations for 2a − [001] chains of lengths ranging from 9 to 19 Fe atoms with ferromagnetic spin alignment (Supplementary Note 7 and Supplementary Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Exemplarily, the calculated LDOS along a Fe9 chain is shown in Figure 4a and can be directly compared to the measured line profile in Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The band formation of the YSR states can clearly be observed in a wide range of the substrate gap in the form of LDOS lines with a well-defined number of maxima along the chain as indicated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' In Figure 4b we present the corresponding spatial distributions of the LDOS of the Fe9 chain in the form of two-dimensional maps for a selection of confined BdG states with the indicated dominant orbital characters and numbers of maxima (see the Methods section for calculation details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The states closest to the substrate’s coherence peaks with nyz = 2 (and admixed nyz = 4) and nz2 = 3 maxima have dyz and dz2 orbital characters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' They reside in a very narrow band formed by the weakly interacting α YSR states of the Fe atoms (Fig- ure 2), which explains the low dispersion of this band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' On the contrary, wide bands are formed by the strong hybridization of the β YSR states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' a dx2−y2 YSR band (between -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='2 meV and +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='1 meV) having high intensities on both sides along the longitudinal axis of the chain and a dxz YSR band (between -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='8 meV and 0 meV) characterized by high intensities between the atoms of the chain (Figure 4a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' In order to deduce the dispersions of these YSR bands from the theoretical calculations we applied the same 1D-FFT method as in the experiment (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 23), and averaged over chains containing 9, 11, 13, 14, 17 and 19 Fe atoms (Supplementary Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The result is plotted in Figure 4c and can be compared to the experimental dispersion in Figure 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The most characteristic, broad bands are the dx2−y2 and dxz YSR bands between -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='2 meV and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='1 meV and between -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='8 and 0 meV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' While the energy range of the dxz YSR band agrees reasonably well with that of the experimental β band, the dx2−y2 YSR band is probably not detected significantly in the experimental data (Figure 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The latter might be explained by the small intensity of the experimental +β state (Figure 1d) which is not reproduced by the calcula- tions (Figure 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Most importantly, the theoretical study confirms the lack of a detectable minigap at EF in the YSR bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 7 Minigap and end states in spin spiral Fe chains on Au monolayer At first sight, the missing minigap seems surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' For similar ferromagnetic chains on the lighter substrates Nb(110) and Ta(110) there are already clear indications for the openings of topological minigaps 23,50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' It is widely accepted that topological minigaps hosting MBSs can open in the quasiparticle spectrum of one-dimensional helical spin systems being proximity-coupled to a con- ventional s-wave superconductor 14,15,18,51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' For ferromagnetic chains, this phenomenon has been attributed to a Rashba-type SOC induced by the substrate52, which is equivalent to a spin spiral structure without SOC in a single-band tight-binding model53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' As outlined in the introduction above, the heavier material Au is well known to exhibit large SOC5,28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' However, as our experi- ments and calculations show, it obviously does not induce a spin-spiral state in the Fe chain, and likewise does not induce a SOC of sufficient strength in the YSR bands of the ferromagnetic chain to open a detectable minigap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' In order to trace whether we can still force the system into a state with a large topological gap ∆ind just by artificially imposing a suitable non-collinear spin state onto the chain, we performed calculations for the same chains as before, but now imposing a helical spin spiral state (Figure 5, Supplementary Note 8, and Supplementary Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The configuration of the spin spiral was such that the first Fe site had its spin pointing along the positive z direction and then each spin is rotated by 90◦ around the chain axis when moving along the chain (Figure 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Indeed, there are two significant features which emerge in the LDOS of the spin-spiral chain with 19 iron atoms (Figure 5b), which were absent in the LDOS of the ferromagnetic chain (Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' First, a minigap at EF opens up between −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='22 meV and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='22 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Second, inside this minigap a single state can be observed at EF with a pronounced intensity localized at the ends of the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' This state has an electron-hole ratio of 1 and is robust against the variation of the chain length from 9 to 19 atoms as illustrated in Figure 5b and Supplementary Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The strongly different spa- tial LDOS distribution of the zero-energy state compared to that of some exemplary higher-energy states is further illustrated in Figure 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The former is localized over a few atoms at the two ends of the chain, while the latter states outside the minigap are extended along the whole chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' It should be mentioned that all these states, both the zero-energy one as well as those outside of the minigap show the same orbital character, indicating that the minigap emerges from the dx2−y2 YSR states of the ferromagnetic chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The induced minigap of 2∆ind = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='44 meV width and the narrow spectral weight around EF stemming from the zero-energy end states are also clearly visible in the dispersion of the scattering wave vectors deduced from the averaged 1D-FFTs of the LDOSs of chains of different lengths (Figure 5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Thus, the calculations show evidence for the formation of a topological, most probably p-wave-like, minigap which hosts a MBS, if the Fe chain on Au(111) is forced into a helical spin spiral state, indicating that the absence of the non-collinear ground state is the limiting factor of this experimental system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 8 Conclusions and outlook In summary, our combined experimental and theoretical investigation shows that in contrast to what might be suggested by simplified tight-binding models52,53, a strong substrate SOC alone generally is not a sufficient condition for the opening of a topological minigap in a ferromagnetic chain in contact to an s-wave superconductor, since the SOC has to exist in the lowest-energy YSR band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' In fact, first-principles calculations of the magnetic interaction parameters in ultrathin film systems have demonstrated that also the connection between the formation of a spin-spiral state and SOC is considerably more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' In particular, the DMI preferring a non-collinear spin alignment is typically weak when a 3d transition metal is deposited on a Au surface compared to other 5d substrates54–56, which may be tentatively attributed to the fully occupied 5d band of Au having a reduced effect on the DMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Proximity to a Au layer is known to give rise to strong Heisenberg exchange interactions and anisotropy32 in the magnetic layer instead, both of which prefer a collinear spin alignment and the latter being induced by the SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Our results indicate that, similarly to the competition between DMI and anisotropy terms in the formation of non-collinear spin structures, the role of SOC may be more complex for inducing topological superconductivity in the YSR bands of ferromagnetic spin chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Our study proves that it is experimentally possible to grow proximitized ultrathin heavy metal lay- ers on a superconductor with a large Tc that can be used as a substrate for the deposition of transi- tion metal atoms and to construct defect-free one-dimensional structures with an excellent quality, enabling the tailoring of YSR bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Further, we presented an ab-initio method that accurately reproduces the main LDOS features observed in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Our work thus demonstrates the theoretical feasibility of an ab-initio screening of other combinations of transition metal chains on heavy metal thin films on bulk superconductors in order to find the optimal conditions for the opening of a large topological minigap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Methods STM and STS measurements The experiments were performed in a custom home-built ultra-high vacuum system, equipped with an STM setup, which was operated at a temperature of 320 mK57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' STM images were obtained by applying a bias voltage Vbias to the sample upon which the tip-sample distance is controlled by a feedback loop such that a constant current I is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' dI/dV spectra were obtained in open feedback mode after stabilizing the tip at Vstab = 6 mV and Istab = 1 nA using a standard lock-in technique with an AC voltage Vmod = 20 µV (rms value) of frequency fmod = 4142 Hz added to the ramped Vbias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' If other stabilization parameters were used for a particular measurement, it is indicated in the respective figure caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' dI/dV maps were obtained by measuring dI/dV spectra 9 on a predefined spatial grid, which was positioned over the structure of interest, and selecting a slice at a given voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Typical measurement parameters are the same as for individual dI/dV spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' dI/dV line profiles are measured similarly to dI/dV maps, with the exception that the spatial grid is one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Constant-contour maps were obtained by repeated scanning of individual lines of STM images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' First, each line is measured as it would be the case in a regular STM image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The z-signal of this sweep is saved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Then, the feedback is turned off, the bias voltage Vbias is set to a predefined value of interest, and for the next sweep on the same scan line the dI/dV signal is measured while restoring the previously recorded z-signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' A mechanically sharpened and in-situ flashed (50 W) bulk Nb tip was used for all measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' While the usage of a superconducting tip is a crucial factor for obtaining a very good energy resolution, it has the downside that the dI/dV spectra are convolutions of the tip and sample DOSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' However, we can determine the superconducting gaps of the tip and the sample, and deconvolute the dI/dV spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' This process is described in Supplementary Note 2 and is performed for every spectrum in the main manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Sample preparation A Nb(110) single crystal with a purity of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='99 % was transferred into the ultra-high vacuum chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The sample was cleaned by cycles of Ar ion sputtering and flashes up to 2400 °C, which results in a clean surface with only few oxygen impurities remaining58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' We established flashing parameters which clean the surface of oxygen, and checked the results by STM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Once this cleaning procedure was reproducible with the given parameters, we evaporated Au from an e-beam evapora- tor (EFM3 by FOCUS GmbH) equipped with a Au rod (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='99 % purity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Following this procedure, we achieved flat and spatially extended films (Figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Fe was evaporated onto the surface from a carefully outgassed Fe rod using a second e-beam evap- orator while keeping the sample temperature below T = 10 K to avoid clustering and diffusion and thus achieve a random distribution of single Fe adatoms (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' From Supplementary Note 1, we conclude that the Au film grows pseudomorphically and that the Fe atoms are adsorbed in the fourfold coordinated hollow sites in the center of four Au atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' This is further supported by the similarity of the spatial distributions of the YSR states of the Fe/Au/Nb(110) system compared to the Mn/Nb(110) system26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' STM tip-induced atom manipulation22,37 is used to position individual Fe atoms and construct artificial structures, such as dimers and chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The structures built in this study have sufficient interatomic spacing to unambiguously identify the positions of the individual atoms forming the structure using STM images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' We restrict the investigations here to Fe atoms positioned on the first ML of Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Fe atoms on the first and second ML can easily be distinguished by their apparent height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' In the top part of Figure 1b one can see that an Fe atom on the second ML appears as a bright spherical protrusion, while an Fe atom on the first ML is more shallow and has a relatively irregular shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Thus, we can be sure that all of the experiments were carried out on the first ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 10 First-principles calculations The calculations were performed in terms of the Screened Korringa-Kohn-Rostoker method (SKKR), based on a fully relativistic Green’s function formalism by solving the Dirac equation for the nor- mal state32 and the Dirac-Bogoliubov-de Gennes (DBdG) equation for the superconducting state within multiple scattering theory (MST)46,59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The impurities are included within an embedding scheme60, being an efficient method to address the electronic and magnetic properties or the in-gap spectra of real-space atomic structures without introducing a supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The system consists of seven atomic layers of Nb, a single atomic layer of Au and four atomic layers of vacuum between semi-infinite bulk Nb and semi-infinite vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The Fe adatoms are placed in the hollow position in the vacuum above the Au layer and relaxed towards the surface by 21%, while the top Au layer is also relaxed inwards by 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The relaxations are obtained from total-energy minimization in a VASP61–63 calculation for a single Fe adatom and are used for the dimer and all the chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' For the potentials we employ the atomic sphere approximation (ASA), the normal state is calculated self-consistently in the local density approximation (LDA) as parametrized by Vosko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='64 The partial waves within MST are treated with an angular momentum cutoff of ℓmax = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' In the self- consistent normal state calculations we used a Brillouin zone (BZ) integration with 253 k points in the irreducible wedge of the BZ and a semicircular energy contour on the upper complex half plane with 16 points for energy integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' In order to take into account charge relaxation around the magnetic sites, the single impurity and the 2a − [001] dimer are calculated by including a neigh- borhood containing 48 and 84 atomic sites, respectively, corresponding to a spherical radius of r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='66 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The atomic chains are calculated with a somewhat smaller neighborhood correspond- ing to 2 atomic shells or a spherical radius of r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='01 a around the Fe atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' This way our largest atomic cluster in the calculation with 19 Fe chain atoms contained 339 atomic sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' After having obtained the self-consistent potentials in the normal state, the superconducting state is simulated within single-shot calculations by solving the DBdG equation with the experimental band gap used as the pairing potential in the Nb layers46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' In the case of the single impurity and the dimer, the BZ integration for the host Green’s function is performed by using the same k mesh as for the normal state, but in order to achieve convergence for the chains we had to increase the number of k points up to 1891 in the irreducible wedge of the BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' A sufficient energy resolution of the LDOS in the superconducting gap is acquired by considering 301 energy points between ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='95 meV with an imaginary part of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='6 µeV related to the smearing of the resulting LDOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Both the electron and the hole components of the LDOS are calculated, but in this paper we present only the electron part leading to the asymmetry of the spectrum as also seen in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Due to the ASA used in our method we obtain the LDOS for each atomic site of the cluster averaged inside the atomic spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The orbital resolution of the YSR states can be determined based on the orbital-resolved LDOS of the Fe atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Since the canonical d orbitals hybridize due to the symmetry of the clus- ter and due to SOC, we assign the labels based on the orbital which has the largest contribution to the given peak in the LDOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' In addition, in order to mimic the constant-contour maps in the experiments, we evaluate the spatial distribution of the LDOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' These LDOS maps are taken from the first vacuum layer above the surface in which the magnetic atoms are embedded, reflecting the orbital characteristics obtained from the resolution of the LDOS of the Fe atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' In order to better 11 reproduce the experimental constant-contour maps taken from the vacuum region, the LDOS of the magnetic sites are replaced by the average LDOS over the two vacuum sites (empty spheres) closest to them in the layer above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' To get a continuous picture for the LDOS maps we applied an interpolation65 scheme on the data calculated as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Data availability The authors declare that all relevant data are included in the paper and its Supplementary Informa- tion files.' metadata={'source': 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of curved surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' IEEE Transactions on Computers 100, 623–629 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Acknowledgements P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', and J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' gratefully acknowledge funding by the Cluster of Excellence ’Advanced Imaging of Matter’ (EXC 2056 - project ID 390715994) of the DFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' gratefully acknowledges funding of the European Union via the ERC Advanced Grant ADMIRE (project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 786020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='Ny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='Sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' acknowledge financial support by the National Research, Development, and Innovation Office (NRDI) of Hungary under Project Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' FK124100 and K131938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='Ny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='Sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' acknowledge support by the Ministry for Innova- tion and Technology and the NRDI Office within the Quantum Information National Laboratory of Hungary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='Ny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' acknowledge the support by the ´UNKP-21-3 and the ´UNKP-21-5 New National Excellence 16 Program of the Ministry for Innovation and Technology from the source of the National Research, Devel- opment and Innovation Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' acknowledges the J´anos Bolyai Research Scholarship of the Hungarian Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Author contributions P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' conceived the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' performed the mea- surements and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' analyzed the experimental data together with J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='. L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' performed the VASP calcu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' performed the SKKR calculations and discussed the data with L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='Sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' also contributed to the spin model calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' prepared the figures and wrote the first version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='Sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' contributed to the discussions and the finalization of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Competing Interests The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Correspondence Correspondence and requests for materials should be addressed to J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Wiebe (email: jwiebe@physnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='uni-hamburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='de).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 17 Figure 1 | Measured YSR states of Fe atoms and dimers on monolayer Au on Nb(110).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' a, STM image of an ultrathin film of Au on Nb(110) with an approximate coverage of 1 ML Au and additionally deposited Fe atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' An extracted line profile along the red line is displayed in the bottom panel (red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Rectangles show the surface composition underneath the line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The white scale bar has a length of 20 nm (Vbias = 50 mV and I = 100 pA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' b, STM image of randomly distributed Fe atoms on 1 ML Au (bottom) and 2 ML Au (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The white scale bar has a length of 4 nm (Vbias = 6 mV and I = 3 nA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' c, Deconvoluted dI/dV spectra measured on the Au/Nb(110) substrate (gray), a single Fe atom (red, top panel), and the center of a dimer of Fe atoms spaced by 2a along the [001] direction (orange, bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Black arrows and Greek letters label YSR states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Gray ticks mark the position of the superconducting energy gap of the sample ∆s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='50 meV as determined in Supplementary Note 2 (Vstab = 6 mV, Istab = 1 nA and Vmod = 20 µV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' d and e, STM images and constant-contour maps of a single Fe atom (d) and a Fe dimer (e) spaced by 2a along the [001] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Constant-contour maps were obtained for every energy for which we identified a peak in the corresponding spectrum of c as indicated by the corresponding Greek letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' White arrows indicate crystallographic directions, red dashed circles depict the positions of the Fe atoms as determined from the topographies, and white scale bars represent a length of 1 nm (Vbias = 6 mV and I = 1 nA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 18 +△s c d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='23 meV + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='27 meV HI Au/Nb(110) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content="5 single Fe atom Fe atom (n'que) ^p/lp 2a-[0011 dimer z (pm) 37 topogr." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' β ^p/p 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='23 meV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='27 meV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='5E 0- 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='13 decon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' +β +α [001], x LO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='0 AU α β Au Nb(110) βs Nb(110) e + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='13 meV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='mev + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='35 meV 0 distance (nm) 28 2a-[001] dimer (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 0 z (pm)72 540 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='4 topogr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 1p//p +βs 20- +α ed- ↑ I+βa z (pm) 10], 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='13 meV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='53 meV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='35 meV +α [001], 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='0 2 -△s -1 0 1 +△s 2 E- E- (meV) 5 20- B βsFigure 2 | Calculated YSR states of Fe atoms and dimers on monolayer Au on Nb(110).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' a, Electron component of the LDOS of the single Fe atom (top panel) and the ferromagnetic 2a − [001] dimer (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Gray dashed vertical lines indicate the superconducting gap of the substrate ∆s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' b, Spatial distributions of the three YSR peaks of the Fe atom with the highest intensities visible in the top panel of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' c, Spatial distributions of the six YSR peaks of the FM 2a − [001] dimer with the highest intensities visible in the bottom panel of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The energies are indicated in the bottom of the panels of b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Red circles indicate the positions of Fe atoms and the white scale bars correspond to a distance of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 19 a Fe atom 2a-[001] dimer b +△ LO LDOS (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=') HI 3500 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='32 meV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='58 meV +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='38 meV 3000 + (n 600个 (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 400 110] LDOS ( [0011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' x 200 0 c 2000 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='32 meV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='53 meV +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='79 meV 1500 LDOS (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=') 400个 200 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='30 meV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='66 meV +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='04 meV 0 [0011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 1 0 +△ E- E (meV)Figure 3 | Measured dispersion of BdG quasiparticles in Fe chains on Au monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' a, Deconvoluted dI/dV line profile (bottom panel) and corresponding topographic line profile (middle panel) measured along the longitudinal axis of a Fe9 2a − [001] chain, as illustrated in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Black arrows mark the energies in the bottom panel at which nβ maxima as indicated are observed along the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The subscript of this label refers to the orbital origin of this state (Vstab = 6 mV, Istab = 1 nA, Vmod = 20 µV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' b, The top panel shows an STM image of a Fe9 2a−[001] chain and the lower panels are dI/dV maps of this chain, obtained at energies indicated in the top right corner of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The maps are labeled by nβ in a similar fashion as the states in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The red lines mark the spatial extent of the chain in the STM image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The white scale bar represents a length of 1 nm (Vstab = −6 mV, Istab = 1 nA, Vmod = 20 µV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' c, Averaged energy- wise 1D-FFT obtained from dI/dV line profiles of FeN 2a − [001] chains with lengths N ranging from seven to fourteen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Prior to the 1D-FFT, the spectra were deconvoluted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' All dI/dV line profiles were obtained with the following parameters: Vstab = 6 mV, Istab = 1 nA, Vmod = 20 µV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 20 a b 76 topogr c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='7 V/FFTI (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='6 (pm) 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='5 (ud) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='42 meV 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='0 N HI nβ= +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='02meV 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content="5 三(n'que) ^p/p nβ = nβ=3 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='02meV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='0 F 2 4 山 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='47meV 33 0 decon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='0 LO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='77 mev TB 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='5 LO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='09 meV [110] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='0 0:5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='0 2 q/2 (元/d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='0 [001] x (nm)Figure 4 | Calculated dispersion of BdG quasiparticles in ferromagnetic Fe chains on Au monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' a, Electron component of the LDOS extracted along the longitudinal axis of a ferromagnetic Fe9 2a − [001] chain, calculated on the Fe and the vacuum sites in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' b, Spatial distributions of the LDOS evaluated in the first vacuum layer above the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' Energies are indicated in the top right corners of each panel, while the numbers of maxima and the orbital origins of the states are indicated in the top left corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' They correspond to the states marked by the same arrows and numbers in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The nyz state has two dominant Fourier components nyz = 2 and nyz = 4, where the former is dominating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The white scale bar has a length of 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' c, Dispersion of scattering wave vectors extracted from the calculated LDOSs of ferromagnetically coupled FeN 2a − [001] chains and averaged for lengths N of 9, 11, 13, 14, 17 and 19 atoms (Supplementary Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The green dashed lines in panels a and c indicate the energy gap of the superconducting substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 21 b a 0 LDOS(arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=') 800 LO HI c LDOS (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=') 2 VIFFTI (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=') 20 Feg 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='30 meV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='22 meV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='0 nz² = 3 nx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='y2 = 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='43 meV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='5 > nx2.' 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| Minigap and MBS enforced by helical spin spirals in Fe chains on Au monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' a, Illustration of the helical spin spiral state with a rotation angle of 90◦ of a chain containing 19 Fe atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' b, Electron component of the LDOS extracted along the longitudinal axis of a Fe19 2a − [001] chain in the helical spin spiral state shown in panel a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' c, The spatial distributions of the LDOS evaluated in the first vacuum layer above the Fe19 2a − [001] chain at the energies indicated in the top right corner of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' d, Dispersion of scattering wave vectors averaged from the calculated LDOSs of the FeN 2a − [001] chains including N = 9, 11, 13, 14, 17 and 19 Fe atoms (Supplementary Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' The green and red dashed lines in b and d indicate the substrate gap and the minigap, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=' 22 b d a 2 V/IFFTI (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE5T4oBgHgl3EQfew_U/content/2301.05622v1.pdf'} +page_content=') 20 0 LDOS (arb.' metadata={'source': 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0000000000000000000000000000000000000000..8d08a66d6c1e48130779b8f442bbb83d08a718cd --- /dev/null +++ b/cdFRT4oBgHgl3EQfSzck/content/tmp_files/2301.13530v1.pdf.txt @@ -0,0 +1,1271 @@ +Domain-Generalizable Multiple-Domain Clustering +Amit Rozner * 1 Barak Battash * 1 Lior Wolf 2 Ofir Lindenbaum 1 +1Faculty of Engineering, Bar Ilan University 2 School of Computer Science, Tel Aviv University +∗ These authors contributed equally. +Abstract +Accurately clustering high-dimensional measure- +ments is vital for adequately analyzing scientific +data. Deep learning machinery has remarkably +improved clustering capabilities in recent years +due to its ability to extract meaningful representa- +tions. In this work, we are given unlabeled sam- +ples from multiple source domains, and we aim +to learn a shared classifier that assigns the exam- +ples to various clusters. Evaluation is done by +using the classifier for predicting cluster assign- +ments in a previously unseen domain. This setting +generalizes the problem of unsupervised domain +generalization to the case in which no supervised +learning samples are given (completely unsuper- +vised). Towards this goal, we present an end-to- +end model and evaluate its capabilities on several +multi-domain image datasets. Specifically, we +demonstrate that our model is more accurate than +schemes that require fine-tuning using samples +from the target domain or some level of supervi- +sion. +1. Introduction +Clustering is a fundamental machine-learning task that cate- +gorizes unlabeled vectors into homogeneous groups. Clus- +tering high dimensional measurements is a difficult task, and +classical methods such as K-means (Lloyd, 1982), Spectral +Clustering (Ng et al., 2001), or density-based methods (Ester +et al., 1996) often fail to group semantically related exam- +ples. Several recent works have developed deep learning- +based frameworks to overcome this limitation by extracting +meaningful features and automatically clustering images. +However, extracting informative features from real-world +images is challenging since, often, the images are collected +from multiple domains with diverse properties. The problem +becomes even more challenging if we want to generalize +the cluster assignments to unseen target domains that may +deviate significantly from the source distribution. +Multi-domain clustering methods simultaneously group +samples in analog domains such that a sample in one domain +is associated with semantically related samples in another +(Menapace et al., 2020). Such methods or others that rely +on labeled observations benefit from the shared knowledge +between the domains (Cheng et al., 2013; Menapace et al., +2020; Zhang et al., 2021; Harary et al., 2022). For example, +if one domain contains grayscale images and the other do- +main contains low-resolution images of the same objects, +one can learn to separate between the images of the different +objects in a way that utilizes the fact that it is easier to sepa- +rate between some objects based on color and between other +objects based on fine details. We do not assume knowledge +about corresponding images between domains, only that all +target classes appear in each domain. +In this work, we study the ability to learn a classifier f that, +given a sample, regardless of its domain, can categorize it to +the matching cluster. The requirement is that f is evaluated +(one sample at a time) in an unseen target domain for which +no sample is seen during training. This task combines the +problem of multiple domain clustering with the aspect of +domain generalization. Both of these are of high value, and +their combination is extremely powerful: multiple source +clustering is core to scientific discovery, e.g., in biology, +where every organism can be a domain, and the application +to new domains is the ultimate test. +Formally, the problem statement is as follows. We are given +a dataset S with N samples from d different domains, where +each domain has Ni, ∀i ∈ {1, ..., d} samples, and we know +which sample belongs to each domain. A classifier f is +trained to map every sample in S to one of K groups. For +evaluation purposes only, a set of labeled samples is then +used, and the K groups are assigned to a set of ground +truth labels using a best-matching method (The Hungarian +method (Kuhn, 1955)) on an unseen single-domain dataset +T. Specifically, f is applied, without further training, to +each sample of T separately, and the assigned labels f out- +puts are compared to the ground truth labels of the samples +in T. Figure 1 illustrates our setting. +To solve this challenge, we propose a two-stage learning +scheme: (i) training a backbone in a self-supervised do- +main invariant fashion. This step leads to a feature extractor +that focuses on semantic features and attenuates the influ- +arXiv:2301.13530v1 [cs.LG] 31 Jan 2023 + +Domain-Generalizable Multiple-Domain Clustering +ence of style. (ii) training multiple clustering heads using +pseudo labels generated using the semantic information. +Our method was tested on various domain generalization +datasets and has shown superior results over several base- +lines. Our framework is at par or better than methods that +rely on some supervision or adaptation to the target domain. +2. Background +Self-supervised learning (SSL) +is an emerging field in +machine learning (ML) that enables extracting useful rep- +resentation from unlabelled data. SSL relies on creating a +pretext task with fictitious labels that can be generated “for +free”. If the pretext task is correlated with a downstream +task of interest, SSL techniques can be compelling for ex- +tracting features useful for many applications. A seminal +work (He et al., 2020) encodes a query image and matches +it to a dictionary of image keys using a contrastive loss. +To stabilize the embedding of keys, they use a momen- +tum encoder to update the representation of keys at a slow +pace. By using queues, the architecture can scale to large +datasets. Another important work was presented by Chen +et al. (2020a). They utilized pairs of data augmentations, +creating feature representations that are then projected and +trained to maximize agreement. After training, the projec- +tion heads are discarded, and the resulting features generate +state-of-the-art self-supervised results. Bootstrap your own +latent (BOYL) (Grill et al., 2020), utilized two neural net- +works for their training. Each model is updated at a different +rate making the interaction beneficial for the learning pro- +cess. Further progress was made by Chen et al. (2020b), +improving contrastive SSL efficiency. +Deep clustering +aims to exploit the strength of neural +networks as feature extractors to identify a representation +that better preserves cluster structures in unlabelled data. In +(Xie et al., 2016), the authors use an autoencoder to extract +features while learning soft cluster assignments by mini- +mizing the KL-divergence between the latent space and a +prior distribution. Chang et al. (2017) propose an algorithm +that iterates between feature extraction and performing a +pairwise classification to predict whether pairs of images +belong to the same cluster. DeepCluster (Caron et al., 2018), +extends this idea by iterating between applying k-means to a +deep feature mapping and training a NN to predict the clus- +ter assignments. Ji et al. (2018) apply content-preserving +image transformations to create pairs of samples with shared +information. Then, they train a NN to maximize the mutual +information between the image pairs in a cluster probabilis- +tic data representation. Recently, Semantic Pseudo-Labeling +for Image Clustering (SPICE) (Niu et al., 2022) obtained +state-of-the-art results on several clustering benchmarks. +SPICE is an iterative deep clustering method that relies on +self-supervision and pseudo-labeling. First, self-supervision +is performed using a contrastive loss to learn informative fea- +tures. Then, they create prototype pseudo labeling to avoid +miss annotations common to pseudo labeling techniques. +Unsupervised domain generalization (UDG) +was re- +cently presented by Zhang et al. (2021); Harary et al. (2022). +UDG is related to our goal but requires some amount of su- +pervision. Specifically, UDG involves: unsupervised train- +ing on a set of source domains, then fitting a classifier using +a small number of labeled images. Finally, evaluating the +model on a set of target domains unseen during training. +Toward this goal Zhang et al. (2021) suggests a method to +ignore domain-related features by selecting negative sam- +ples from a queue based on their similarity to the positive +sample domain. Recently, Harary et al. (2021) presented +BrAD, in which self-supervised pre-training on multiple +source domains is performed in a shared sketch-like domain. +To fine-tune a classifier, they used various amounts of source +domain-labeled samples. In contrast to these works, in our +research, no class labels for either source or target domains +are used for training the model. +Unsupervised Clustering under Domain Shift (UCDS) +was presented by (Menapace et al., 2020). The goal is +to cluster samples from multiple source domains and then +adapt the model to the target domain using multiple unla- +beled samples from that domain. Menapace et al. (2020) +suggested optimizing an information-theoretic loss coupled +with domain-alignment layers. This setting is similar to +ours; however, we aim to design a model that can predict +cluster assignments on multiple source domains and gener- +alize to new unseen domains without any further tuning or +adaptation. To the best of our knowledge, this is the first +work that solves this task without using any labels or sam- +ples from the target domain. This is a big advantage since, +in real-world settings; we often don’t know that a domain +shift occurred, or we can not access a pool of samples from +the test domain. +3. Method +High-level overview +Our method consists of two phases; +in the first, a feature extraction model f1 is trained in a self- +supervised fashion on data from multiple source domains. +This phase bridges the gap between different domains by +extracting semantically related features ui ∈ Re, where e +is the embedding dimension. The second phase focuses +on training a clustering head f2 while the weights of f1 +are frozen. Then our cluster predictions are based on f = +f2 ◦ f1. +We leverage a basic common domain (BCD) to mitigate +the gap between several domains. The BCD is designed +to maintain the sample’s content while removing domain- + +Domain-Generalizable Multiple-Domain Clustering +Figure 1. Problem statement - given unlabeled samples from mul- +tiple source domains, our goal is to learn a deep clustering model +that can accurately assign samples to their cluster in each source +domain. We aim to design a model that, at inference, can predict +cluster assignments on new unseen domains without any labels or +model tuning. +related information. Conceptually, a sketch-like domain can +be considered a suitable BCD for image data with varying +color and texture domains. Transforming an image to a +sketch domain keeps high-level features such as object iden- +tity while decreasing the bias that features such as colors +and image style can induce. +3.1. Pre-training +Our pre-training phase is inspired by MoCoV2 (Chen et al., +2020b). This self-supervised method applies a contrastive +loss to learn a representation invariant to a set of strong +augmentations defined in the appendix A.1. Each training +example xi is strongly augmented, then compared using +a contrastive loss to a positive x+ +i (another strongly aug- +mented) example and a negative example x− +j , where i ̸= j. +The negative example x− +j is a strong augmentation of xj, +which is stored in a queue as explained in section 3.1. The +pre-training process aims to generate valuable features for +the downstream tasks. We adapt MoCoV2 for the problem +of domain-generalizable multiple-domain clustering by pre- +senting the four components described below and illustrated +in figure 2. +1. Adversarial domain classifier: Using adversarial training +for domain adaptation was presented in the seminal work of +Ganin & Lempitsky (2015) and subsequently used by many +(Liu et al., 2022; Zhang, 2019; Wilson & Cook, 2018; Zhao +et al., 2020). Here, we leverage this idea and introduce a do- +main classifier to the features extracted from our backbone +f1 to learn features invariant to the domain identity. The +weights of the domain classifier fd are denoted as θd, and +the gradient of fd is: +∂Lfd +∂θd . We train f1 in an adversarial +fashion by updating its weights using a constant λd ≥ 0 as +in −λd +∂Lfd +∂θ1 , which is known as gradient reversal layer. +2. Domain balancing: Since we are given samples from +several source domains, quite often, the population of sam- +ples from each domain is different. Such domain imbalance +can cause poor generalization since the model may learn +only from the highly populated domain. We use a simple +approach to mitigate this phenomenon by choosing a do- +main uniformly and then picking a sample from the chosen +domain. +3. Multi queue: Multiple-domain self-supervised feature +extraction is prone to focus on domain-related features re- +sulting in poor performance. Specifically, a single neg- +ative sample queue has shown inferior results compared +to multiple domain-specific queues (Harary et al., 2022). +Following this work, we employ a domain-specific queue +Q = [Q1, Q2, ..., Qd] each with size Nq for each of the d +domains. The negative examples u− +i are drawn from the +same domain as the positive sample, which makes the dis- +crimination more challenging and encourages the model to +focus on the content rather than the domain. For efficiency, +positive samples u+ +i are stored in the relevant negative do- +main queue for later use. +4. Style transfer augmentation: We want our backbone to +learn semantic features that are invariant to the “style” of the +input image. To encourage this property, we use a style trans- +fer neural network model (Huang & Belongie, 2017). We +perform a style transfer augmentation ST (xi) with proba- +bility pst as a replacement for the strong augmentation used +in MoCoV2. The style of xi is replaced by the style of +an image from a different domain xj, s.t. dxi ̸= dxj. By +augmenting with varying styles of domains, we further en- +hance the ability of our backbone to ignore domain specific +features. +Algorithmic overview +Given an input batch of images +x = [x1, x2, .., xB]T ∈ RB×C×W ×H, each image xi is +transformed twice. First, it is augmented using a strong aug- +mentation xs +i = S(xi). Second, the image is transformed +using the following: +xst +i = +� +ST (xi), w.p pst, +S(xi) , +w.p 1 − pst. +(1) +Where ST (xi) replaces the style of xi with another do- +main’s style with probability pst. +Otherwise, a strong +augmentation S(xi) is applied to generate the positive + +Training +Multiple source domains +Domain index: +Cartoon +Clustering model +Unsupervised training +Domain index: +Photo +Inference +Target domain: Art Painting +Clustering model +→Class 1 +→Class 5Domain-Generalizable Multiple-Domain Clustering +sample. Both xs +i and xst +i +are passed through the back- +bone to create the embeddings us +i = P(f1(xs +i, θ1), θp), +ust +i = P(f1(xst +i , θ +′ +1), θ +′ +p), where P and θp are the projection +head and its weights respectively. θ +′ +1 = µθ +′ +1 + (1 − µ)θ1 +and θ +′ +p = µθ +′ +p + (1 − µ)θp are the moving average ver- +sions of θ1 and θp respectively. Finally, negative samples +u− +i , ∀i ∈ [0, Nq] are sampled from a domain queue of the +same domain dxi as xi. The contrastive loss is used: +Lfproj = −log +exp((us)T ust) +�Nq +i=1 exp((us)T u− +i ) + exp((us)T ust) +, +(2) +where us += +[us +1, us +2, .., us +B]T +∈ +RBxe, +ust += +[ust +1 , ust +2 , .., ust +B]T ∈ RBxe, are the embeddings for the trans- +formed input batch xs, xst. +To further remove domain-specific information, we use an +additional domain loss term using cross-entropy (Kullback +& Leibler, 1951) loss: Lfd = −Lce(fd(us), θd). Hence our +final loss objective: +Lf1 = Lfproj + Lfd. +(3) +The contrastive and domain-adversarial loss terms com- +plement one another in finding content-related, domain- +invariant features. +3.2. Clustering head +The pre-training phase provides a solid and robust backbone +f1 on which a clustering head f2 can be trained. Then we +apply a clustering head on top of the backbone designed +to predict the assignments for each sample. The clustering +head is trained in a two-step iterative manner. Each iteration +begins by assigning pseudo labels based on the clustering +head’s predictions (logits) l(y|x) = f2 ◦ f1(x), and the +semantic features (embeddings) u = f1(x). In the second +step, the clustering heads are trained using the pseudo labels +with cross-entropy loss. Since biases are much more likely +to arise in multi-domain tasks, using both embeddings and +logits in domain generalization proves crucial, as demon- +strated in the ablation study 4. An illustration of the training +process is available in figure 3. +Training of the clustering head is initiated by sampling a +batch of images x with size B from all source domains. +Each sample xi passes through the backbone in three dif- +ferent versions. Based on the original image, and using +two transformations: a strong augmentation xs +i = E(xi), +and style transfer to our BCD xbcd +i += C(xi). Where C() +represents a style transfer of the input image xi to an image +with a sketch-like style. E() is defined as: +xst +i = +� +� +� +� +� +C(xi) +, w.p pstpbcd, +ST (xi), w.p pst(1 − pbcd), +S(xi) , w.p 1 − pst. +(4) +The BCD transformed, and the original images are used to +define the pseudo labels while the strong augmentations are +used to train the clustering head f2. +First, features are extracted from the original image: +u = f1(x, θ1). +(5) +Then, logits are extracted in the BCD based on: +l(y|xbcd) = f(xbcd) = f2 ◦ f1(xbcd). +(6) +The top γ := +B +2K samples are chosen from l(y|xbcd +i +) as +the set of most confident samples of class k based on the +clustering head’s score on samples from the BCD. Thus, the +selected samples are denoted as follows: +Mk = {ui|i ∈ argtopγ(l(k|xbcd)), ∀i ∈ {1, ..., B}}, +(7) +where argtopγ(l(k|xbcd)) ∈ Nγ is a vector of indexes that +chooses the γ most confident samples, based on their corre- +sponding BCD score l(k|xbcd). Mk is a set of γ embedding +vectors. Using Mk, the center of class k is determined by: +Gk = 1 +γ +� +ui∈Mk +ui. +(8) +One can calculate the similarity between each sample and +each of the centers: +simk = ⟨ ¯Gk, ¯u⟩. +(9) +Where ¯Gk = +Gk +∥Gk∥, and ¯u = +u +∥u∥ are the normalized feature +and center vectors, respectively. In plain words, simk ∈ RB +holds the information of how close each sample in the batch +is to the center of cluster k. +Samples that are closest to the center are used as pseudo +labels for cluster k; thus, the set of strongly augmented data +samples with pseudo labels is formulated as follows: +ˆZk = {xs[argtopγ(simk)], k} +(10) +Where xs[argtopγ(simk)] means indexing xs using +argtopγ(simk), i.e., choosing the samples that will be used +as pseudo-labels for class k using the similarity in the em- +bedding domain. While Mk are chosen based on the heads +predictions, which are in the logits space, ˆZk are pseudo +labels chosen based on information from both the semantic +space Eqs. 8,10 and in the logits space Eq. 7. Using only +the logit values to infer the pseudo labels results in poor +cluster assignments, as examined in section 4. The entire +set of representatives and pseudo-label pairs will be denoted +as ˆZ. +In the second phase, the clustering head f2 is trained using +ˆZ to minimize cross-entropy loss using the pseudo labels. A + +Domain-Generalizable Multiple-Domain Clustering +Figure 2. The proposed pre-training procedure. Each image is transformed using strong augmentations and/or style transfer augmentation. +The features us (strong) and ust (style) are extracted using the backbone f1. Then we use a domain head fd to classify the domain +identity of each sample, minimizing the domain loss Lfd; we use gradient reversal to update the backbone to fool the domain head in an +adversarial fashion. The contrastive loss Lfproj is minimized by the projection head’s output of us, ust, and u− (negative samples). The +losses complement each other in training the backbone. +batch of samples and pseudo labels (xs +pl, ypl) ∈ ˆZ, are prop- +agated through the backbone and heads, and the objective +can be formulated as: +L = 1 +B Lce(f2 ◦ f1(xpl), ypl). +(11) +During this training phase, domain balancing is used as +detailed in section 3.1. +Multiple clustering heads +Clustering is inherently unsta- +ble, especially when dealing with many classes or high- +dimensional datasets. Several authors have proposed using +feature selection (Solorio-Fern´andez et al., 2020; Shaham +et al., 2022; Lindenbaum et al., 2021) to improve clustering +capabilities by removing nuisance features in tabular data. +We are interested in stabilizing clustering performance on +diverse high-dimensional image data. Therefore, we pro- +pose training multiple clustering heads simultaneously and +selecting a reliable head based on an unsupervised criterion. +This allows us to handle many categories and overcome +the instability that stems from the clustering heads weights’ +initialization. For more details about the source of ran- +domization between heads, please see appendix A.2. The +number of clustering heads is denoted as h, hence the objec- +tive in Eq. 11, L can now be formulated as the average of +the h head specific losses: +L = +1 +Bh +h +� +i=1 +Li +ce(f2 ◦ f1(xpl), ypl). +(12) +Next, we define the diversification of head j as: +dvj = unique +B +argmax +k∈K +lj(y|x)/K. +(13) +First, argmax reduces the prediction lj(y|x) of the j-th +head to a cluster index. Next, we use the unique operator +to extract the number of clusters that head j predicts; this +process is done using the entire dataset in parallel, i.e., there +is no parameter update during this evaluation. +Due to high variability in the training procedure between +heads, some are better than others; we leverage this vari- +ability by keeping only the most diversified heads (MDH). +Two MDH are chosen out of h clustering heads based on +higher dvj values compared to the other heads. The heads +with lower dvj are discarded, and we replace the weights of +the non-MDH with a linear combination of the two MDH +weights. Mathematically speaking, let us define θ2i as the +weights of the i-th head, and let us assume that j, k are the +MDH indices; hence the weights of the non-MDH heads are +overridden in the following manner: +θ2i = rk +θ2k +∥θ2k∥ + rj +θ2j +∥θ2j∥, ∀i ̸= k, j. +(14) +Where rk ∼ U(0, 1) and rj = 1 − rk. This removes the +influence of non-diverse heads and maintains some degree +of variability for the following optimization steps. +In cases where there is equality in dvj between several +heads, which results in more than two MDHs, we limit the + +Domain head +Backbone +f d +Features +Doamin loss +Gradient reversal +Projection head +J proj +Contrastive +positive keys stored in +loss +per domain queue +Negative examples queue +Negative keys are taken from the +same domain as the positive examples +Domain 1 negative queue +u +Domain d negative queueDomain-Generalizable Multiple-Domain Clustering +Figure 3. Clustering head training scheme. The image is passed through the backbone in its original, strongly augmented, and BCD form. +The weights of the backbone are frozen and used to produce the features. Representatives are selected from the original image features +based on the clustering head’s predictions over the BCD images. The class representatives are used as pseudo labels for the CE loss. +number of MDHs to five. The rationale behind this limi- +tation can be elucidated through the following illustrative +scenario, w.l.o.g., assume that the first head does not predict +one class, and the other heads do not predict five classes; if +the number of MDH kept is not limited, the advantage of the +first head is not utilized. Since all heads will perform poorly +in the early training phase, MDH selection is initiated after +a few epochs. Furthermore, to allow the heads to make +gradual learning, the process repeats every n epochs. +4. Experiments +Experiments are conducted using three datasets commonly +used for evaluation of domain generalization methods. Rep- +resentative images from several datasets and domains appear +in appendix A.3. The Office31 dataset (Saenko et al., 2010) +consists of images collected from three domains: Amazon, +Webcam, and DSLR, with 2817, 795, and 498 images, re- +spectively. The dataset includes 31 different classes shared +across all domains. The samples consist of objects com- +monly encountered in an office setting. The PACS dataset +(Li et al., 2017) consists of four domains: Sketch, Cartoon, +Photo, and Artpainting with 3929, 2344, 1670, and 2048 im- +ages, respectively. It includes seven different classes, which +are shared across all domains. The Officehome dataset +(Venkateswara et al., 2017) contains four domains: Art, +Product, Realworld, and Clipart, with 2427, 4439, 4357, and +4365 images, respectively. It includes 65 different classes, +which are shared across all domains. The large number of +domains, and classes, make the task challenging. In par- +ticular, since we aim to cluster the data without access to +labeled observations. Existing state-of-the-art results on this +data (Menapace et al., 2020) corroborate this claim. +Implementation details +Our work is implemented using +PyTorch (Paszke et al., 2019). We use Resnet18 (He et al., +2016) as our backbone for a fair comparison with the results +of Menapace et al. (2020); Harary et al. (2022); Wang et al. +(2022). The models in the pre-training were trained using +SGD with momentum 0.9 and weight decay 1e − 4. We +use a batch size of 8 and train the model for 500 epochs. +To train the clustering head, we use the same optimizer +with batches of size 256 for 100 epochs for Office31 and +Officehome datasets and 50 epochs for the PACS dataset. +The reason for this difference is the small number of classes +in the PACS dataset, which enables the model to converge +much faster. To create style transfer augmentations, we +use a pre-trained AdaIN model (Huang & Belongie, 2017). +The most diversified head selection mechanism initiates at +epoch 30 and is repeated every n = 10 epochs. For more +information on the head selection mechanism, see section +3.2. An important regularization for diversified training is +label smoothing (Szegedy et al., 2016). By using pseudo- +labels, we assume that there is a high ratio of mislabeled +samples; label smoothing helps preventing the model from +predicting the training samples too confidently. Empiric +evidence of label smoothing importance in our task can be +seen in the ablation study. +Comparison with other methods +To evaluate the capa- +bilities of our model, we focus on the following scheme: +train the model using d unlabelled source domains, then eval- +uate our model on the unseen and unlabelled target domain. +We compare our approach to several recently proposed deep +learning-based models for multi-domain clustering. +When evaluating Office31 and Officehome datasets, we com- +pare with the baselines from (Menapace et al., 2020): popu- + +Backbone +Features +f1 +Clustering heads +l(y|αbced) +Select +Representitives +Pseudo labels for +selected representatives +CE lossDomain-Generalizable Multiple-Domain Clustering +Table 1. Results on the Office31 dataset (31 classes) upon all three domain combinations, each of the letters A, W, D represent the +domains Amazon, Webcam, and DSLR, respectively. The notation X, Y → Z, means the model was trained on X, Y domain and tested +on the Z domain. Target fine-tuned means the method was trained or adapted to the test domain. In K-means, we first pre-trained the +MocoV2 model and trained K-means on top of its embeddings. +Method +Target fine-tuned +Supervision +D, W → A +A, W → D +A, D → W +Avg +DeepCluster (Caron et al., 2018) +✓ +- +19.6 +18.7 +18.9 +19.1 +IIC (Ji et al., 2018) +✓ +- +31.9 +34.0 +37.0 +34.3 +IIC-Merge (Ji et al., 2018) +✓ +- +29.1 +36.1 +33.5 +32.9 +IIC + DIAL (Ji et al., 2018) +✓ +- +28.1 +35.3 +30.9 +31.4 +Continuous DA (Mancini et al., 2019) +✓ +- +20.5 +28.8 +30.6 +26.6 +ACIDS (Menapace et al., 2020) +✓ +- +33.4 +36.1 +37.5 +35.6 +K-means (Lloyd, 1982) +✓ +- +14.9 +24.3 +20.8 +29.9 +Ours +- +- +23.1 +49.2 +45.2 +39.2 +Table 2. Results on the Officehome dataset (65 classes) upon all four domain combinations, each of the letters A, P, R, C represent +the domains Art, Product, Realworld, and Clipart, respectively. The notation W, X, Y → Z, means the model was trained on W, X, Y +domain and tested on the Z domain. Target fine-tuned means the method was trained or adapted to the test domain. In K-means, we +pre-train the MocoV2 model and then train K-means on top of its embeddings. +Method +Target fine-tuned +Supervision +C, P, R → A +A, P, R → C +A, C, R → P +A, C, P → R +Avg +DeepCluster (Caron et al., 2018) +✓ +- +8.9 +11.1 +16.9 +13.3 +12.6 +IIC (Ji et al., 2018) +✓ +- +12.0 +15.2 +22.5 +15.9 +16.4 +IIC-Merge(Ji et al., 2018) +✓ +- +11.3 +13.1 +16.2 +12.4 +13.3 +IIC + DIAL(Ji et al., 2018) +✓ +- +10.9 +12.9 +15.4 +12.8 +13.0 +Continuous DA (Mancini et al., 2019) +✓ +- +10.2 +11.5 +13.0 +11.7 +11.6 +ACIDS (Menapace et al., 2020) +✓ +- +12.0 +16.2 +23.9 +15.7 +17.0 +K-means (Lloyd, 1982) +✓ +- +9.1 +11.3 +13.8 +10.6 +11.2 +Ours +- +- +20.8 +25.5 +27.9 +25.6 +25.0 +Table 3. Results on the PACS dataset (7 classes) upon all four domain combinations, each of the letters A, P, S, C represents the domains: +Art painting, Photo, Sketch, and Cartoon, respectively. The notation W, X, Y → Z, means the model was trained on W, X, Y domain and +tested on the Z domain. Target fine-tuned means the method was trained or adapted to the test domain. In K-means, we first pre-trained +the MocoV2 model and trained K-means on top of its embeddings. +Method +Target fine-tuned +Supervision +C, P, S → A +A, P, S → C +A, C, S → P +A, C, P → S +Avg +DeepCluster(Caron et al., 2018) +✓ +- +22.2 +24.4 +27.9 +27.1 +25.4 +IIC (Ji et al., 2018) +✓ +- +39.8 +39.6 +70.6 +46.6 +49.1 +IIC-Merge (Ji et al., 2018) +✓ +- +32.2 +33.2 +56.4 +30.4 +38.1 +IIC + DIAL(Ji et al., 2018) +✓ +- +30.2 +30.5 +50.7 +30.7 +35.3 +Continuous DA (Mancini et al., 2019) +✓ +- +35.2 +34.0 +44.2 +42.9 +39.1 +ACIDS (Menapace et al., 2020) +✓ +- +42.1 +44.5 +64.4 +51.1 +50.5 +K-means (Lloyd, 1982) +✓ +- +17.7 +18.5 +21.1 +22.4 +19.9 +Ours +- +- +46.7 +44.7 +66.8 +49.2 +51.9 +BrAD (Harary et al., 2021) +- +1% +33.6 +43.5 +61.8 +36.4 +43.8 +BrAD-KNN (Harary et al., 2021) +- +1% +35.5 +38.1 +55.0 +34.1 +40.7 +BrAD (Harary et al., 2021) +- +5% +41.4 +50.9 +65.2 +50.7 +52.0 +BrAD-KNN (Harary et al., 2021) +- +5% +39.1 +45.4 +58.7 +46.1 +47.3 +BrAD (Harary et al., 2021) +- +10% +44.2 +50.0 +72.2 +55.7 +55.5 +BrAD-KNN (Harary et al., 2021) +- +10% +42.0 +45.3 +67.2 +50.0 +51.1 + +Domain-Generalizable Multiple-Domain Clustering +lar deep clustering papers Invariant Information Clustering +for Unsupervised Image Classification and Segmentation +(IIC) (Ji et al., 2018), and DeepCluster (Caron et al., 2018). +Importantly, they were both trained directly on the target do- +main before predicting the clusters. Menapace et al. (2020) +used two variations of IIC. Specifically, IIC-Merge involves +training IIC on all domains, including the target domain; +IIC+DIAL: IIC, which contains a domain-specific batch +norm layer jointly trained on all domains. Continuous DA: +continuous domain adaptation strategy used in (Mancini +et al., 2019) using the method presented in (Menapace et al., +2020) denoted as Adaptive Clustering of Images under Do- +main Shift (ACIDS), i.e., train on d domains, adapt on the +target domain and then test on the target domain. Note +that all the former baselines compared with our work were +trained on the target domain. We added another baseline, +training MoCoV2 on all the source domains and then fitting +the K-means clustering algorithm on the target domain. +On PACS dataset, we also compared ourselves to BrAD +(Harary et al., 2021) with various amounts of source domain +labels. This comparison is very challenging as we do not +use any class labels. +Results +Table 1 depicts the results on all three domain +combinations of the Office31 dataset. On both DSLR and +Webcam as target domains, our method outperforms the cur- +rent state-of-the-art (SOTA) by a large margin, even without +adaptation to the target domain. Our method performance on +the Amazon domain is inferior to the current SOTA; we be- +lieve this is due to the very limited source domain data. The +target fine-tuned method (unsupervised fine-tuning on the +target domain) relies on 317% more data for their training +scheme. Overall, on average, our method is better by 10.1% +than the method that uses the target domain for adaptation +and 31.1% over the baseline with the same conditions. +Results on the Officehome dataset can be seen in Table 2. +This dataset is more challenging than the former and consists +of four domains. Our method outperforms the baselines on +all four domain combinations and is better on average by +47.1% percent than the previous SOTA. +On the PACS dataset (Table 3), our method is compared +to both target fine-tuned and limited source domain label +settings. Our method outperforms the current SOTA on 3 +out of 4 target domains for the target fine-tuned case. On the +fourth domain, Sketch, our method achieves slightly lower +results than the current target fine-tuned SOTA. This can +be explained by a large amount of additional data (65%) +from the Sketch domain, exploited by baselines that are fine- +tuned on the target domain. Our method performs much +better than the baseline on all domains compared to the +same setting. When comparing the two variations of BrAD +(Harary et al., 2021) with 1% source domain labels, we +achieve superior results on all domains. Overall, even when +using 10% of source domain label, our method is better than +BrAD-KNN (Harary et al., 2021). +Ablation study +We use the PACS dataset to perform an +ablation study. The first variant of our model, which we +term “Plain pre-training”, uses a standard MoCoV2 back- +bone followed by training the clustering heads using our +best setup. We omit the style transfer augmentation in pre- +training and clustering heads training in the second ablation. +A third ablation, ”no domain head,” is performed using the +full training procedure except for the domain head and its +adversarial loss. The fourth ablation on the PACS dataset +was done by removing the label smoothing (using 1 as the +smoothing value). As indicated by the results presented in +Table 4, our model performs better than all of its ablated +versions. This suggests that all proposed components con- +tribute to our ability to generalize to unseen domains. In +cases where not all clusters were predicted and thus, no +clustering accuracy can be calculated (NA). +Table 4. Ablation study on PACS dataset upon all four domain +combinations, each of the letters A, P, S, C represent the domains +Art painting, Photo, Sketch, and Cartoon, respectively. The no- +tation W, X, Y → Z, means the model was trained on W, X, Y +domain and tested on the Z domain. We denote by NA cases in +which some of the classes were not predicted by the model, which +makes calculating clustering accuracy unavailable. +Method +C, P, S A, P, S A, C, S A, C, P Avg +→ A +→ C +→ P +→ S +Logits only +NA +NA +NA +NA +NA +Plain pre-training +25.0 +22.7 +29.8 +NA +NA +No style transfer +40.6 +37.2 +50.2 +45.8 +43.5 +No domain head +40.2 +41.5 +58.6 +42.8 +45.8 +No smoothing +46.4 +44.5 +65.8 +43.9 +50.4 +Ours +46.7 +44.7 +66.8 +47.2 +51.9 +5. Conclusions +This paper presents a novel framework for completely unsu- +pervised multi-source domain generalized clustering. 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IEEE Transactions +on Neural Networks and Learning Systems, 2020. + +Domain-Generalizable Multiple-Domain Clustering +A. Further implementation details +A.1. Strong augmentations +In pre-training phase, strong augmentations include: random resized crop, color jitter with probability of 0.8, and parameters +brightness=0.4, contrast=0.4, saturation=0.4, hue=0.1, random grayscale (p=0.2), horizontal flip (p=0.5) and Gaussian blur +(p=0.5). Strong augmentation for the clustering head training was done with same strategies used in SCAN (Van Gansbeke +et al., 2020): Cutout (DeVries & Taylor, 2017) and four randomly transformations from RandAugment (Cubuk et al., 2020). +A.2. Source of randomization between heads +Each head weight is initialized randomly using PyTorch(Paszke et al., 2019) default initialization. Since the pseudo labels +are determined by the classifier prediction, each head will be trained using different pseudo labels; this variability will keep +propagating as training proceeds. +A.3. Additional Results +In Figure 4, we present example images from the original domains and our BCD. Next, In Figure 5 we present sample +images from datasets used in our paper. +Figure 4. Sample images from BCD domain for Officehome dataset. The right and left columns show the original image and its BCD +transform, respectively. + +Art +BCD +Product +BCD +12 +Clipart +BCD +Real World +BCDDomain-Generalizable Multiple-Domain Clustering +Figure 5. Sample images from the datasets used in the paper. + +Sample images from each dataset +Art Painting +Photo +Sketch +Cartoon +PACS +Amazon +Webcam +DSLR +Office 31 +Art +Clipart +Product +Real World +Officehome \ No newline at end of file diff --git a/cdFRT4oBgHgl3EQfSzck/content/tmp_files/load_file.txt b/cdFRT4oBgHgl3EQfSzck/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..40444a0ce852ba35bff7d14d07ad840aafb0f675 --- /dev/null +++ b/cdFRT4oBgHgl3EQfSzck/content/tmp_files/load_file.txt @@ -0,0 +1,996 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf,len=995 +page_content='Domain-Generalizable Multiple-Domain Clustering Amit Rozner * 1 Barak Battash * 1 Lior Wolf 2 Ofir Lindenbaum 1 1Faculty of Engineering, Bar Ilan University 2 School of Computer Science, Tel Aviv University ∗ These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Abstract Accurately clustering high-dimensional measure- ments is vital for adequately analyzing scientific data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Deep learning machinery has remarkably improved clustering capabilities in recent years due to its ability to extract meaningful representa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' In this work, we are given unlabeled sam- ples from multiple source domains, and we aim to learn a shared classifier that assigns the exam- ples to various clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Evaluation is done by using the classifier for predicting cluster assign- ments in a previously unseen domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' This setting generalizes the problem of unsupervised domain generalization to the case in which no supervised learning samples are given (completely unsuper- vised).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Towards this goal, we present an end-to- end model and evaluate its capabilities on several multi-domain image datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Specifically, we demonstrate that our model is more accurate than schemes that require fine-tuning using samples from the target domain or some level of supervi- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Introduction Clustering is a fundamental machine-learning task that cate- gorizes unlabeled vectors into homogeneous groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Clus- tering high dimensional measurements is a difficult task, and classical methods such as K-means (Lloyd, 1982), Spectral Clustering (Ng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2001), or density-based methods (Ester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 1996) often fail to group semantically related exam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Several recent works have developed deep learning- based frameworks to overcome this limitation by extracting meaningful features and automatically clustering images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' However, extracting informative features from real-world images is challenging since, often, the images are collected from multiple domains with diverse properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The problem becomes even more challenging if we want to generalize the cluster assignments to unseen target domains that may deviate significantly from the source distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Multi-domain clustering methods simultaneously group samples in analog domains such that a sample in one domain is associated with semantically related samples in another (Menapace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Such methods or others that rely on labeled observations benefit from the shared knowledge between the domains (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Menapace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Harary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' For example, if one domain contains grayscale images and the other do- main contains low-resolution images of the same objects, one can learn to separate between the images of the different objects in a way that utilizes the fact that it is easier to sepa- rate between some objects based on color and between other objects based on fine details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' We do not assume knowledge about corresponding images between domains, only that all target classes appear in each domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' In this work, we study the ability to learn a classifier f that, given a sample, regardless of its domain, can categorize it to the matching cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The requirement is that f is evaluated (one sample at a time) in an unseen target domain for which no sample is seen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' This task combines the problem of multiple domain clustering with the aspect of domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Both of these are of high value, and their combination is extremely powerful: multiple source clustering is core to scientific discovery, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', in biology, where every organism can be a domain, and the application to new domains is the ultimate test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Formally, the problem statement is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' We are given a dataset S with N samples from d different domains, where each domain has Ni, ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', d} samples, and we know which sample belongs to each domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' A classifier f is trained to map every sample in S to one of K groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' For evaluation purposes only, a set of labeled samples is then used, and the K groups are assigned to a set of ground truth labels using a best-matching method (The Hungarian method (Kuhn, 1955)) on an unseen single-domain dataset T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Specifically, f is applied, without further training, to each sample of T separately, and the assigned labels f out- puts are compared to the ground truth labels of the samples in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Figure 1 illustrates our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' To solve this challenge, we propose a two-stage learning scheme: (i) training a backbone in a self-supervised do- main invariant fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' This step leads to a feature extractor that focuses on semantic features and attenuates the influ- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='13530v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='LG] 31 Jan 2023 Domain-Generalizable Multiple-Domain Clustering ence of style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (ii) training multiple clustering heads using pseudo labels generated using the semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Our method was tested on various domain generalization datasets and has shown superior results over several base- lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Our framework is at par or better than methods that rely on some supervision or adaptation to the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Background Self-supervised learning (SSL) is an emerging field in machine learning (ML) that enables extracting useful rep- resentation from unlabelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' SSL relies on creating a pretext task with fictitious labels that can be generated “for free”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' If the pretext task is correlated with a downstream task of interest, SSL techniques can be compelling for ex- tracting features useful for many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' A seminal work (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2020) encodes a query image and matches it to a dictionary of image keys using a contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' To stabilize the embedding of keys, they use a momen- tum encoder to update the representation of keys at a slow pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' By using queues, the architecture can scale to large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Another important work was presented by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' They utilized pairs of data augmentations, creating feature representations that are then projected and trained to maximize agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' After training, the projec- tion heads are discarded, and the resulting features generate state-of-the-art self-supervised results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Bootstrap your own latent (BOYL) (Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2020), utilized two neural net- works for their training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Each model is updated at a different rate making the interaction beneficial for the learning pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Further progress was made by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (2020b), improving contrastive SSL efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Deep clustering aims to exploit the strength of neural networks as feature extractors to identify a representation that better preserves cluster structures in unlabelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' In (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2016), the authors use an autoencoder to extract features while learning soft cluster assignments by mini- mizing the KL-divergence between the latent space and a prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (2017) propose an algorithm that iterates between feature extraction and performing a pairwise classification to predict whether pairs of images belong to the same cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' DeepCluster (Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2018), extends this idea by iterating between applying k-means to a deep feature mapping and training a NN to predict the clus- ter assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (2018) apply content-preserving image transformations to create pairs of samples with shared information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Then, they train a NN to maximize the mutual information between the image pairs in a cluster probabilis- tic data representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Recently, Semantic Pseudo-Labeling for Image Clustering (SPICE) (Niu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2022) obtained state-of-the-art results on several clustering benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' SPICE is an iterative deep clustering method that relies on self-supervision and pseudo-labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' First, self-supervision is performed using a contrastive loss to learn informative fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Then, they create prototype pseudo labeling to avoid miss annotations common to pseudo labeling techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Unsupervised domain generalization (UDG) was re- cently presented by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Harary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' UDG is related to our goal but requires some amount of su- pervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Specifically, UDG involves: unsupervised train- ing on a set of source domains, then fitting a classifier using a small number of labeled images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Finally, evaluating the model on a set of target domains unseen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Toward this goal Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (2021) suggests a method to ignore domain-related features by selecting negative sam- ples from a queue based on their similarity to the positive sample domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Recently, Harary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (2021) presented BrAD, in which self-supervised pre-training on multiple source domains is performed in a shared sketch-like domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' To fine-tune a classifier, they used various amounts of source domain-labeled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' In contrast to these works, in our research, no class labels for either source or target domains are used for training the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Unsupervised Clustering under Domain Shift (UCDS) was presented by (Menapace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The goal is to cluster samples from multiple source domains and then adapt the model to the target domain using multiple unla- beled samples from that domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Menapace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (2020) suggested optimizing an information-theoretic loss coupled with domain-alignment layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' This setting is similar to ours;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' however, we aim to design a model that can predict cluster assignments on multiple source domains and gener- alize to new unseen domains without any further tuning or adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' To the best of our knowledge, this is the first work that solves this task without using any labels or sam- ples from the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' This is a big advantage since, in real-world settings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' we often don’t know that a domain shift occurred, or we can not access a pool of samples from the test domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Method High-level overview Our method consists of two phases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' in the first, a feature extraction model f1 is trained in a self- supervised fashion on data from multiple source domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' This phase bridges the gap between different domains by extracting semantically related features ui ∈ Re, where e is the embedding dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The second phase focuses on training a clustering head f2 while the weights of f1 are frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Then our cluster predictions are based on f = f2 ◦ f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' We leverage a basic common domain (BCD) to mitigate the gap between several domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The BCD is designed to maintain the sample’s content while removing domain- Domain-Generalizable Multiple-Domain Clustering Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Problem statement - given unlabeled samples from mul- tiple source domains, our goal is to learn a deep clustering model that can accurately assign samples to their cluster in each source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' We aim to design a model that, at inference, can predict cluster assignments on new unseen domains without any labels or model tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' related information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Conceptually, a sketch-like domain can be considered a suitable BCD for image data with varying color and texture domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Transforming an image to a sketch domain keeps high-level features such as object iden- tity while decreasing the bias that features such as colors and image style can induce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Pre-training Our pre-training phase is inspired by MoCoV2 (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' This self-supervised method applies a contrastive loss to learn a representation invariant to a set of strong augmentations defined in the appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Each training example xi is strongly augmented, then compared using a contrastive loss to a positive x+ i (another strongly aug- mented) example and a negative example x− j , where i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The negative example x− j is a strong augmentation of xj, which is stored in a queue as explained in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The pre-training process aims to generate valuable features for the downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' We adapt MoCoV2 for the problem of domain-generalizable multiple-domain clustering by pre- senting the four components described below and illustrated in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Adversarial domain classifier: Using adversarial training for domain adaptation was presented in the seminal work of Ganin & Lempitsky (2015) and subsequently used by many (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Zhang, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Wilson & Cook, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Here, we leverage this idea and introduce a do- main classifier to the features extracted from our backbone f1 to learn features invariant to the domain identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The weights of the domain classifier fd are denoted as θd, and the gradient of fd is: ∂Lfd ∂θd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' We train f1 in an adversarial fashion by updating its weights using a constant λd ≥ 0 as in −λd ∂Lfd ∂θ1 , which is known as gradient reversal layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Domain balancing: Since we are given samples from several source domains, quite often, the population of sam- ples from each domain is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Such domain imbalance can cause poor generalization since the model may learn only from the highly populated domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' We use a simple approach to mitigate this phenomenon by choosing a do- main uniformly and then picking a sample from the chosen domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Multi queue: Multiple-domain self-supervised feature extraction is prone to focus on domain-related features re- sulting in poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Specifically, a single neg- ative sample queue has shown inferior results compared to multiple domain-specific queues (Harary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Following this work, we employ a domain-specific queue Q = [Q1, Q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', Qd] each with size Nq for each of the d domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The negative examples u− i are drawn from the same domain as the positive sample, which makes the dis- crimination more challenging and encourages the model to focus on the content rather than the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' For efficiency, positive samples u+ i are stored in the relevant negative do- main queue for later use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Style transfer augmentation: We want our backbone to learn semantic features that are invariant to the “style” of the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' To encourage this property, we use a style trans- fer neural network model (Huang & Belongie, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' We perform a style transfer augmentation ST (xi) with proba- bility pst as a replacement for the strong augmentation used in MoCoV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The style of xi is replaced by the style of an image from a different domain xj, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' dxi ̸= dxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' By augmenting with varying styles of domains, we further en- hance the ability of our backbone to ignore domain specific features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Algorithmic overview Given an input batch of images x = [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='., xB]T ∈ RB×C×W ×H, each image xi is transformed twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' First, it is augmented using a strong aug- mentation xs i = S(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Second, the image is transformed using the following: xst i = � ST (xi), w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='p pst, S(xi) , w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='p 1 − pst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (1) Where ST (xi) replaces the style of xi with another do- main’s style with probability pst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Otherwise, a strong augmentation S(xi) is applied to generate the positive Training Multiple source domains Domain index: Cartoon Clustering model Unsupervised training Domain index: Photo Inference Target domain: Art Painting Clustering model →Class 1 →Class 5Domain-Generalizable Multiple-Domain Clustering sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Both xs i and xst i are passed through the back- bone to create the embeddings us i = P(f1(xs i, θ1), θp), ust i = P(f1(xst i , θ ′ 1), θ ′ p), where P and θp are the projection head and its weights respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' θ ′ 1 = µθ ′ 1 + (1 − µ)θ1 and θ ′ p = µθ ′ p + (1 − µ)θp are the moving average ver- sions of θ1 and θp respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Finally, negative samples u− i , ∀i ∈ [0, Nq] are sampled from a domain queue of the same domain dxi as xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The contrastive loss is used: Lfproj = −log exp((us)T ust) �Nq i=1 exp((us)T u− i ) + exp((us)T ust) , (2) where us = [us 1, us 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='., us B]T ∈ RBxe, ust = [ust 1 , ust 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='., ust B]T ∈ RBxe, are the embeddings for the trans- formed input batch xs, xst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' To further remove domain-specific information, we use an additional domain loss term using cross-entropy (Kullback & Leibler, 1951) loss: Lfd = −Lce(fd(us), θd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Hence our final loss objective: Lf1 = Lfproj + Lfd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (3) The contrastive and domain-adversarial loss terms com- plement one another in finding content-related, domain- invariant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Clustering head The pre-training phase provides a solid and robust backbone f1 on which a clustering head f2 can be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Then we apply a clustering head on top of the backbone designed to predict the assignments for each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The clustering head is trained in a two-step iterative manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Each iteration begins by assigning pseudo labels based on the clustering head’s predictions (logits) l(y|x) = f2 ◦ f1(x), and the semantic features (embeddings) u = f1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' In the second step, the clustering heads are trained using the pseudo labels with cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Since biases are much more likely to arise in multi-domain tasks, using both embeddings and logits in domain generalization proves crucial, as demon- strated in the ablation study 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' An illustration of the training process is available in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Training of the clustering head is initiated by sampling a batch of images x with size B from all source domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Each sample xi passes through the backbone in three dif- ferent versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Based on the original image, and using two transformations: a strong augmentation xs i = E(xi), and style transfer to our BCD xbcd i = C(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Where C() represents a style transfer of the input image xi to an image with a sketch-like style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' E() is defined as: xst i = � � � � � C(xi) , w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='p pstpbcd, ST (xi), w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='p pst(1 − pbcd), S(xi) , w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='p 1 − pst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (4) The BCD transformed, and the original images are used to define the pseudo labels while the strong augmentations are used to train the clustering head f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' First, features are extracted from the original image: u = f1(x, θ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (5) Then, logits are extracted in the BCD based on: l(y|xbcd) = f(xbcd) = f2 ◦ f1(xbcd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (6) The top γ := B 2K samples are chosen from l(y|xbcd i ) as the set of most confident samples of class k based on the clustering head’s score on samples from the BCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Thus, the selected samples are denoted as follows: Mk = {ui|i ∈ argtopγ(l(k|xbcd)), ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', B}}, (7) where argtopγ(l(k|xbcd)) ∈ Nγ is a vector of indexes that chooses the γ most confident samples, based on their corre- sponding BCD score l(k|xbcd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Mk is a set of γ embedding vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Using Mk, the center of class k is determined by: Gk = 1 γ � ui∈Mk ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (8) One can calculate the similarity between each sample and each of the centers: simk = ⟨ ¯Gk, ¯u⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (9) Where ¯Gk = Gk ∥Gk∥, and ¯u = u ∥u∥ are the normalized feature and center vectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' In plain words, simk ∈ RB holds the information of how close each sample in the batch is to the center of cluster k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Samples that are closest to the center are used as pseudo labels for cluster k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' thus, the set of strongly augmented data samples with pseudo labels is formulated as follows: ˆZk = {xs[argtopγ(simk)], k} (10) Where xs[argtopγ(simk)] means indexing xs using argtopγ(simk), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', choosing the samples that will be used as pseudo-labels for class k using the similarity in the em- bedding domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' While Mk are chosen based on the heads predictions, which are in the logits space, ˆZk are pseudo labels chosen based on information from both the semantic space Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' 8,10 and in the logits space Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Using only the logit values to infer the pseudo labels results in poor cluster assignments, as examined in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The entire set of representatives and pseudo-label pairs will be denoted as ˆZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' In the second phase, the clustering head f2 is trained using ˆZ to minimize cross-entropy loss using the pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' A Domain-Generalizable Multiple-Domain Clustering Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The proposed pre-training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Each image is transformed using strong augmentations and/or style transfer augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The features us (strong) and ust (style) are extracted using the backbone f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Then we use a domain head fd to classify the domain identity of each sample, minimizing the domain loss Lfd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' we use gradient reversal to update the backbone to fool the domain head in an adversarial fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The contrastive loss Lfproj is minimized by the projection head’s output of us, ust, and u− (negative samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The losses complement each other in training the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' batch of samples and pseudo labels (xs pl, ypl) ∈ ˆZ, are prop- agated through the backbone and heads, and the objective can be formulated as: L = 1 B Lce(f2 ◦ f1(xpl), ypl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (11) During this training phase, domain balancing is used as detailed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Multiple clustering heads Clustering is inherently unsta- ble, especially when dealing with many classes or high- dimensional datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Several authors have proposed using feature selection (Solorio-Fern´andez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Shaham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Lindenbaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2021) to improve clustering capabilities by removing nuisance features in tabular data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' We are interested in stabilizing clustering performance on diverse high-dimensional image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Therefore, we pro- pose training multiple clustering heads simultaneously and selecting a reliable head based on an unsupervised criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' This allows us to handle many categories and overcome the instability that stems from the clustering heads weights’ initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' For more details about the source of ran- domization between heads, please see appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The number of clustering heads is denoted as h, hence the objec- tive in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' 11, L can now be formulated as the average of the h head specific losses: L = 1 Bh h � i=1 Li ce(f2 ◦ f1(xpl), ypl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (12) Next, we define the diversification of head j as: dvj = unique B argmax k∈K lj(y|x)/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (13) First, argmax reduces the prediction lj(y|x) of the j-th head to a cluster index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Next, we use the unique operator to extract the number of clusters that head j predicts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' this process is done using the entire dataset in parallel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', there is no parameter update during this evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Due to high variability in the training procedure between heads, some are better than others;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' we leverage this vari- ability by keeping only the most diversified heads (MDH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Two MDH are chosen out of h clustering heads based on higher dvj values compared to the other heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The heads with lower dvj are discarded, and we replace the weights of the non-MDH with a linear combination of the two MDH weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Mathematically speaking, let us define θ2i as the weights of the i-th head, and let us assume that j, k are the MDH indices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' hence the weights of the non-MDH heads are overridden in the following manner: θ2i = rk θ2k ∥θ2k∥ + rj θ2j ∥θ2j∥, ∀i ̸= k, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (14) Where rk ∼ U(0, 1) and rj = 1 − rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' This removes the influence of non-diverse heads and maintains some degree of variability for the following optimization steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' In cases where there is equality in dvj between several heads, which results in more than two MDHs, we limit the Domain head Backbone f d Features Doamin loss Gradient reversal Projection head J proj Contrastive positive keys stored in loss per domain queue Negative examples queue Negative keys are taken from the same domain as the positive examples Domain 1 negative queue u Domain d negative queueDomain-Generalizable Multiple-Domain Clustering Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Clustering head training scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The image is passed through the backbone in its original, strongly augmented, and BCD form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The weights of the backbone are frozen and used to produce the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Representatives are selected from the original image features based on the clustering head’s predictions over the BCD images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The class representatives are used as pseudo labels for the CE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' number of MDHs to five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The rationale behind this limi- tation can be elucidated through the following illustrative scenario, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', assume that the first head does not predict one class, and the other heads do not predict five classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' if the number of MDH kept is not limited, the advantage of the first head is not utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Since all heads will perform poorly in the early training phase, MDH selection is initiated after a few epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Furthermore, to allow the heads to make gradual learning, the process repeats every n epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Experiments Experiments are conducted using three datasets commonly used for evaluation of domain generalization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Rep- resentative images from several datasets and domains appear in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The Office31 dataset (Saenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2010) consists of images collected from three domains: Amazon, Webcam, and DSLR, with 2817, 795, and 498 images, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The dataset includes 31 different classes shared across all domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The samples consist of objects com- monly encountered in an office setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The PACS dataset (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2017) consists of four domains: Sketch, Cartoon, Photo, and Artpainting with 3929, 2344, 1670, and 2048 im- ages, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' It includes seven different classes, which are shared across all domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The Officehome dataset (Venkateswara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2017) contains four domains: Art, Product, Realworld, and Clipart, with 2427, 4439, 4357, and 4365 images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' It includes 65 different classes, which are shared across all domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The large number of domains, and classes, make the task challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' In par- ticular, since we aim to cluster the data without access to labeled observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Existing state-of-the-art results on this data (Menapace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2020) corroborate this claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Implementation details Our work is implemented using PyTorch (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' We use Resnet18 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2016) as our backbone for a fair comparison with the results of Menapace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Harary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The models in the pre-training were trained using SGD with momentum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 and weight decay 1e − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' We use a batch size of 8 and train the model for 500 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' To train the clustering head, we use the same optimizer with batches of size 256 for 100 epochs for Office31 and Officehome datasets and 50 epochs for the PACS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The reason for this difference is the small number of classes in the PACS dataset, which enables the model to converge much faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' To create style transfer augmentations, we use a pre-trained AdaIN model (Huang & Belongie, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The most diversified head selection mechanism initiates at epoch 30 and is repeated every n = 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' For more information on the head selection mechanism, see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' An important regularization for diversified training is label smoothing (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' By using pseudo- labels, we assume that there is a high ratio of mislabeled samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' label smoothing helps preventing the model from predicting the training samples too confidently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Empiric evidence of label smoothing importance in our task can be seen in the ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Comparison with other methods To evaluate the capa- bilities of our model, we focus on the following scheme: train the model using d unlabelled source domains, then eval- uate our model on the unseen and unlabelled target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' We compare our approach to several recently proposed deep learning-based models for multi-domain clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' When evaluating Office31 and Officehome datasets, we com- pare with the baselines from (Menapace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2020): popu- Backbone Features f1 Clustering heads l(y|αbced) Select Representitives Pseudo labels for selected representatives CE lossDomain-Generalizable Multiple-Domain Clustering Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Results on the Office31 dataset (31 classes) upon all three domain combinations, each of the letters A, W, D represent the domains Amazon, Webcam, and DSLR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The notation X, Y → Z, means the model was trained on X, Y domain and tested on the Z domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Target fine-tuned means the method was trained or adapted to the test domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' In K-means, we first pre-trained the MocoV2 model and trained K-means on top of its embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Method Target fine-tuned Supervision D, W → A A, W → D A, D → W Avg DeepCluster (Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2018) ✓ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='7 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 IIC (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2018) ✓ 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='0 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='3 IIC-Merge (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2018) ✓ 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='5 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 IIC + DIAL (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2018) ✓ 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='3 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4 Continuous DA (Mancini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2019) ✓ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='5 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='8 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='6 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='6 ACIDS (Menapace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2020) ✓ 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='6 K-means (Lloyd, 1982) ✓ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='3 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='8 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 Ours 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Results on the Officehome dataset (65 classes) upon all four domain combinations, each of the letters A, P, R, C represent the domains Art, Product, Realworld, and Clipart, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The notation W, X, Y → Z, means the model was trained on W, X, Y domain and tested on the Z domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Target fine-tuned means the method was trained or adapted to the test domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' In K-means, we pre-train the MocoV2 model and then train K-means on top of its embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Method Target fine-tuned Supervision C, P, R → A A, P, R → C A, C, R → P A, C, P → R Avg DeepCluster (Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2018) ✓ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='6 IIC (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2018) ✓ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4 IIC-Merge(Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2018) ✓ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='3 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='3 IIC + DIAL(Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2018) ✓ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='0 Continuous DA (Mancini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2019) ✓ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='7 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='6 ACIDS (Menapace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2020) ✓ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='7 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='0 K-means (Lloyd, 1982) ✓ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='3 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 Ours 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='8 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='6 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='0 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Results on the PACS dataset (7 classes) upon all four domain combinations, each of the letters A, P, S, C represents the domains: Art painting, Photo, Sketch, and Cartoon, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The notation W, X, Y → Z, means the model was trained on W, X, Y domain and tested on the Z domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Target fine-tuned means the method was trained or adapted to the test domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' In K-means, we first pre-trained the MocoV2 model and trained K-means on top of its embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Method Target fine-tuned Supervision C, P, S → A A, P, S → C A, C, S → P A, C, P → S Avg DeepCluster(Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2018) ✓ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4 IIC (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2018) ✓ 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='8 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='6 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='6 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 IIC-Merge (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2018) ✓ 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 IIC + DIAL(Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2018) ✓ 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='5 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='7 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='3 Continuous DA (Mancini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2019) ✓ 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='0 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 ACIDS (Menapace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2020) ✓ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='5 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='5 K-means (Lloyd, 1982) ✓ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='7 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 Ours 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='7 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='7 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 BrAD (Harary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2021) 1% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='6 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='5 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='8 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='8 BrAD-KNN (Harary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2021) 1% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='5 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='0 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='7 BrAD (Harary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2021) 5% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='7 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='0 BrAD-KNN (Harary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2021) 5% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='7 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='3 BrAD (Harary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2021) 10% 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='7 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='5 BrAD-KNN (Harary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2021) 10% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='0 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='3 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1 Domain-Generalizable Multiple-Domain Clustering lar deep clustering papers Invariant Information Clustering for Unsupervised Image Classification and Segmentation (IIC) (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2018), and DeepCluster (Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Importantly, they were both trained directly on the target do- main before predicting the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Menapace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' (2020) used two variations of IIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Specifically, IIC-Merge involves training IIC on all domains, including the target domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' IIC+DIAL: IIC, which contains a domain-specific batch norm layer jointly trained on all domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Continuous DA: continuous domain adaptation strategy used in (Mancini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2019) using the method presented in (Menapace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2020) denoted as Adaptive Clustering of Images under Do- main Shift (ACIDS), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', train on d domains, adapt on the target domain and then test on the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Note that all the former baselines compared with our work were trained on the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' We added another baseline, training MoCoV2 on all the source domains and then fitting the K-means clustering algorithm on the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' On PACS dataset, we also compared ourselves to BrAD (Harary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2021) with various amounts of source domain labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' This comparison is very challenging as we do not use any class labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Results Table 1 depicts the results on all three domain combinations of the Office31 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' On both DSLR and Webcam as target domains, our method outperforms the cur- rent state-of-the-art (SOTA) by a large margin, even without adaptation to the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Our method performance on the Amazon domain is inferior to the current SOTA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' we be- lieve this is due to the very limited source domain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The target fine-tuned method (unsupervised fine-tuning on the target domain) relies on 317% more data for their training scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Overall, on average, our method is better by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1% than the method that uses the target domain for adaptation and 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1% over the baseline with the same conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Results on the Officehome dataset can be seen in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' This dataset is more challenging than the former and consists of four domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Our method outperforms the baselines on all four domain combinations and is better on average by 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1% percent than the previous SOTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' On the PACS dataset (Table 3), our method is compared to both target fine-tuned and limited source domain label settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Our method outperforms the current SOTA on 3 out of 4 target domains for the target fine-tuned case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' On the fourth domain, Sketch, our method achieves slightly lower results than the current target fine-tuned SOTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' This can be explained by a large amount of additional data (65%) from the Sketch domain, exploited by baselines that are fine- tuned on the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Our method performs much better than the baseline on all domains compared to the same setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' When comparing the two variations of BrAD (Harary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2021) with 1% source domain labels, we achieve superior results on all domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Overall, even when using 10% of source domain label, our method is better than BrAD-KNN (Harary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Ablation study We use the PACS dataset to perform an ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The first variant of our model, which we term “Plain pre-training”, uses a standard MoCoV2 back- bone followed by training the clustering heads using our best setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' We omit the style transfer augmentation in pre- training and clustering heads training in the second ablation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' A third ablation, ”no domain head,” is performed using the full training procedure except for the domain head and its adversarial loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The fourth ablation on the PACS dataset was done by removing the label smoothing (using 1 as the smoothing value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' As indicated by the results presented in Table 4, our model performs better than all of its ablated versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' This suggests that all proposed components con- tribute to our ability to generalize to unseen domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' In cases where not all clusters were predicted and thus, no clustering accuracy can be calculated (NA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Ablation study on PACS dataset upon all four domain combinations, each of the letters A, P, S, C represent the domains Art painting, Photo, Sketch, and Cartoon, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The no- tation W, X, Y → Z, means the model was trained on W, X, Y domain and tested on the Z domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' We denote by NA cases in which some of the classes were not predicted by the model, which makes calculating clustering accuracy unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Method C, P, S A, P, S A, C, S A, C, P Avg → A → C → P → S Logits only NA NA NA NA NA Plain pre-training 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='7 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='8 NA NA No style transfer 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='6 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='8 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='5 No domain head 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='5 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='6 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='8 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='8 No smoothing 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='8 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4 Ours 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='7 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='7 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='8 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Conclusions This paper presents a novel framework for completely unsu- pervised multi-source domain generalized clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' To the best of our knowledge, this paper presents, for the first time, a framework where no class labels are used for either source or target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Further, no adaptation to the target do- main is required, which demonstrates better generalization abilities to unseen domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Our solution outperforms all existing baselines while being evaluated in a more stringent (and realistic) setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' We consider several future directions, specifically extending our model to other modalities, such as audio and text, and generalizing the clustering task to other important unsupervised learning tasks, such as anomaly de- tection or feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' We believe our idea has great potential to advance the unsupervised multi-domain regime and be applied in other future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Domain-Generalizable Multiple-Domain Clustering References Caron, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', Bojanowski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', Joulin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', and Douze, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Deep clustering for unsupervised learning of visual features.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' arXiv preprint arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='06219, 2(3):4, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', Yue, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', Zhao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', Wu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', Kr- ishna, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', Gonzalez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', Sangiovanni-Vincentelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', Seshia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' A review of single-source deep un- supervised visual domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' IEEE Transactions on Neural Networks and Learning Systems, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Domain-Generalizable Multiple-Domain Clustering A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Further implementation details A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Strong augmentations In pre-training phase, strong augmentations include: random resized crop, color jitter with probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='8, and parameters brightness=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4, contrast=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4, saturation=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='4, hue=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='1, random grayscale (p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2), horizontal flip (p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='5) and Gaussian blur (p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Strong augmentation for the clustering head training was done with same strategies used in SCAN (Van Gansbeke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2020): Cutout (DeVries & Taylor, 2017) and four randomly transformations from RandAugment (Cubuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Source of randomization between heads Each head weight is initialized randomly using PyTorch(Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=', 2019) default initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Since the pseudo labels are determined by the classifier prediction, each head will be trained using different pseudo labels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' this variability will keep propagating as training proceeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Additional Results In Figure 4, we present example images from the original domains and our BCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Next, In Figure 5 we present sample images from datasets used in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Sample images from BCD domain for Officehome dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' The right and left columns show the original image and its BCD transform, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Art BCD Product BCD 12 Clipart BCD Real World BCDDomain-Generalizable Multiple-Domain Clustering Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Sample images from the datasets used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} +page_content=' Sample images from each dataset Art Painting Photo Sketch Cartoon PACS Amazon Webcam DSLR Office 31 Art Clipart Product Real World Officehome' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFRT4oBgHgl3EQfSzck/content/2301.13530v1.pdf'} diff --git a/eNE1T4oBgHgl3EQfyQUU/content/tmp_files/2301.03430v1.pdf.txt b/eNE1T4oBgHgl3EQfyQUU/content/tmp_files/2301.03430v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a62e13fa59bbbe84b26503991314c075847721f7 --- /dev/null +++ b/eNE1T4oBgHgl3EQfyQUU/content/tmp_files/2301.03430v1.pdf.txt @@ -0,0 +1,340 @@ +Thermal diffuse scattering: elastic and coherent +Yuan Yao +Beijing National Laboratory of Condensed Matter Physics, Institute of Physics, Chinese Academy +of Sciences, Beijing 100190, People’s Republic of China +Corresponding authors: yaoyuan@iphy.ac.cn + +Abstract +The thermal diffuse scattering (TDS) by the interaction between high energy electrons and +phonon vibration has been re-investigated. The elastic scattering potential of an oscillating atom +conserves the scattering ability though the vibration changes its spatial distribution. Moreover, the +thermal scattered electrons are also coherent to form the diffuse interference patterns under the +time-average assumption. Elastic and coherent TDS contributes most contrast in Kikuchi lines. + +1. Introduction +Thermal diffuse scattering (TDS) is a historic topic in scattering physics of transmission +electron microscopy (TEM). As same as in X-ray diffraction, this conception has been introduced +to explain the intensity dilution of the Bragg reflections which correspond to the ordered lattice in +specimen. TDS originates from the thermal vibration of the atoms that breaks the regular period of +the atomic arrangement, weakens the strength of Bragg diffraction and spreads a diffuse intensity +in the background. Yoshioka1 proposed a general theory to treat the electron scattering taking +account of inelastic process. He deduced a coefficient representing the energy transfer to excited +atoms based on Bloch waves and attributed the intensity decreasing of Bragg spots to an inelastic +scattering. Later, Takagi2 pointed out that the difference ΔV(𝑟⃗) between the scattering potential of +the atoms V(𝑟⃗) and its time-average should be responsible for the diffuse scattering. That +scattering potential variation is equal to an absorption which perturbs Bloch theorem so that the +Bragg reflections reduce their intensity if considering ΔV(𝑟⃗) as absorption potential. Because ΔV(𝑟⃗) +is determined by the atom departure from its equilibrium position, the scattering potential loss +correlates to the phonon vibration modes. This assumption directly bridges the atom scattering +potential and thermal fluctuation so that ΔV(r) or phonon frequency spectrum can be formally used +to calculate diffuse scattering by the perturbation transition matrix with Bloch waves. Although such +treatment did not regard ΔV(r) as the energy transfer from incident electrons to phonons, Takagi +thought the diffuse scattering accompanies the emission or absorption of phonon vibration, implying +that the scattering is inelastic and ΔV(r) should be equivalent to the phono absorption or a decay +absorption potential. Yoshioka and Kainuma3 found that the calculated intensity by the inelastic +phonon scattering was much less than the dilution of Bragg scattering, but they still introduced an +absorption potential originated from the lattice fluctuation into the scattering potential to describe +the thermal influence on diffraction. Whelan4 developed the inelastic scattering theory and explicitly +included the phonon scattering into the inelastic framework to calculate the attenuation of +corresponding reflections. He selected the phono absorption potentials defined by Takagi and related +it to Debye-Waller (DW) factor. Einstein model was chosen to simplify the phonon vibration mode. +So an inelastic framework containing absorption potential (imaginary potential) which connects to +DW factor has been established. It is an energy transfer process and therefore incoherent scattering. +Rez et al5 extended it to involve the partial coherence effect in the inelastic phonon scattering +process. + +But Hall and Hirsch6 believed elastic scattering by the vibrating atoms dominates the diffuse +feature. They claimed that TDS is essential an elastic scattering though the scattering potential +should be modified by the phonon-related mechanism. Then they employed an effective absorption +potential and Einstein model to predict the fade of Bragg reflexion and interpreted the diffuse +scattering as the coherent summation of the Bloch waves with DW factor. Here, absorption potential +derived from DW factor or Einstein model is just a tool to expect the attenuation of Bragg spots, not +means a true energy transfer between incident electrons and phonons. The opinion actually laid the +theory foundation that simulates the diffuse intensity as a summation of elastic scattering from +various atom configurations with frozen lattice (FL) or frozen phonon (FP) model in multislice +algorithm. +Another important issue is whether the electrons scattered by phonons are coherent. In +principle, inelastic scattered electrons should be incoherent in most cases while elastic scattered +electrons may or may not be coherent, depending on the specific scattering process. Van Dyck7 +studied coherence of inelastic scattered electrons and concluded that TDS should be incoherent +owing to its inelastic property. Wang8 used “quasi-elastic” scattering to describe TDS and proved +that the diffuse scattering is incoherent based on Yoshioka inelastic theory even with Einstein model. +He also barrowed the schematic from Takagi in which the instantaneous scattering potential V(𝑟⃗) is +larger than the time-averaged static potential to illuminate the contrast from TDS.9 Ten years +later Van Dyck10 tried to solve the contradiction between the elastic nature in multislice program +with FL model and the inelastic property of phonon absorption. He claimed that the calculated +diffuse intensity from the integration of various elastic configurations with quantum-mechanical +treatment should be in fact equal to the inelastic distribution generated by phonon vibration just +because they are caused by the potential different ΔV(𝑟⃗), with an implicit assumption that ΔV(𝑟⃗) +correlates the energy transfer to lattice vibration. Van Duck’s statement ensures the adequacy of FL +or FP model in image simulation without inelastic mechanism or absorption potential and the +quantitatively interpretation has been achieved for the experimental high angle annular dark field +(HAADF) images where TDS dominates the contrast.11 +Here, we will re-argue the physical basis of TDS and refer it to elastic and coherent process. + +2. Theory +2.1 Elastic or inelastic TDS? +High energy incident electrons can evoke many inelastic scattering processes, such as plasmon +oscillation, ionic absorption and phonon exciting etc. Plasmon oscillation and shell transition +seldom occur in room temperature but phonon vibrations always exist because of the environment +thermal fluctuation. An essential fact is that the population of phonons at given temperature is not +determined by the incident electrons even though high energy electrons may produce additional +phonons within specimen. DW factor is the gauge for intrinsic phonon distribution in environment +temperature, not the probe for the electron beam created phonons. Thus it is not correct to use DW +factor to estimate the energy absorption of the excited phonon because it means that all phonons in +specimen are excited by electron bombard, which obviously greatly exaggerates the electron- +phonon process. Actually the temperature increasing by incident beam is limited and restricted +locally. So the calculations with phonon population or DW factor are not the available approaches +for inelastic phonon scattering. + +a) + b) + c) + +d) +e) + +Fig. 1 a) Project potential V(𝑟⃗) of a static Si atom. b) Average project potential of a vibrating +Si atom. c) ΔV(𝑟⃗) = V(𝑟⃗) - . d) Comparison of the line profiles in a) and b). d) Magnification +of the outside part of d). Red line is the data for thermal fluctuation effect. + +The discrepancy ΔV(𝑟⃗) between the scattering potential V(𝑟⃗) of atom and its time-average +static really exists. Takagi and followers regarded it as a true loss of scattering ability and +directly implement ΔV(𝑟⃗) as an inelastic perturbation to measure the TDS intensity. However, this +assumption is questionable since the spatial movement of atomic potential will definitely +redistribute the elastic scattering. To reveal the atom vibration effect on scattering potential, I +simulated the spatial distribution of project scattering potential of a Si atom with and without +thermal oscillation (Fig. 1a and Fig. 1b). DW factor is 0.5 and 100 random positions determined by +Gaussian function are employed to mimic the thermal oscillation. Fig. 1c and Fig. 1d compare the +difference of two potential maps and line profiles, respectively. At first glance, it is apparent that +ΔV(𝑟⃗) is much positive near the equilibrium position, indicating that atomic vibration attenuates the +scattering potential. Similar picture was also demonstrated by Wang9 to deduce that the decayed +potential represents the energy transfer to phonons. However, the total intensity of Fig. 1c is 52.4942, +almost zero if considering that the total intensity of Fig. 1a and Fig. 1b are 1.093×108. Fig. 1c and +Fig. 1e reveal that the time-average potential has a larger value outside the central region. Although +the amount of excess is tiny on each pixel near the edge, the large number of outside pixels +compensates the potential loss in the center. This property cannot be figured by the potential map +of Wang because a DW factor decay there failed to portray the excess part away from the center. +Indeed, it is easy to recognize that the vibrating atom should spread the coming electrons to a wider +area by sacrificing the central strength. ΔV(𝑟⃗ ) is positive around the equilibrium position but +negative in other places. So the total scattering potential does not loss as expected, it is just +redistributed in real space! A rational deduction from the potential conservation is that the scattering +event by vibrating atom is still elastic. In fact, the experiment of the electron energy loss spectrum +(EELS) verified that the elastic scattered electrons overwhelm the inelastic phonon scattered +electrons at the diffuse region.12 It convinces the validity of the elastic scattering assumption in +multislice method with FL or FP model to calculate TDS. + +2.2 Spatial Coherence or incoherence? + +Without TDSWith TDS0.7 +0.71200 +noTDS +1100 +TDS +1000 +900 +800 +700 +600 +500 +400 +-0.2 +0.1 +0.0 +0.1 +0.2 +A416.5 +no TDS +TDS +416.0 +415.5 +415.0 +414.5 +414.0 +413.5 +-0.25 +-0.24 +-0.23 +-0.22 +-0.21 +Aa) + b) + +c) +d) + +Fig. 2Simulated diffraction of artificial cubic structure a) containing static Si atoms and b) with +FL model, respectively. Comparison of the diffraction intensity of static diffraction and thermal +vibrating diffraction by c) linear and d) logarithm scale, respectively. Red line is the profile of +diffraction intensity containing TDS. + +Fig. 2a and Fig. 2b show the simulated diffraction patterns for an artificial crystal with and +without thermal diffuse effect, respectively. Si atoms arrayed in a 10 × 10 cubic lattice with distance +3 Å. For simplifying the calculation, one layer atoms were used for generating the projection +potential under sampling 0.01 Å. The absolution of fast Fourier transform (FFT) of project potential +was seemed as the intensity of diffraction. FL model with DW factor 0.5 were implemented and +1000 configurations were averaged to achieve TDS feature. It is obvious that the FL model can +reproduce the diffuse intensity in the diffraction image (Fig. 2b). +A key point is that the diffuse quality does not naturally relate to the incoherence. That the +disordered structure, such as amorphous state or defects, can lead the coherent diffuse intensity is a +general phenomenon in diffraction physics. So it is wrong to discriminate an inelastic or elastic +characteristic based on whether a scattered electron falls on the Bragg point. Unfortunately, this +mistake appeared in some literature which directly judge TDS as the inelastic scattering just for its +diffuse feature. + +cabic-s T35 +150 +100 +50 +6 +41x1011- +Ixio10. +1x109. +1x10E +1α107 +1x10f- +1x105. +1x104- +1000 +108 +10 +-7 +-4 +-5 +-4 +-1 +-2 +-1 +2 +6a) + b) + +c) + d) + +Fig. 3 Simulated diffraction patterns for independent electron events. a) and b) for static structure, +c) and d) for vibration lattice. Numbers are the total accumulating electrons. + +In order to illuminate coherent or incoherent behavior of thermal scattered electrons, we +simulated a virtual “single electron scattering” experiment. The intensity of each pixel on Fourier +transformed exit wave map was converted to the emergency probability of electrons in that pixel. +Time-average scattering is a statistical buildup for many scattering events within a given time. So it +is implicitly equivalent to the summation of snapshot of single electron scattering. One snapshot of +the electron distribution in reciprocal space (g space) of cause captured the coherent scattering +characteristic. Thousands of snapshots were imposed to disclose the diffraction features. The +integration of snapshots was certainly an incoherent summation. Fig. 3a is the data for the static +atom configuration which is undoubtedly elastic scattering. However, without pre-knowledge of the +specimen structure, one cannot judge which point in g space is the Bragg point when few scattering +events are acquired because there is no regular distribution. Actually, all point in g pace is equal, no +point particularly correlates to a Bragg reflection or elastic scattering. As number of electrons +increases, the Bragg features appear gradually (Fig. 3b). From the quantum mechanism viewpoint, +single electron also carries the probability of coherent exit wave to the imaging plane. “Incoherent +summation” of the coherent snapshots still displays the property of coherence, forming the Bragg +reflections in Fig. 3b. Single electron interference experiment achieved by Tonomura et al attested +that long-exposure imaging of the independent electrons still appeared the Young’s interference + +Total 12Total 615Total 33Total 24010fringes, verifying that time-averaging does not damage the interference behavior of scattering.13 +Now for TDS simulation with FL model, similar phenomenon repeats. It should be emphasized +that each snapshot is the Fourier transform of the elastic potential array carrying a thermal +fluctuation. Thus each snapshot still mirrors the “coherent scattering” of the vibrating elastic +potential for the corresponding lattice configuration. For low does image, one cannot distinguish +the Bragg reflection (Fig. 3c). Sharp spots emerge when enough electrons accumulate (Fig. 3d). As +mentioned above, all points in g space are equal and one should not appoint which spot corresponds +a coherent or incoherent scattering. The diffuse intensity beyond the sharp spots also reflects the +“coherent scattering” from an atom array, to be precise, from the vibration effect, although it is an +“incoherent summation” or a time-average product. + +3. Discussion +High energy electrons can certainly excite extra atom vibration in specimen. The creation of +new phonons is the true energy absorption for incident beam and leads to the inelastic scattering. +However, the contribution of this inelastic event to TDS is small comparing to the elastic scattering +by the vibrating atoms. Using the phonon population, for example, in room temperature to estimate +the inelastic scattering ability by phonons with perturbation theory is obviously wrong because in +reality the energy transferred to phonon is not so much. The assumption that thermal fluctuation of +the atom will reduce its elastic scattering ability is not correct. The elastic potential conserves the +scattering ability so that the inelastic treatment with the energy loss ΔV(𝑟⃗) or DW factor should be +abandoned. +The perturbation approach suggested by Hall and Hirsch in elastic framework is a valid way to +solve the scattering attenuation of Bragg reflections by recombining the elastic Bloch waves with +DW factor. The assistant “absorption potential” to calculate TDS intensity should be stated with +caution since it is a virtual concept to describe the dilution of Bragg spots, not corresponding to a +true energy transfer in the usual sense. The multislice method including the FL or FP model is +practical to implement TDS simulation because it depicts the actual physical picture of thermal +elastic scattering and the coherence characteristic. DW factor in the simulation does not weaken the +scattering ability, but breaks the structure order to result in the diffuse feature. +In principle, the elastic scattered electrons by vibrating atoms are coherent to forge the +interference patterns. Interference patterns are not always regular sharp Bragg spots while diffuse +intensity cannot be taken for granted as incoherent effect. The contrast of time-average diffraction +in real physical process or in incoherent buildup also demonstrates the coherence characteristic of +the elastic scattered electrons. Bragg reflections are the coherent interference of electrons scattered +by periodic equilibrium potential and TDS reflects the coherent interference of the electrons +scattered by the fast variation of the potential. From a statistic point of view, the probability of all +atoms locating on the equilibrium position is very small, but this configuration can result in a large +in-phase consequence which instead tremendously improves the chance of electron emergency in +Bragg spots. The out-of-phase electrons scattered by the atom arrangement deviated from the +periodic position should display a weak contrast because of the low appearance probability though +those configurations are large probability events. From the perspective of signal processing, various +configurations contain common frequency components corresponding to the regular lattice so that +superimpose of the Fourier transform result in the enhancement of Bragg reflections. It may be the +reason why Bragg reflections are very sharp in the real diffraction patterns. + +The inelastic scattering with energy absorption by phonons really happens during electrons +penetrate the specimen and contributes diffuse contrast. The ratio of this inelastic scattering is small +if checking the EELS shape in the literature which contains the phonon absorption.12 The discussion +about phonon absorption is out of the scope of this report. How many phonons are excited by +incident electrons or what is the true phonon spectrum concerning the energy exchange between +electrons and phonons needs to be thought with great caution. + + +Fig. 4 Simulated CBED patterns of [111] Si. Numbers are the summations of configurations. +Accelerate energy: 100 kV, convergent angle: 8 mrad, Cs: 3.3 mm, defocuse: -140 nm, sampling: +0.05 Å/pixel, image size: 1024 × 1024, thickness: 489 Å. + +Another interesting phenomenon related to TDS is Kikuchi lines. The multislice simulations +of convergent beam electron diffraction (CBED) with and without FL model for [111] Si are shown +in Fig. 4. As number of configures increases, Kikuchi lines become stronger. It at least proves that +elastic potential and Einstein model can reproduce the Kikuchi patterns. Some literature also +confirmed this result.12, 14 + +Conclusion +The summary is simple. The spatial distribution of elastic scattering potential is changed by +the thermal vibration of atoms but the total scattering ability still conserves. TDS is predominated +by oscillated elastic scattering potential. The diffuse feature of TDS reflects the coherent +interference of the electrons scattered by fluctuating lattice in spite of the time-averaging assumption. + +Acknowledgement +This work was supported the National Key R&D Program of China (No. 2022YFB3803900). + + +No TDS +5 +10 +25References +1. H. Yoshioka, Effect of Inelastic Waves on electron diffraction, J. Phys. Soc. Jpn. 12 (1957) 618- +628. +2. S. Takagi, On the temperature diffuse scattering of electrons (1-2), J. Phys. Soc. Jpn. 13 (1958) +278-296. +3. H. Yoshioka, Y. Kainuma, The effect of thermal vibrations on electron diffraction, J. Phys. Soc. +Jpn. 17 (1962) 134-136. +4. M. J. Whelan, Inelastic Scattering of Fast Electrons by Crystals. II. Phonon Scattering, J. Appl. +Phys. 36 (1965) 2103-2110. +5. P. Rez, C. J. Humphreys and M. J. Whelan, The distribution of intensity in electron diffraction +patterns due to phonon scattering, Philos. Mag. 35 (1997) 81-96. +6. C. R. Hall and P. B. Hirsch, Effect of thermal diffuse scattering on crystals propagation of high +energy electrons through, Proc. R. Soc. A 286 (1965) 158-177. +7. D. Van Dyck, Inelastic Scattering and Interference, Microsc. Microanal. 3(suppl. 2) (1997) +1033-1034. +8. Z. L. Wang, The ‘frozen-lattice’ approach for incoherent phonon excitation in electron +scattering. How accurate is it? Acta Cryst. A 54 (1998) 460-467. +9. Z. L. Wang, Phonon scattering: How does it affect the image contrast in high-resolution +transmission electron microscopy? Philos. Mag. B 79 (1999) 37-48. +10. D. Van Dyck, Is the frozen phonon model adequate to describe inelastic phonon scattering? +Ultramicroscopy 109 (2009) 677-682. +11. J. M. LeBeau, S. D. Findlay, L. J. Allen and S. Stemmer, Quantitative atomic resolution +scanning transmission electron microscopy, Phys. Rev. Lett. 100 (2008) 206101. +12. F. S. Hage, D. M. Kepaptsoglou, Q. M. Ramasse and L. J. Allen, Phonon spectroscopy at atomic +resolution, Phys. Rev. Lett. 122 (2019) 016103 +13. A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, and H. Ezawa, Demonstration of single- +electron buildup of an interference pattern, Am. J. Phys. 57 (1989) 117-120. +14. E. J. Kirkland, Advanced computing in electron microscopy (Second Edition, 2010), Springer + diff --git a/eNE1T4oBgHgl3EQfyQUU/content/tmp_files/load_file.txt b/eNE1T4oBgHgl3EQfyQUU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f0ad6712a0878c2aa663ee8b61438294e46870a3 --- /dev/null +++ b/eNE1T4oBgHgl3EQfyQUU/content/tmp_files/load_file.txt @@ -0,0 +1,311 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf,len=310 +page_content='Thermal diffuse scattering: elastic and coherent Yuan Yao Beijing National Laboratory of Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China Corresponding authors: yaoyuan@iphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='cn Abstract The thermal diffuse scattering (TDS) by the interaction between high energy electrons and phonon vibration has been re-investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The elastic scattering potential of an oscillating atom conserves the scattering ability though the vibration changes its spatial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Moreover, the thermal scattered electrons are also coherent to form the diffuse interference patterns under the time-average assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Elastic and coherent TDS contributes most contrast in Kikuchi lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Introduction Thermal diffuse scattering (TDS) is a historic topic in scattering physics of transmission electron microscopy (TEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' As same as in X-ray diffraction, this conception has been introduced to explain the intensity dilution of the Bragg reflections which correspond to the ordered lattice in specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' TDS originates from the thermal vibration of the atoms that breaks the regular period of the atomic arrangement, weakens the strength of Bragg diffraction and spreads a diffuse intensity in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Yoshioka1 proposed a general theory to treat the electron scattering taking account of inelastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' He deduced a coefficient representing the energy transfer to excited atoms based on Bloch waves and attributed the intensity decreasing of Bragg spots to an inelastic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Later, Takagi2 pointed out that the difference ΔV(𝑟⃗) between the scattering potential of the atoms V(𝑟⃗) and its time-average should be responsible for the diffuse scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' That scattering potential variation is equal to an absorption which perturbs Bloch theorem so that the Bragg reflections reduce their intensity if considering ΔV(𝑟⃗) as absorption potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Because ΔV(𝑟⃗) is determined by the atom departure from its equilibrium position, the scattering potential loss correlates to the phonon vibration modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' This assumption directly bridges the atom scattering potential and thermal fluctuation so that ΔV(r) or phonon frequency spectrum can be formally used to calculate diffuse scattering by the perturbation transition matrix with Bloch waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Although such treatment did not regard ΔV(r) as the energy transfer from incident electrons to phonons, Takagi thought the diffuse scattering accompanies the emission or absorption of phonon vibration, implying that the scattering is inelastic and ΔV(r) should be equivalent to the phono absorption or a decay absorption potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Yoshioka and Kainuma3 found that the calculated intensity by the inelastic phonon scattering was much less than the dilution of Bragg scattering, but they still introduced an absorption potential originated from the lattice fluctuation into the scattering potential to describe the thermal influence on diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Whelan4 developed the inelastic scattering theory and explicitly included the phonon scattering into the inelastic framework to calculate the attenuation of corresponding reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' He selected the phono absorption potentials defined by Takagi and related it to Debye-Waller (DW) factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Einstein model was chosen to simplify the phonon vibration mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' So an inelastic framework containing absorption potential (imaginary potential) which connects to DW factor has been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' It is an energy transfer process and therefore incoherent scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Rez et al5 extended it to involve the partial coherence effect in the inelastic phonon scattering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' But Hall and Hirsch6 believed elastic scattering by the vibrating atoms dominates the diffuse feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' They claimed that TDS is essential an elastic scattering though the scattering potential should be modified by the phonon-related mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Then they employed an effective absorption potential and Einstein model to predict the fade of Bragg reflexion and interpreted the diffuse scattering as the coherent summation of the Bloch waves with DW factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Here, absorption potential derived from DW factor or Einstein model is just a tool to expect the attenuation of Bragg spots, not means a true energy transfer between incident electrons and phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The opinion actually laid the theory foundation that simulates the diffuse intensity as a summation of elastic scattering from various atom configurations with frozen lattice (FL) or frozen phonon (FP) model in multislice algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Another important issue is whether the electrons scattered by phonons are coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' In principle, inelastic scattered electrons should be incoherent in most cases while elastic scattered electrons may or may not be coherent, depending on the specific scattering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Van Dyck7 studied coherence of inelastic scattered electrons and concluded that TDS should be incoherent owing to its inelastic property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Wang8 used “quasi-elastic” scattering to describe TDS and proved that the diffuse scattering is incoherent based on Yoshioka inelastic theory even with Einstein model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' He also barrowed the schematic from Takagi in which the instantaneous scattering potential V(𝑟⃗) is larger than the time-averaged static potential to illuminate the contrast from TDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='9 Ten years later Van Dyck10 tried to solve the contradiction between the elastic nature in multislice program with FL model and the inelastic property of phonon absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' He claimed that the calculated diffuse intensity from the integration of various elastic configurations with quantum-mechanical treatment should be in fact equal to the inelastic distribution generated by phonon vibration just because they are caused by the potential different ΔV(𝑟⃗), with an implicit assumption that ΔV(𝑟⃗) correlates the energy transfer to lattice vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Van Duck’s statement ensures the adequacy of FL or FP model in image simulation without inelastic mechanism or absorption potential and the quantitatively interpretation has been achieved for the experimental high angle annular dark field (HAADF) images where TDS dominates the contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='11 Here, we will re-argue the physical basis of TDS and refer it to elastic and coherent process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Theory 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='1 Elastic or inelastic TDS?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' High energy incident electrons can evoke many inelastic scattering processes, such as plasmon oscillation, ionic absorption and phonon exciting etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Plasmon oscillation and shell transition seldom occur in room temperature but phonon vibrations always exist because of the environment thermal fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' An essential fact is that the population of phonons at given temperature is not determined by the incident electrons even though high energy electrons may produce additional phonons within specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' DW factor is the gauge for intrinsic phonon distribution in environment temperature, not the probe for the electron beam created phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Thus it is not correct to use DW factor to estimate the energy absorption of the excited phonon because it means that all phonons in specimen are excited by electron bombard, which obviously greatly exaggerates the electron- phonon process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Actually the temperature increasing by incident beam is limited and restricted locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' So the calculations with phonon population or DW factor are not the available approaches for inelastic phonon scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' a) b) c) d) e) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 1 a) Project potential V(𝑟⃗) of a static Si atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' b) Average project potential of a vibrating Si atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' c) ΔV(𝑟⃗) = V(𝑟⃗) - .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' d) Comparison of the line profiles in a) and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' d) Magnification of the outside part of d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Red line is the data for thermal fluctuation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The discrepancy ΔV(𝑟⃗) between the scattering potential V(𝑟⃗) of atom and its time-average static really exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Takagi and followers regarded it as a true loss of scattering ability and directly implement ΔV(𝑟⃗) as an inelastic perturbation to measure the TDS intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' However, this assumption is questionable since the spatial movement of atomic potential will definitely redistribute the elastic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' To reveal the atom vibration effect on scattering potential, I simulated the spatial distribution of project scattering potential of a Si atom with and without thermal oscillation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 1a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' DW factor is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='5 and 100 random positions determined by Gaussian function are employed to mimic the thermal oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 1c and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 1d compare the difference of two potential maps and line profiles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' At first glance, it is apparent that ΔV(𝑟⃗) is much positive near the equilibrium position, indicating that atomic vibration attenuates the scattering potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Similar picture was also demonstrated by Wang9 to deduce that the decayed potential represents the energy transfer to phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' However, the total intensity of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 1c is 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='4942, almost zero if considering that the total intensity of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 1a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 1b are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='093×108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 1c and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 1e reveal that the time-average potential has a larger value outside the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Although the amount of excess is tiny on each pixel near the edge, the large number of outside pixels compensates the potential loss in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' This property cannot be figured by the potential map of Wang because a DW factor decay there failed to portray the excess part away from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Indeed, it is easy to recognize that the vibrating atom should spread the coming electrons to a wider area by sacrificing the central strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' ΔV(𝑟⃗ ) is positive around the equilibrium position but negative in other places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' So the total scattering potential does not loss as expected, it is just redistributed in real space!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' A rational deduction from the potential conservation is that the scattering event by vibrating atom is still elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' In fact, the experiment of the electron energy loss spectrum (EELS) verified that the elastic scattered electrons overwhelm the inelastic phonon scattered electrons at the diffuse region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='12 It convinces the validity of the elastic scattering assumption in multislice method with FL or FP model to calculate TDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='2 Spatial Coherence or incoherence?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Without TDSWith TDS0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='71200 noTDS 1100 TDS 1000 900 800 700 600 500 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='2 A416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='5 no TDS TDS 416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='0 415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='5 415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='0 414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='5 414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='0 413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='21 Aa) b) c) d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 2Simulated diffraction of artificial cubic structure a) containing static Si atoms and b) with FL model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Comparison of the diffraction intensity of static diffraction and thermal vibrating diffraction by c) linear and d) logarithm scale, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Red line is the profile of diffraction intensity containing TDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 2a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 2b show the simulated diffraction patterns for an artificial crystal with and without thermal diffuse effect, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Si atoms arrayed in a 10 × 10 cubic lattice with distance 3 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' For simplifying the calculation, one layer atoms were used for generating the projection potential under sampling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='01 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The absolution of fast Fourier transform (FFT) of project potential was seemed as the intensity of diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' FL model with DW factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='5 were implemented and 1000 configurations were averaged to achieve TDS feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' It is obvious that the FL model can reproduce the diffuse intensity in the diffraction image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' A key point is that the diffuse quality does not naturally relate to the incoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' That the disordered structure, such as amorphous state or defects, can lead the coherent diffuse intensity is a general phenomenon in diffraction physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' So it is wrong to discriminate an inelastic or elastic characteristic based on whether a scattered electron falls on the Bragg point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Unfortunately, this mistake appeared in some literature which directly judge TDS as the inelastic scattering just for its diffuse feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' cabic-s T35 150 100 50 6 41x1011- Ixio10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 1x109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 1x10E 1α107 1x10f- 1x105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 1x104- 1000 108 10 7 4 5 4 1 2 1 2 6a) b) c) d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 3 Simulated diffraction patterns for independent electron events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' a) and b) for static structure, c) and d) for vibration lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Numbers are the total accumulating electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' In order to illuminate coherent or incoherent behavior of thermal scattered electrons, we simulated a virtual “single electron scattering” experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The intensity of each pixel on Fourier transformed exit wave map was converted to the emergency probability of electrons in that pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Time-average scattering is a statistical buildup for many scattering events within a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' So it is implicitly equivalent to the summation of snapshot of single electron scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' One snapshot of the electron distribution in reciprocal space (g space) of cause captured the coherent scattering characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Thousands of snapshots were imposed to disclose the diffraction features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The integration of snapshots was certainly an incoherent summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 3a is the data for the static atom configuration which is undoubtedly elastic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' However, without pre-knowledge of the specimen structure, one cannot judge which point in g space is the Bragg point when few scattering events are acquired because there is no regular distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Actually, all point in g pace is equal, no point particularly correlates to a Bragg reflection or elastic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' As number of electrons increases, the Bragg features appear gradually (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' From the quantum mechanism viewpoint, single electron also carries the probability of coherent exit wave to the imaging plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' “Incoherent summation” of the coherent snapshots still displays the property of coherence, forming the Bragg reflections in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Single electron interference experiment achieved by Tonomura et al attested that long-exposure imaging of the independent electrons still appeared the Young’s interference Total 12Total 615Total 33Total 24010fringes, verifying that time-averaging does not damage the interference behavior of scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='13 Now for TDS simulation with FL model, similar phenomenon repeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' It should be emphasized that each snapshot is the Fourier transform of the elastic potential array carrying a thermal fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Thus each snapshot still mirrors the “coherent scattering” of the vibrating elastic potential for the corresponding lattice configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' For low does image, one cannot distinguish the Bragg reflection (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Sharp spots emerge when enough electrons accumulate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' As mentioned above, all points in g space are equal and one should not appoint which spot corresponds a coherent or incoherent scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The diffuse intensity beyond the sharp spots also reflects the “coherent scattering” from an atom array, to be precise, from the vibration effect, although it is an “incoherent summation” or a time-average product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Discussion High energy electrons can certainly excite extra atom vibration in specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The creation of new phonons is the true energy absorption for incident beam and leads to the inelastic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' However, the contribution of this inelastic event to TDS is small comparing to the elastic scattering by the vibrating atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Using the phonon population, for example, in room temperature to estimate the inelastic scattering ability by phonons with perturbation theory is obviously wrong because in reality the energy transferred to phonon is not so much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The assumption that thermal fluctuation of the atom will reduce its elastic scattering ability is not correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The elastic potential conserves the scattering ability so that the inelastic treatment with the energy loss ΔV(𝑟⃗) or DW factor should be abandoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The perturbation approach suggested by Hall and Hirsch in elastic framework is a valid way to solve the scattering attenuation of Bragg reflections by recombining the elastic Bloch waves with DW factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The assistant “absorption potential” to calculate TDS intensity should be stated with caution since it is a virtual concept to describe the dilution of Bragg spots, not corresponding to a true energy transfer in the usual sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The multislice method including the FL or FP model is practical to implement TDS simulation because it depicts the actual physical picture of thermal elastic scattering and the coherence characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' DW factor in the simulation does not weaken the scattering ability, but breaks the structure order to result in the diffuse feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' In principle, the elastic scattered electrons by vibrating atoms are coherent to forge the interference patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Interference patterns are not always regular sharp Bragg spots while diffuse intensity cannot be taken for granted as incoherent effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The contrast of time-average diffraction in real physical process or in incoherent buildup also demonstrates the coherence characteristic of the elastic scattered electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Bragg reflections are the coherent interference of electrons scattered by periodic equilibrium potential and TDS reflects the coherent interference of the electrons scattered by the fast variation of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' From a statistic point of view, the probability of all atoms locating on the equilibrium position is very small, but this configuration can result in a large in-phase consequence which instead tremendously improves the chance of electron emergency in Bragg spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The out-of-phase electrons scattered by the atom arrangement deviated from the periodic position should display a weak contrast because of the low appearance probability though those configurations are large probability events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' From the perspective of signal processing, various configurations contain common frequency components corresponding to the regular lattice so that superimpose of the Fourier transform result in the enhancement of Bragg reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' It may be the reason why Bragg reflections are very sharp in the real diffraction patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The inelastic scattering with energy absorption by phonons really happens during electrons penetrate the specimen and contributes diffuse contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The ratio of this inelastic scattering is small if checking the EELS shape in the literature which contains the phonon absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='12 The discussion about phonon absorption is out of the scope of this report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' How many phonons are excited by incident electrons or what is the true phonon spectrum concerning the energy exchange between electrons and phonons needs to be thought with great caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 4 Simulated CBED patterns of [111] Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Numbers are the summations of configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Accelerate energy: 100 kV, convergent angle: 8 mrad, Cs: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='3 mm, defocuse: -140 nm, sampling: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='05 Å/pixel, image size: 1024 × 1024, thickness: 489 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Another interesting phenomenon related to TDS is Kikuchi lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The multislice simulations of convergent beam electron diffraction (CBED) with and without FL model for [111] Si are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' As number of configures increases, Kikuchi lines become stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' It at least proves that elastic potential and Einstein model can reproduce the Kikuchi patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Some literature also confirmed this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content='12, 14 Conclusion The summary is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The spatial distribution of elastic scattering potential is changed by the thermal vibration of atoms but the total scattering ability still conserves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' TDS is predominated by oscillated elastic scattering potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' The diffuse feature of TDS reflects the coherent interference of the electrons scattered by fluctuating lattice in spite of the time-averaging assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Acknowledgement This work was supported the National Key R&D Program of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' 2022YFB3803900).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' No TDS 5 10 25References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Yoshioka, Effect of Inelastic Waves on electron diffraction, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfyQUU/content/2301.03430v1.pdf'} +page_content=' Phys.' 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of accuracy the type of activity they are engaged in. In +this paper we investigate the ability of modern machine learning +algorithms in inferring basic offline activities, e.g., shopping and +dining, from location data. Using anonymized data of thousands of +users of a prominent location-based social network, we empirically +demonstrate that not only state-of-the-art machine learning excels +at the task at hand (Macro-F1>0.9) but also tabular models are +among the best performers. The findings we report here not only +fill an existing gap in the literature, but also highlight the potential +risks of such capabilities given the ubiquity of location data and +the high accessibility of tabular machine learning models. +CCS CONCEPTS +• Information systems → Global positioning systems; Loca- +tion based services; • Security and privacy → Social aspects +of security and privacy. +KEYWORDS +location data, activity inference, privacy +ACM Reference Format: +Alameen Najjar and Kyle Mede. 2023. Where You Are Is What You Do: On +Inferring Offline Activities From Location Data. In Proceedings of Knowledge +discovery and Data Mining (KDD’23). ACM, New York, NY, USA, 9 pages. +https://doi.org/XXXXXXX.XXXXXXX +1 +INTRODUCTION +With the proliferation of social media, smartphones, Internet-of- +things (IOT) devices and low Earth orbit (LEO) satellites, we live +in a world where location data (Data with reference to a physical +location) is ubiquitous. Recent estimates [11] suggest that loca- +tion data make up over 80% of the data created on a daily basis. +While location data can be and have been used for widely agreed +on, positive applications — such as understanding the spread of +infectious diseases [15] — it can be easily misused. For example, +misleading users of a navigation smartphone app about collecting +and/or selling their data [14]. +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than ACM +must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, +to post on servers or to redistribute to lists, requires prior specific permission and/or a +fee. Request permissions from permissions@acm.org. +KDD’23, August 6–10, 2023, Long Beach, CA +© 2023 Association for Computing Machinery. +ACM ISBN 978-1-4503-XXXX-X/18/06...$15.00 +https://doi.org/XXXXXXX.XXXXXXX +It is well established that a user’s location is indicative of the +type of real-world, day-to-day activities, such as shopping and +dining (Hereafter referred to as “offline activities”) they perform. +Research has shown that basic offline activities can be inferred +from GPS traces using both conventional statistical methods [22– +24] and machine learning algorithms [19, 26, 28, 29]. The same has +been demonstrated using mobile phone data collected city-wide +at the base station level [25, 30]. Other relevant sources of data, +such as geotagged social media posts [20, 38], WiFi signals [40], +and ride-hailing app data [10] have also been successfully used to +infer offline activities. In short, there is an abundance of evidence +that points to the correlation between a person’s location and the +offline activity they are engaged in. +As with any type of location technology, opportunities exist for +malicious targeting. For example, the same algorithm used in [26] +to help patients with alcohol use disorder recover can be exploited +to target users of a smartphone app most vulnerable to alcoholism. +Such a risk is further amplified given the recent widespread avail- +ability of powerful machine learning algorithms [9] that require +little to no knowledge of statistics and/or algorithm design to con- +figure and employ. +In this paper, we attempt to answer the following question: How +well can modern machine learning algorithms infer user’s offline ac- +tivity given their location data? To this end, we empirically evaluate +the performance of 6 models trained to infer 9 basic offline activi- +ties using anonymized data collected over a period of 18 months +from ≈15k users of a prominent location-based social network ac- +tive in 6 major cities spread over 4 different continents. +The findings we report in this paper not only fill an existing gap +in the literature, but also highlight the potential risks of applying +machine learning to location data in a time where powerful machine +learning models are easily accessible. The following is a summary +of our most interesting findings: +• Our experiments show that modern machine learning algo- +rithms are well capable of inferring basic offline activities +with the best performing model achieving an average Macro- +F1 score of over 0.9. +• We also found that “Nightlife” is the most and “At home” is +the least challenging activities to infer on average. +• Finally, we found that tabular models that require minimal +machine learning knowledge to configure and limited re- +sources to run not only excel at the task at hand but could +also outperform sophisticated models trained end-to-end. +The remainder of this paper is organized as follows. Previous rel- +evant works are briefly overviewed in Section 2. The methodology +we follow to infer offline activities from location data is described in +arXiv:2301.13537v1 [cs.CY] 31 Jan 2023 + +KDD’23, August 6–10, 2023, Long Beach, CA +Alameen Najjar and Kyle Mede +Section 3. The results of our extensive empirical analysis are given +in Section 4. And finally the paper is summarized in Section 5. +2 +PREVIOUS WORKS +The existing literature on inferring offline activities is vast, and it is +beyond the scope of this paper to review it in its entirety. Instead we +have organized collections of representative works into 4 categories +based on the data used; those being: ’mobility diaries’, ’Call Detail +Records’, ’social check-ins’ and ’others’. +The bulk of the literature use self-reported mobility diaries +recorded using specialized hardware and/or software [19, 22–24, 26, +28, 29]. The data used is dense however limited in terms of number +of subjects, temporal span and spatial coverage. Early works [22– +24] used Conditional Random Fields (CRFs) and Relational Markov +Networks (RMNs) to infer basic offline activities from GPS traces +of a few subjects. In the same vain, [19] proposed a personalized +framework that leverages similarities among users to infer offline +activities from GPS traces of 50 subjects. In [29], Random Forests +are used to infer the purpose of trips (High-level offline activities) of +GPS traces of 156 subjects collected over a one week period around +Zurich, Switzerland. Similarly in [28] graph convolutional neural +networks (GCNs) [18] are used to classify GPS traces of 139 sub- +jects into 5 offline activities. GCNs are also used recently in [26] to +infer 8 offline activities in GPS diaries collected and labeled by 167 +subjects. +A second subset of the literature [25, 30] infer offline activities at +the base station level from Call Detail Records (CDR) provided by +mobile network operators. In [30], conventional machine learning +algorithms are used to infer 8 offline activities from CDR data col- +lected in Barcelona and Madrid. Similarly [25] uses an ensemble of +models to classify CDR data of 80 users into 5 basic offline activities. +A third subset of the literature [20, 38] use social check-in data +to infer user’s offline activities at the Point Of Interest (POI) level. +In [20], CRFs combined with unsupervised clustering is used to +infer 7 offline activities from DianPing1 check-in data of 83 users +collected in Beijing, China. Similarly [38] uses non-zero matrix +factorization to infer 9 offline activities from Foursquare2 check-in +data of ≈2000 users collected in Tokyo and New York city. +The fourth and final subset of the literature are works that use +data sources other than those listed above. For example, [40] infers 8 +offline activities from WiFi traces of 13 subjects moving around a +university campus, [6, 34] infer offline activities from microblogs, +and [10] infers 13 high-level activities from ride-hailing app data +collected around the city of Toronto, Canada. +Data wise, our work mostly resembles that of [20, 38]. However, +we are not aware of any previous work that attempted to infer +offline activities at the scale we report here (6 cities, ≈15k users, and +6 models). Finally, it is worth mentioning that works, such as [21, +36, 39] that “predict” future offline activities using data similar to +ours are beyond the scope of this survey as we are interested in +inferring the user’s current rather than future activities. +1https://www.dianping.com/ +2https://foursquare.com/ +3 +METHODOLOGY +In this section we explain the methodology we follow to infer offline +activities from location data. +3.1 +Preliminaries +Definition 1 (Check-in record). A check-in record is a tuple ⟨𝑢,𝑙,𝑡⟩ +indicating that user 𝑢 is present at location 𝑙 at time 𝑡, where 𝑙 is the +location of a uniquely identified POI. +Definition 2 (Offline activity). The offline activity associated with a +given check-in record is the category of the POI at which the check- +in takes places. For example, “Dining” is the activity associated +with “Food” POIs, and “At home” is the activity associated with +“Residence” POIs. It is worth noting that using POI categories as +activities is a common practice as it has been done previously +in [6, 20, 21, 30, 34, 38, 39]. +Problem (Activity inference). Given a check-in record ⟨𝑢,𝑙,𝑡⟩ of +user 𝑢, the objective is to infer the activity user 𝑢 is engaged in +at time 𝑡 and location 𝑙. Let X = {𝑥1,𝑥2, · · ·,𝑥𝑛} represent the +set of check-in records and Y = {𝑦1,𝑦2, · · ·,𝑦𝑚} represent the set +of activities, the goal is to find a function 𝑓 (·) that assigns each +record in X with one activity in Y while satisfying the following +condition: +arg min +𝑓 ∈F +∥𝑓 (𝑥𝑖) − 𝑦𝑖 ∥, 𝑓 (𝑥𝑖) ∈ Y,𝑦𝑖 ∈ Y, +(1) +where 𝑦𝑖 is the true label (Activity) associated with 𝑥𝑖, ∥·∥ is an +evaluation operator (That evaluates to 0 when 𝑓 (𝑥𝑖) = 𝑦𝑖 and 1 +otherwise), and F is the hypothetical space of the task at hand. +3.2 +Enrichment +To account for meaningful contextual information, we enrich check- +in records with two sets of features as follows. +Relative location. By relative location we mean location with +respect to the center of the city. More specifically, 1) distance to city +center, and 2) bearing angle with respect to city center. Our intuition +behind including these two features is based on the assumption +that POIs of the same category are more likely to share similar +spatial distribution patterns with respect to the center of the city. +For example, camping grounds are more likely to be found in the +periphery of the city. +Distance to city center (𝛿) is calculated using the Haversine +formula as follows: +𝑎 = sin2( +Δ𝜙 +2 ) + cos𝜙1 · cos𝜙2 · sin2( Δ𝜆 +2 ), +(2) +𝑐 = 2 · atan2(√𝑎, +√ +1 − 𝑎), +(3) +𝛿 = 𝑅 · 𝑐, +(4) +where 𝜙 is latitude, 𝜆 is longitude, Δ𝜙 is the difference between +two latitudes, Δ𝜆 is the difference between two longitudes, and 𝑅 +is Earth’s radius. +On the other hand, bearing angle with respect to city center (𝜃) +is calculated such as: + +Where You Are Is What You Do: On Inferring Offline Activities From Location Data +KDD’23, August 6–10, 2023, Long Beach, CA +𝐴 = sin Δ𝜆 · cos𝜙2, +(5) +𝐵 = cos𝜙1 · sin𝜙2 − sin𝜙1 · cos𝜙2 · cos Δ𝜆, +(6) +𝜃 = atan2(𝐴, 𝐵), +(7) +where 𝜙 is latitude, 𝜆 is longitude, and Δ𝜆 is the difference between +two longitudes. +Grid statistics. We extract three POI-related statistics calculated at +the grid cell level, namely POI count, unique user count and check- +in count. Our intuition behind including these features is to help +the model capture meaningful patterns on the spatial distribution +of different POI categories around the city. For example, the number +of POIs around residential areas is likely to be less than that around +commercial areas. Or grid cells with high check-in activity are less +likely to be in a residential area, and so forth. Each of the features +can be described as follows: +𝜓𝑐 = +∑︁ +𝑖 ∈𝑀 +𝑖,𝑐 ∈ 𝑁, +(8) +where𝜓𝑐 is the extracted feature at cell𝑐,𝑖 is the𝑖-th POI/user/check- +in per cell 𝑐, 𝑀 is the set of unique POIs/users/check-in per cell 𝑐, +and 𝑁 is the set of all cells in the target city. +Multi-scale & multi-grid feature extraction. To account for a richer +feature set, we extract the aforementioned features at multiple +spatial scales using multiple hierarchical grids. Implementation +details can be found in Appendix A. +3.3 +Encoding & Classification +After enrichment, each check-in is represented with a vector 𝑣 ∈ R𝑑 +made from concatenating the check-in attributes (User ID, aggre- +gated location, and timestamp-related attributes) and the enriched +features, such that: +𝑣 = (𝑢,𝑡,𝑙,𝛿,𝜙,𝜓1, · · ·,𝜓𝑘) ∈ R𝑑, +(9) +where 𝑢 is the user ID, 𝑡 is the timestamp, 𝑙 is the aggregated +location, 𝛿 and 𝜃 are the relative location features, 𝜓 is the grid +statistics, and 𝑘 is the number of scales and/or grids used in the +check-in enrichment step. +The representation obtained in Equation 9 is used next for classi- +fication. To this end, we experimented with 6 classifiers organized +into 3 groups: conventional classifiers including both 𝑘-Nearest +Neighbours (𝑘-NN) [7] and Support Vector Machines (SVM) [5]. +Tabular classifiers including Extreme Gradient Boosting (XGB) [4] +and TabNet [1]. And Multilayer Perceptrons [32] implemented in +two flavors: vanilla/plain (MLP) and regularized (rMLP). See Ap- +pendix A for model implementation details. +4 +RESULTS +In this section we present the validation results of inferring offline +activities from location data. +4.1 +Data +We used the Foursquare dataset [37] since it is one of the most +widely used check-in dataset in the research community. The dataset +consists of over 33M check-ins made by over 266k users at over +3.6M venues in 415 cities worldwide over a period of 18 months. +For privacy concerns we opted out of using the user’s actual location. +Instead we aggregated location using a grid, such as Uber H33 and +Geohash4. In other words, check-ins within the same grid cell are +indistinguishable to one another from the location point of view. Venue +names and user names, on the other hand, are already anonymized +by the source therefore we used them as they are. +Next, we kept check-ins that belong to 6 major cities spanning +a wide range of latitudes and longitudes, and representing 6 sub- +regions as defined by the United Nations Geoscheme5. Each check- +in is assigned a city based on its distance to the city center’s co- +ordinates as provided in the dataset. See Figure 1 for a map of the +selected cities. Target cities have good data coverage with little data +gaps. See Figure 2 for a visualization of data coverage. +Figure 1: Map of the target cities: Los Angeles, Tokyo, Mum- +bai, Sydney, Paris and Milan. +Figure 2: Data coverage per target city. Missing polygons in- +dicate data gaps. +Finally, we assigned each venue a category out of nine parent +categories as defined by the Foursquare API6. In other words, each +3https://github.com/uber/h3 +4http://geohash.org/ +5https://en.wikipedia.org/wiki/United_Nations_geoscheme +6http://foursquare-categories.herokuapp.com/ + +40 +20 +0 +-20 +(C) OpenStreetMap +contributors +-100 +-50 +0 +50 +100 +150Los Angeles +Tokyo +Paris +34.1 +49.0 +35.8 +48.9 +34.0 +35.7 +48.8 +33.9 +(C) +35.6 +[contributors +(C) +48.7 +Icontributors +139.8 +OpenStreetMap +139.6 +140.0 +2.2 +2.4 +2.6 +33.8 -contributors +-118.4-118.3-118.2-118.1 +Mumbai +Sydney +-33.7 +Milan +19.2 +-33.8 +45.6 +19.1 +45.5 +-33.9 +45.4 +19.0 - +-34.0 +C +OpenStreetMap +C +45.3 contributors +OpenStreetMap +8 +9.0 +contributors +9.2 +9.4 +18.9 +OpenStreetMap +contributors +150.9 151.0 151.1 151.2 +72.8 +72.9 +73.0KDD’23, August 6–10, 2023, Long Beach, CA +Alameen Najjar and Kyle Mede +Table 1: Summary of the generated datasets. +Dataset/City +Check-ins +Venues +Users +Los Angeles +97274 +18685 +2476 +Tokyo +708863 +64761 +8360 +Mumbai +25248 +6088 +525 +Sydney +31934 +7739 +733 +Paris +59816 +13603 +1973 +Milan +40641 +8642 +897 +Total +963776 +119518 +14964 +venue is assigned one of the following categories: “Arts & Entertain- +ment,” “College & University,” “Food,” “Nightlife Spot,” “Outdoors +& Recreation,” “Professional & Other Places,” “Residence,” “Shop & +Service,” and “Travel & Transport”. These categories are the target +variables we aim to infer given a check-in. +Cities are treated as separate datasets and analyzed indepen- +dently as reported in the following. See Table 1 for a summary of +the statistics of the final datasets. +4.2 +Evaluation +We used logarithmic loss (Log loss) to compare different models +in a 3-fold cross validation evaluation scheme. On the other hand, +we evaluate the best model’s classification performance on the test +data by reporting the Macro-F1 score which is the harmonic mean +of both precision (Macro-P) and recall (Macro-R) averaged over all +classes, such that: +Macro-F1 = 2 · Macro-P · Macro-R +Macro-P + Macro-R , +(10) +where Macro-P and Macro-R are given as: +Macro-P = +1 +|Y| +∑︁ +𝑦∈Y +TP𝑦 +TP𝑦 + FP𝑦 +, +(11) +Macro-R = +1 +|Y| +∑︁ +𝑦∈Y +TP𝑦 +TP𝑦 + FN𝑦 +, +(12) +where 𝑦 is activity label, TP𝑦, FP𝑦, and FN𝑦 are the number of true +positives, false positives and false negatives for class/activity 𝑦, +respectively. +4.3 +Model Comparison +In Table 2 we report the validation loss of all models on all 6 datasets. +See Appendix A for experiment implementation details. +Key takeaways can be summarized as follows. XGB dominates +all models on every single dataset. On average, XGB reduces the +naive model’s (𝑘-NN) loss by 59%. Interestingly, XGB’s performance +is on average 20.9% better than that of TabNet which is Google’s +deep-learning model designed for tabular data; although inline with +previous studies [16, 33]. +Second in place is rMLP which cuts the naive model’s loss +by 51.7%. Still rMLP performs 18% worse than XGB. Regulariza- +tion boosts plain MLPs by an average of 11.7% which we take as +a demonstration of the potential regularization has for MLPs. It is +worth noting that rMLP outperforms TabNet on all datasets except +Los Angeles and on average it provides a 6% performance boost +over TabNet. This finding is inline with the results in [16]. +Moreover, TabNet performs better than plain MLPs on all datasets +except Tokyo. On average TabNet outperforms plain MLPs by ≈ 6%. +Given the above results, we consider XGB the winning model +and therefore we use it hereafter in our experiments. Next, we em- +pirically attempt to understand how individual features contribute +to the model’s performance. +4.4 +Ablation Study +In order to understand how different features contribute to the +model’s performance in the following we present the results of a +series of ablation experiments we conducted. +Starting with location, in Figure 3 we plotted the per-category +classification performance (Macro-F1) against a decreasing grid +resolution. The general trend indicates that higher grid resolution +yields better performance. City wise, performance degrades by an +average of 57% when grid resolution goes from highest to lowest +with Milan and Tokyo being the least and most impacted cities. +Category wise on the other hand, “Travel & Transport” and “Arts & +Entertainment” are the least and most impacted. +While the above results indicate that location data play an essen- +tial role in the model’s performance what is more interesting is the +observation that even when features are extracted using the lowest +resolution grid the model is still able to correctly infer the user +activity up to 36% of the time. This is indicative of the importance +of non-location features. +Figure 3: Location granularity and performance: Classifica- +tion performance (Y axis) plotted against decreasing grid res- +olution (X Axis). +Moving on to relative location, in Figure 4, we plotted the change +in performance obtained when different relative-location features +are removed. Different cities and/or categories are impacted differ- +ently by relative location. However, on average excluding relative in- +formation degrades performance by 2% to 9% with Paris and Tokyo +being the least and most impacted cities. Moreover, “Residence” and +“Outdoors & Recreation” are the least and most impacted categories. +Individually, “Nightlife Spot” and “Arts & Entertainment” benefit +the most from bearing angle and distance-to-center, respectively. + +Paris +Milan +Tokyo +1.0 +0.8 +0.8 +0.8 +0.6 +0.6 +0.6 +0.4 +0.4 +0.4 +0.2 +0.2 +Mumbai +Sydney +Los Angeles +0.8 +0.8 +0.8 +0.6 +0.6 +0.6 +Arts & Entertainment +0.4 +College & University +0.4 +Fooc +Nightlife Spot +0.4 +Outdoors & Recreation +Professional & Other Places +0.2 +Residence +0.2 +Shop & Service +Travel & Transport +macro avg +12 11 10 +9 +8 +6 +12 11 10 +9 +8 +6 +12 11 10 +9 +8 +7 +6Where You Are Is What You Do: On Inferring Offline Activities From Location Data +KDD’23, August 6–10, 2023, Long Beach, CA +Table 2: Validation loss (Log loss ± standard deviation) obtained by different models on all datasets. Bold and underline indicate +best and second best results, respectively. +Mumbai +Sydney +Milan +Paris +Los Angeles +Tokyo +Average +𝑘-NN +1.968±0.023 +1.891±0.043 +1.888±0.019 +1.963±0.04 +1.914±0.022 +1.596±0.011 +1.870±0.003 +SVM +1.742±0.009 +1.725±0.002 +1.787±0.002 +1.882±0.004 +1.894±0.002 +1.556±0.004 +1.764±0.004 +XGB +0.868±0.011 +0.832±0.003 +0.709±0.009 +0.811±0.01 +0.755±0.005 +0.598±0.001 +0.762±0.006 +TabNet +0.951±0.062 +1.099±0.006 +1.007±0.049 +1.003±0.007 +0.895±0.031 +0.813±0.016 +0.961±0.028 +MLP +1.110±0.009 +1.191±0.021 +1.047±0.025 +1.125±0.005 +0.94±0.011 +0.72±0.003 +1.022±0.012 +rMLP +0.937±0.01 +0.994±0.011 +0.869±0.002 +0.931±0.004 +0.924±0.024 +0.757±0.004 +0.902±0.009 +Figure 4: Relative location and performance: Percentage of +performance change (Y axis) resulted from excluding differ- +ent relative-location features (X axis). +Next we evaluated grid statistics in Figure 5. Removing grid +statistics degrades performance by and average of 9.6% with Milan +and Paris being the least and most impacted cities. Category wise, +“Travel & Transport” and “Arts & Entertainment” are the least and +most impacted categories. Different statistics contribute differently +to the model’s performance with check-in count being the most +important among the three. Followed by user count and finally POI +count with a very small margin in between. +The obtained results demonstrate that both relative information +and grid statistics are important to the model’s performance and +thus confirm the assumptions we made earlier. +In Figure 6 we compared the model’s performance when features +are extracted using one (H3) versus two (H3 and Geohash) grids (See +Appendix A for implementation details). Using two grids instead +of one boosts performance by an average of 7%. Tokyo and Los +Angeles benefit the most while Milan and Sydney benefit the least. +Category wise, “Nightlife Spot” and “Residence” are the most and +least impacted categories. In fact, “Residence” in Tokyo is negatively +impacted when two grids are used instead of one. +Finally, in Figure 7 we studied the impact multi-scale features +have on the model’s performance. It is worth noting that both +models (Single-/multi-scale) have the same location granularity (See +Appendix A for implementation details). The obtained results show +that all categories across all cities benefit from multi-scale feature +Figure 5: Grid statistics and performance: Percentage of per- +formance change (Y axis) resulted from excluding different +grid statistics (X axis). +Figure 6: Multi-grid feature extraction and performance: +Per-category classification performance (Y axis) plotted +against number of grids (X axis). +extraction. On average performance is boosted by 5.4%. Mumbai +and Milan are the most and least impacted cities. On the other +hand, category wise, “Arts & Entertainment” and “Residence” are +the most and least impacted categories. + +College & +Nightlife +Outdoors & +Professional +Shop & +Travel & +Arts & Ent. +University +Food +Spot +Recreation +& Others +Residence +Service +Transport +0.0 +Tokyo +0.1 + Angeles +0.0 +0.1 +so7 +0.0 +Paris +0.1 +Mumbai +0.0 +0.1 +Sydney +0.1 +0.0 +Milan +0.1College & +Nightlife +Outdoors & +Professional +Shop & +Travel & +Arts & Ent. +University +Food +Spot +Recreation +& Others +Residence +Service +Iransport +kyo +0.00 +-0.25 +les +Angel +0.00 +-0.25 +Los +. +0.00 +Par +-0.25 +Mumbai +0.00 +-0.25 +ley +0.00 +Sydne +0.25 +Milan +0.00 +-0.25Paris +Milan +Tokyo +1.0 +0.75 +0.75 +0.50 +0.5 +0.50 +0.25 +0.25 +0.00 +0.0 +0.00 +Mumbai +Sydney +Los Angeles +0.75 +0.75 +0.75 +0.50 +0.50 +Arts&Entertainnent +0.50 +College& University +Food +Nightlife Spot +Outdoors&Recreation +0.25 +0.25 +0.25 +Professional & Other Places +Residence +Shop & Service +Travel&Transport +macroavg +0.00 +0.00 +0.00 +H3 H3+Geohash +H3 H3+Geohash +H3 H3+GeohashKDD’23, August 6–10, 2023, Long Beach, CA +Alameen Najjar and Kyle Mede +Figure 7: Multi-scale feature extraction and performance: +Per-category classification performance (Y axis) plotted +against number of grid scales (X axis). +In the following we build upon the insights we gained from the +results above to evaluate our final model. +4.5 +Best Model Evaluation +For a final evaluation we retrained the winning model using all +features extracted using 2 grids at 7 different scales. Evaluation +results of this model are reported in Table 3. +On average and over all cities, the best model achieves a Macro- +F1 score of 0.904. City-wise, the model’s performance is comparable +with Mumbai and Los Angeles being on the opposite ends of per- +formance. The same observation holds true at the category level +with “Nightlife Spot” and “Residence” being the most and least chal- +lenging categories. In general, the model performance is consistent +across city and category with a standard deviation of 2.4% and 4.6%, +respectively. +To better understand how the model performs across category we +plotted the confusion matrix in Figure 8. While the matrix shows +mostly clear separation between categories, it is clear that that +“Nightlife Spot” check-ins are largely misclassified as “Food” in +almost all cities. The same behaviour is observed with “Shop & +Service” however to a lesser degree. +Finally, for a subjective evaluation we mapped in Figure 9 the +aggregated inferences made by the best model next to the ground- +truth data for the city of Los Angeles. The visualizations clearly +indicate that the inferred maps preserve to a high degree the spatial +distribution of the data for the majority of classes. It is worth noting +that we obtained similar results on the other cities. And that our +choice of Los Angeles is based on both data coverage and model +performance on the test data. +4.6 +Summary +The following is a summary of the reported results. First, XGB is the +best performing model by a large margin followed by regularized +MLPs and TabNet, respectively. Second, the winning model is well +capable of inferring offline activities with an average Macro-F1 +score of 0.904. Third, performance is consistent across city and +Figure 8: Best model evaluation (2): Normalized confusion +matrix for all cities. +activity with a standard deviation value of 2.4% and 4.6%, respec- +tively. Fourth, we found that “Nightlife” is the most and “At home” +is the least challenging offline activities to infer with the winning +model achieving an average Macro-F1 score of 0.844 and 0.967, re- +spectively. Finally, the ablation study demonstrated that location +granularity, relative location and grid statistics each on its own +plays a significant role in the model’s performance. +5 +DISCUSSION +Thanks to recent software and hardware advances, we live in a +world where location data is ubiquitous. Previous research has well +demonstrated that location is an effective proxy for the type of +activity a person is engaged in in the real world. +In this paper, we attempted to answer the following question: +How well can modern machine learning algorithms infer offline +activities from location data? To this end, we empirically evalu- +ated the performance of 6 models trained to infer 9 basic offline +activities using anonymized data collected from ≈15k Foursquare +users active in 6 major cities spread across 4 continents. Our exper- +iments show that not only modern machine learning algorithms +are well capable of inferring basic offline activities (Macro-F1>0.9) +given location data, but also tabular models which require minimal +knowledge to configure and limited resources to run are among the +best performers. +As with the majority of studies, ours is subject to limitations. +First, social check-in data is subject to bias since it comes from +public social media posts shared willingly by individuals who may +be less concerned with privacy and not representative of the whole +population. Second, it is not always the case that activities match +the category of the POI at which a person checks in. For example, +checking in at a “Residence” POI does not always mean engaging in +“At home” activity. It could also mean “Work” if the person works +from home. Similar arguments could be made about other POI +categories. Therefore, the empirical findings reported in this work +should be seen in the light of such limitations. + +Paris +Milan +Tokyo +1.0 +0.75 +0.75 +0.50 +0.5 +0.50 +0.25 +0.25 +0.00 +0.0 +0.00 +Mumbai +Sydney +Los Angeles +0.75 +0.75 +0.75 +0.50 +Arts&Entertainnent +0.50 +0.50 +College&University +FOOd +Nightlife Spot +Outdoors&Recreation +0.25 +0.25 +0.25 +Professional & Other Places +Residence +Shop& Service +Travel &Transport +0.00 +0.00 +0.00 +macroavg +10 +6-10 +10 +6-10 +10 +6-10Los Angeles +Tokyo +Milan +Arts & Entertainment -0.910.00 0.03 0.00 0.01 0.02 0.00 0.02 0.00 +0.910.00 0.04 0.00 0.00 0.01 0.00 0.03 0.00 +0.800.000.120.010.020.030.000.010.01 +College & University +0.000.93 +0.03 0.00 0.00 0.01 0.00 0.02 0.00 +0.00 +0.95 +0.02 0.00 0.00 0.01 0.00 0.01 0.00 +0.00 +0.940.02 0.00 0.01 0.01 0.00 0.01 0.00 +000000 +1O'050'0000T0000'0TO'0Z60 +000000 +0.90 +0.000.010.030.000.040.01 +Nightlife Spot -0.00 0.00 0.06 +50.890.000.010.000.030.01 +-0.01 0.000.190.75 +10000000100000 +0.00 0.000.080.88 +10'0T0O00OT0'0T0'0 +Outdoors & Recreation -0.00 0.00 0.02 0.000.95 +50.010.000.020.00 +0.00 0.00 0.02 0.000.940.01 0.00 0.02 0.01 +0.00 0.00 0.03 0.000.930.01 0.00 0.02 0.00 +Professional & Other Places -0.01 0.00 0.05 0.00 0.01 +0.870.000.050.01 +0.00 0.00 0.05 0.00 0.010.900.00 0.03 0.01 +0.00 0.000.050.00 0.019 +0.91 +0.000.020.00 +Residence -0.00 0.00 0.01 0.00 0.01 0.010.970.01 0.00 +0.00 0.00 0.02 0.00 0.01 0.01 +0.95 +50.01.0.00 +0'000000'0T0'000000'0 +10.970.00 0.00 +Shop Service -0.00 0.00 0.07 0.00 0.00 0.01 0.00 +0.910.00 +0.00 0.00 0.05 0.00 0.00 0.01 0.000.930.01 +0.00 0.000.090.00 0.02 0.04 0.00 +0.00 +0.00 0.000.030.00 0.01 0.01 0.000.01 +0.94 +Sydney +Paris +Mumbai +Arts &Entertainment-0.860.000.080.000.02 0.01 0.00 0.01 0.01 +0.830.000.100.01 0.01 0.03 0.00 0.01 0.02 +0.800.000.080.00 0.02 0.04 0.00 0.03 0.03 +College & University +0.00 +D0.91 +00'010'000010'0100100S00 +-0.000.92 +0.00 +Food -0.00 0.00 0.91 +0.00 0.00 +0.01 0.01 +0.88 +30.01 0.010.030.010.040.01 +-0.01 0.000.210.70 +0.00 0.02 0.00 0.03 0.03 +0.010.000.180.67 +0.03 0.03 0.02 0.05 0.02 +Outdoors Recreation -0.00 0.00 0.04 0.00 +0.91 +10.010.000.010.01 +0.00 0.00 0.04 0.00 +0.91 +0.020.000.010.01 +0.00 0.00 0.03 0.000.930.01 0.01 0.01 0.01 +Professional &Other Places-0.00 0.00 0.07 0.00 0.030.83 +0.000.040.02 +0.01 0.00 0.050.00 0.010.900.00 0.02 0.01 +0.00 0.00 0.04 0.00 0.010.91 +0.010.020.01 +Residence -0.00 0.00 0.01 0.00 0.02 0.00 +0.960.000.01 +10000000000000000 +0.940.01 0.01 +0.00 0.00 0.03 0.00 0.00 0.02 +0.930.01 0.01 +Shop & Service +0.810.01 +0.00 0.000.150.01 0.01 0.020.00 +00.780.02 +0.01 0.000.130.00 0.010.04 0.010.790.01 +Travel &Transport +0.00 0.00 0.02 0.00 0.01 0.00 0.00 0.00 +0.95 +0.00 0.00 0.04 0.00 0.01 0.01 0.00 0.01 +0.93 +0.00 0.00 0.03 0.00 0.01 0.02 0.01 0.01 +0.92 +inment + University + Spot +eation +Residence + Service +Entertainment +University +Spot +tion +Places +: Service +I & Transport . +Entertainment +Spot +Recreation +Places +Residence ++ & Service +Travel & Transport , +Recreat +Residen +htlife : +Nightlife : +Nightlife : + Entertai +Recr +other +Other +ler +Shop +Shop +2 +Shop i +College i +College +College i +Outdoors +Outdoors i +Outdoors +Professional +Professional +ProfessionalWhere You Are Is What You Do: On Inferring Offline Activities From Location Data +KDD’23, August 6–10, 2023, Long Beach, CA +Table 3: Best model evaluation (1): Per-category Macro-F1 score for all cities. +Mumbai +Sydney +Milan +Paris +Los Angeles +Tokyo +Average +Arts & Entertainment +0.849 +0.883 +0.861 +0.889 +0.932 +0.934 +0.891 +College & University +0.885 +0.935 +0.956 +0.952 +0.949 +0.965 +0.94 +Food +0.834 +0.859 +0.853 +0.796 +0.91 +0.89 +0.857 +Nightlife Spot +0.767 +0.882 +0.915 +0.776 +0.92 +0.807 +0.844 +Outdoors & Recreation +0.929 +0.896 +0.938 +0.927 +0.958 +0.948 +0.933 +Professional & Other Places +0.9 +0.869 +0.908 +0.895 +0.878 +0.916 +0.894 +Residence +0.937 +0.976 +0.982 +0.956 +0.979 +0.974 +0.967 +Shop & Service +0.812 +0.839 +0.847 +0.807 +0.9 +0.925 +0.855 +Travel & Transport +0.933 +0.955 +0.954 +0.934 +0.96 +0.993 +0.955 +Average +0.872 +0.899 +0.913 +0.881 +0.932 +0.928 +0.904 +Figure 9: Best model evaluation (3): Aggregated inferences compared to ground-truth data for the city of Los Angeles. 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The journal of machine learning research 15, 1 (2014), 1929–1958. +Table 4: 𝑘-Nearest Neighbours (𝑘-NN) hyper-parameters +search space. “𝑘” is the number of nearest neighbours. “L1” +and “L2” are Manhattan and Euclidean distances, respec- +tively. +Hyper-parameter +Type +Range +𝑘 +Integer +[1, 33] +Distance metric +Nominal +{L1, L2} +[36] Diem To, Dong Si, and Ying Chen. 2019. Traveler’s Next Activity Predication with +Location-Based Social Network Data. In Proceedings of the 3rd ACM SIGSPATIAL +International Workshop on Prediction of Human Mobility. 15–23. +[37] Dingqi Yang, Daqing Zhang, and Bingqing Qu. 2016. Participatory cultural +mapping based on collective behavior data in location-based social networks. +ACM Transactions on Intelligent Systems and Technology (TIST) 7, 3 (2016), 1–23. +[38] Dingqi Yang, Daqing Zhang, Vincent W Zheng, and Zhiyong Yu. 2014. Modeling +user activity preference by leveraging user spatial temporal characteristics in +LBSNs. IEEE Transactions on Systems, Man, and Cybernetics: Systems 45, 1 (2014), +129–142. +[39] Jihang Ye, Zhe Zhu, and Hong Cheng. 2013. What’s your next move: User activity +prediction in location-based social networks. In Proceedings of the 2013 SIAM +International Conference on Data Mining. SIAM, 171–179. +[40] Vincent W Zheng and Qiang Yang. 2011. User-dependent aspect model for +collaborative activity recognition. In Twenty-Second International Joint Conference +on Artificial Intelligence. +A +IMPLEMENTATION DETAILS +Data. We used 80% of each dataset for training and validation, and +20% for testing. The training/validation subset is used for hyper- +parameters tuning following a 3-fold cross validation evaluation +scheme. Moreover, once the best hyper-parameters are found, the +whole of the training/validation subset is used to train the final +model. On the other hand, the test subset is used to evaluate the +performance of the winning model on the target classification task. +Feature extraction. Unless otherwise mentioned, we used resolu- +tion 10 Uber H3 grid to extract all features. For multi-grid models +we extracted features using two grids: 1) Uber H3 (Resolution 10) +and, 2) Geohash (7 digits). Both grids have cells with an area of +the same order of magnitude (105𝑚2). For multiscale models we +extracted features using only Uber H3 grid at 7 different scales +(Resolutions 6 to 12). +Hyper-parameters tuning. We used Hyperopt library [2] to tune +the hyper-parameters of every model we experimented with. Unlike +traditional hyper-parameter tuning methods that blindly explore +the search space, such as Grid Search, Hyperopt takes the results +of the previous runs into consideration when sampling the search +space for the next run. For all models we limited the search to 100 +runs or 48 hours (Whichever comes first). Search space configu- +rations for 𝑘-NN, SVM, XGB, TabNet, MLP and rMLP models are +detailed in Tables 4, 5, 6, 7, 8, and 9, respectively. +Training. We trained all models by minimizing the logarithmic +loss for a maximum of 500 epochs. We used early stopping with +1𝑒 − 3 tolerance and patience of 10 epochs whenever possible. All +models were trained using the same virtual machine equipped with +128 GB of RAM, 16 × 2.4 GHz CPUs and 2 × NVIDIA Tesla V100 +GPUs. + +Where You Are Is What You Do: On Inferring Offline Activities From Location Data +KDD’23, August 6–10, 2023, Long Beach, CA +Table 5: Support Vector Machine (SVM) hyper-parameters +search space. “𝐶” and “gamma” are the regularization term +and the kernel coefficient, respectively. +Hyper-parameter +Type +Range +Log scale +𝐶 +Continuous +[2−5, 215] +✓ +gamma +Continuous +[2−15, 23] +✓ +Table +6: +Extreme +Gradient +Boosting +(XGB) +hyper- +parameters search space. See XGB’s official documentation +for more on the hyper-parameters. +Hyper-parameter +Type +Range +Log scale +eta +Continuous +[1𝑒 − 3, 1] +✓ +lambda +Continuous +[1𝑒 − 10, 1] +✓ +alpha +Continuous +[1𝑒 − 10, 1] +✓ +gamma +Continuous +[1𝑒 − 1, 1] +✓ +num_round +Integer +[1, 100] +- +max_depth +Integer +[1, 20] +- +max_delta_step +Integer +[0, 10] +- +min_child_weight +Continuous +[0.1, 20] +✓ +subsample +Continuous +[0.01, 1] +- +colsample_bylevel +Continuous +[0.1, 1] +- +colsample_bynode +Continuous +[0.1, 1] +- +colsample_bytree +Continuous +[0.5, 1] +- +Table 7: TabNet hyper-parameters search space. See [1] for +more details on the hyper-parameters. +Hyper-parameter +Type +Range +𝑛𝑎 +Integer +{8, 16, 24, 32, 64, 128} +𝑛𝑠𝑡𝑒𝑝𝑠 +Integer +[3, 10] +batch_size +Integer +{256, 512,1024, 2048, 4096} +virtual_batch_size +Integer +{256,512,1024,2048, 4096} +learning_rate +Continuous +{0.005, 0.01, 0.02, 0.025} +gamma +Continuous +{1.0, 1.2, 1.5, 2.0} +𝜆𝑠𝑝𝑎𝑟𝑠𝑒 +Continuous +{0, 10−6, 10−4, 10−3, 10−2, 10−1} +momentum +Continuous +{0.6, 0.7, 0.8, 0.9, 0.95, 0.98} +Table 8: Multilayer Perceptron (MLP) hyper-parameters +search space. “units” is the number of neurons per hidden +layer. +Hyper-parameter +Type +Range +Log scale +Hidden layers +Integer +{3, 6, 9} +- +units +Integer +{128, 256, 512} +- +learning_rate +Continuous +[1𝑒 − 3, 1𝑒 − 1] +✓ +𝑘-NN. We used the Scikit-learn [31] implementation of 𝑘-NN with +the search algorithm set to “auto.” We tuned two hyper-parameters, +namely number of nearest neighbours (𝑘) and distance metric. See +Table 4 for more details. +Table 9: Regularized Multilayer Perceptron (rMLP) hyper- +parameters search space. “units”, “stddev”, “SC”, and “SWA” +are number of neurons per hidden layer, Gaussian noise +standard deviation, Skip Connections and Stochastic Weight +Averaging, respectively. +Hyper-parameter +Type +Range +Log scale +Hidden layers +Integer +{3, 6, 9} +- +units +Integer +{128, 256, 512} +- +learning_rate +Continuous +[1𝑒 − 3, 1𝑒 − 1] +✓ +dropout_rate +Continuous +[0.0, 0.5] +- +weight_decay +Continuous +[1𝑒 − 6, 1𝑒 − 1] +✓ +stddev +Continuous +[0.0, 0.5] +✓ +SC +Binary +[False, True] +- +SWA +Binary +[False, True] +- +SVM. We used the Scikit-learn implementation of SVM with the +default settings on. We tuned the kernel coefficient (gamma) and the +algorithm’s regularization term (𝐶). See Table 5 for more details. +XGB. Using the official Python implementation of XGB7, we used +the gbtree booster paired with the multi:softprob objective func- +tion and tuned the hyper-parameters shown in Table 6. +TabNet. Using the unofficial PyTorch implementation of TabNet8 +we tuned the hyper-parameters shown in Table 7. +MLP. We used the Keras framework9 to implement all MLP mod- +els. Our MLP block consists of a dense layer followed by a ReLu +activation layer. We set batch size to 128 and trained the network +using the Adam optimizer [17]. See Table 8 for a complete list of +the hyper-parameters we tuned. +rMLP. We applied implicit (Batch normalization [12] and Stochastic +weight averaging [13]), ensemble (Dropout [35]), structural (Skip +connections [8]), and data augmentation (Gaussian noise [3]) tech- +niques to regularize vanilla MLPs. Implemented in Keras, our rMLP +block consists of a dense layer followed by a ReLu activation layer, +a Gaussian noise layer, a batch normalization layer, a dropout layer +and a concatenation layer. We set batch size to 128 and trained +the network using the AdamW optimizer [27]. To implement SWA +we used an unofficial Python implementation10. See Table 9 for a +complete list of the hyper-parameters we tuned. +7https://pypi.org/project/xgboost/ +8https://pypi.org/project/pytorch-tabnet/ +9https://keras.io/ +10https://github.com/simon-larsson/keras-swa + diff --git a/eNFRT4oBgHgl3EQfUzcf/content/tmp_files/load_file.txt b/eNFRT4oBgHgl3EQfUzcf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe5c36bd620158683df69bb6099075850de9b3fd --- /dev/null +++ b/eNFRT4oBgHgl3EQfUzcf/content/tmp_files/load_file.txt @@ -0,0 +1,1287 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf,len=1286 +page_content='Where You Are Is What You Do: On Inferring Offline Activities From Location Data Alameen Najjar Rakuten Institute of Technology Tokyo, Japan alameen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='najjar@rakuten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='com Kyle Mede Rakuten Institute of Technology Tokyo, Japan kyle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='mede@rakuten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='com ABSTRACT Studies have shown that a person’s location can reveal to a high degree of accuracy the type of activity they are engaged in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In this paper we investigate the ability of modern machine learning algorithms in inferring basic offline activities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=', shopping and dining, from location data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Using anonymized data of thousands of users of a prominent location-based social network, we empirically demonstrate that not only state-of-the-art machine learning excels at the task at hand (Macro-F1>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='9) but also tabular models are among the best performers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The findings we report here not only fill an existing gap in the literature, but also highlight the potential risks of such capabilities given the ubiquity of location data and the high accessibility of tabular machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' CCS CONCEPTS Information systems → Global positioning systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Loca- tion based services;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' • Security and privacy → Social aspects of security and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' KEYWORDS location data, activity inference, privacy ACM Reference Format: Alameen Najjar and Kyle Mede.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Where You Are Is What You Do: On Inferring Offline Activities From Location Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In Proceedings of Knowledge discovery and Data Mining (KDD’23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' ACM, New York, NY, USA, 9 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='XXXXXXX 1 INTRODUCTION With the proliferation of social media, smartphones, Internet-of- things (IOT) devices and low Earth orbit (LEO) satellites, we live in a world where location data (Data with reference to a physical location) is ubiquitous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Recent estimates [11] suggest that loca- tion data make up over 80% of the data created on a daily basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' While location data can be and have been used for widely agreed on, positive applications — such as understanding the spread of infectious diseases [15] — it can be easily misused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' For example, misleading users of a navigation smartphone app about collecting and/or selling their data [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' KDD’23, August 6–10, 2023, Long Beach, CA © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' ACM ISBN 978-1-4503-XXXX-X/18/06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='XXXXXXX It is well established that a user’s location is indicative of the type of real-world, day-to-day activities, such as shopping and dining (Hereafter referred to as “offline activities”) they perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Research has shown that basic offline activities can be inferred from GPS traces using both conventional statistical methods [22– 24] and machine learning algorithms [19, 26, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The same has been demonstrated using mobile phone data collected city-wide at the base station level [25, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Other relevant sources of data, such as geotagged social media posts [20, 38], WiFi signals [40], and ride-hailing app data [10] have also been successfully used to infer offline activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In short, there is an abundance of evidence that points to the correlation between a person’s location and the offline activity they are engaged in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' As with any type of location technology, opportunities exist for malicious targeting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' For example, the same algorithm used in [26] to help patients with alcohol use disorder recover can be exploited to target users of a smartphone app most vulnerable to alcoholism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Such a risk is further amplified given the recent widespread avail- ability of powerful machine learning algorithms [9] that require little to no knowledge of statistics and/or algorithm design to con- figure and employ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In this paper, we attempt to answer the following question: How well can modern machine learning algorithms infer user’s offline ac- tivity given their location data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' To this end, we empirically evaluate the performance of 6 models trained to infer 9 basic offline activi- ties using anonymized data collected over a period of 18 months from ≈15k users of a prominent location-based social network ac- tive in 6 major cities spread over 4 different continents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The findings we report in this paper not only fill an existing gap in the literature, but also highlight the potential risks of applying machine learning to location data in a time where powerful machine learning models are easily accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The following is a summary of our most interesting findings: Our experiments show that modern machine learning algo- rithms are well capable of inferring basic offline activities with the best performing model achieving an average Macro- F1 score of over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' We also found that “Nightlife” is the most and “At home” is the least challenging activities to infer on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Finally, we found that tabular models that require minimal machine learning knowledge to configure and limited re- sources to run not only excel at the task at hand but could also outperform sophisticated models trained end-to-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Previous rel- evant works are briefly overviewed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The methodology we follow to infer offline activities from location data is described in arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='13537v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='CY] 31 Jan 2023 KDD’23, August 6–10, 2023, Long Beach, CA Alameen Najjar and Kyle Mede Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The results of our extensive empirical analysis are given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' And finally the paper is summarized in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 2 PREVIOUS WORKS The existing literature on inferring offline activities is vast, and it is beyond the scope of this paper to review it in its entirety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Instead we have organized collections of representative works into 4 categories based on the data used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' those being: ’mobility diaries’, ’Call Detail Records’, ’social check-ins’ and ’others’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The bulk of the literature use self-reported mobility diaries recorded using specialized hardware and/or software [19, 22–24, 26, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The data used is dense however limited in terms of number of subjects, temporal span and spatial coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Early works [22– 24] used Conditional Random Fields (CRFs) and Relational Markov Networks (RMNs) to infer basic offline activities from GPS traces of a few subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In the same vain, [19] proposed a personalized framework that leverages similarities among users to infer offline activities from GPS traces of 50 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In [29], Random Forests are used to infer the purpose of trips (High-level offline activities) of GPS traces of 156 subjects collected over a one week period around Zurich, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Similarly in [28] graph convolutional neural networks (GCNs) [18] are used to classify GPS traces of 139 sub- jects into 5 offline activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' GCNs are also used recently in [26] to infer 8 offline activities in GPS diaries collected and labeled by 167 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' A second subset of the literature [25, 30] infer offline activities at the base station level from Call Detail Records (CDR) provided by mobile network operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In [30], conventional machine learning algorithms are used to infer 8 offline activities from CDR data col- lected in Barcelona and Madrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Similarly [25] uses an ensemble of models to classify CDR data of 80 users into 5 basic offline activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' A third subset of the literature [20, 38] use social check-in data to infer user’s offline activities at the Point Of Interest (POI) level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In [20], CRFs combined with unsupervised clustering is used to infer 7 offline activities from DianPing1 check-in data of 83 users collected in Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Similarly [38] uses non-zero matrix factorization to infer 9 offline activities from Foursquare2 check-in data of ≈2000 users collected in Tokyo and New York city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The fourth and final subset of the literature are works that use data sources other than those listed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' For example, [40] infers 8 offline activities from WiFi traces of 13 subjects moving around a university campus, [6, 34] infer offline activities from microblogs, and [10] infers 13 high-level activities from ride-hailing app data collected around the city of Toronto, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Data wise, our work mostly resembles that of [20, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' However, we are not aware of any previous work that attempted to infer offline activities at the scale we report here (6 cities, ≈15k users, and 6 models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Finally, it is worth mentioning that works, such as [21, 36, 39] that “predict” future offline activities using data similar to ours are beyond the scope of this survey as we are interested in inferring the user’s current rather than future activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='dianping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='com/ 2https://foursquare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='com/ 3 METHODOLOGY In this section we explain the methodology we follow to infer offline activities from location data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='1 Preliminaries Definition 1 (Check-in record).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' A check-in record is a tuple ⟨𝑢,𝑙,𝑡⟩ indicating that user 𝑢 is present at location 𝑙 at time 𝑡, where 𝑙 is the location of a uniquely identified POI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Definition 2 (Offline activity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The offline activity associated with a given check-in record is the category of the POI at which the check- in takes places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' For example, “Dining” is the activity associated with “Food” POIs, and “At home” is the activity associated with “Residence” POIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' It is worth noting that using POI categories as activities is a common practice as it has been done previously in [6, 20, 21, 30, 34, 38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Problem (Activity inference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Given a check-in record ⟨𝑢,𝑙,𝑡⟩ of user 𝑢, the objective is to infer the activity user 𝑢 is engaged in at time 𝑡 and location 𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Let X = {𝑥1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='𝑥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='𝑥𝑛} represent the set of check-in records and Y = {𝑦1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='𝑦2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='𝑦𝑚} represent the set of activities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' the goal is to find a function 𝑓 (·) that assigns each record in X with one activity in Y while satisfying the following condition: arg min 𝑓 ∈F ∥𝑓 (𝑥𝑖) − 𝑦𝑖 ∥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 𝑓 (𝑥𝑖) ∈ Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='𝑦𝑖 ∈ Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' (1) where 𝑦𝑖 is the true label (Activity) associated with 𝑥𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' ∥·∥ is an evaluation operator (That evaluates to 0 when 𝑓 (𝑥𝑖) = 𝑦𝑖 and 1 otherwise),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' and F is the hypothetical space of the task at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='2 Enrichment To account for meaningful contextual information, we enrich check- in records with two sets of features as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Relative location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' By relative location we mean location with respect to the center of the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' More specifically, 1) distance to city center, and 2) bearing angle with respect to city center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Our intuition behind including these two features is based on the assumption that POIs of the same category are more likely to share similar spatial distribution patterns with respect to the center of the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' For example, camping grounds are more likely to be found in the periphery of the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Distance to city center (𝛿) is calculated using the Haversine formula as follows: 𝑎 = sin2( Δ𝜙 2 ) + cos𝜙1 · cos𝜙2 · sin2( Δ𝜆 2 ), (2) 𝑐 = 2 · atan2(√𝑎, √ 1 − 𝑎), (3) 𝛿 = 𝑅 · 𝑐, (4) where 𝜙 is latitude, 𝜆 is longitude, Δ𝜙 is the difference between two latitudes, Δ𝜆 is the difference between two longitudes, and 𝑅 is Earth’s radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' On the other hand, bearing angle with respect to city center (𝜃) is calculated such as: Where You Are Is What You Do: On Inferring Offline Activities From Location Data KDD’23, August 6–10, 2023, Long Beach, CA 𝐴 = sin Δ𝜆 · cos𝜙2, (5) 𝐵 = cos𝜙1 · sin𝜙2 − sin𝜙1 · cos𝜙2 · cos Δ𝜆, (6) 𝜃 = atan2(𝐴, 𝐵), (7) where 𝜙 is latitude, 𝜆 is longitude, and Δ𝜆 is the difference between two longitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Grid statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' We extract three POI-related statistics calculated at the grid cell level, namely POI count, unique user count and check- in count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Our intuition behind including these features is to help the model capture meaningful patterns on the spatial distribution of different POI categories around the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' For example, the number of POIs around residential areas is likely to be less than that around commercial areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Or grid cells with high check-in activity are less likely to be in a residential area, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Each of the features can be described as follows: 𝜓𝑐 = ∑︁ 𝑖 ∈𝑀 𝑖,𝑐 ∈ 𝑁, (8) where𝜓𝑐 is the extracted feature at cell𝑐,𝑖 is the𝑖-th POI/user/check- in per cell 𝑐, 𝑀 is the set of unique POIs/users/check-in per cell 𝑐, and 𝑁 is the set of all cells in the target city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Multi-scale & multi-grid feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' To account for a richer feature set, we extract the aforementioned features at multiple spatial scales using multiple hierarchical grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Implementation details can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='3 Encoding & Classification After enrichment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' each check-in is represented with a vector 𝑣 ∈ R𝑑 made from concatenating the check-in attributes (User ID,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' aggre- gated location,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' and timestamp-related attributes) and the enriched features,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' such that: 𝑣 = (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='𝑙,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='𝛿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='𝜙,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='𝜓1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='𝜓𝑘) ∈ R𝑑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' (9) where 𝑢 is the user ID,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 𝑡 is the timestamp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 𝑙 is the aggregated location,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 𝛿 and 𝜃 are the relative location features,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 𝜓 is the grid statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' and 𝑘 is the number of scales and/or grids used in the check-in enrichment step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The representation obtained in Equation 9 is used next for classi- fication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' To this end, we experimented with 6 classifiers organized into 3 groups: conventional classifiers including both 𝑘-Nearest Neighbours (𝑘-NN) [7] and Support Vector Machines (SVM) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Tabular classifiers including Extreme Gradient Boosting (XGB) [4] and TabNet [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' And Multilayer Perceptrons [32] implemented in two flavors: vanilla/plain (MLP) and regularized (rMLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' See Ap- pendix A for model implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 4 RESULTS In this section we present the validation results of inferring offline activities from location data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='1 Data We used the Foursquare dataset [37] since it is one of the most widely used check-in dataset in the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The dataset consists of over 33M check-ins made by over 266k users at over 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='6M venues in 415 cities worldwide over a period of 18 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' For privacy concerns we opted out of using the user’s actual location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Instead we aggregated location using a grid, such as Uber H33 and Geohash4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In other words, check-ins within the same grid cell are indistinguishable to one another from the location point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Venue names and user names, on the other hand, are already anonymized by the source therefore we used them as they are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Next, we kept check-ins that belong to 6 major cities spanning a wide range of latitudes and longitudes, and representing 6 sub- regions as defined by the United Nations Geoscheme5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Each check- in is assigned a city based on its distance to the city center’s co- ordinates as provided in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' See Figure 1 for a map of the selected cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Target cities have good data coverage with little data gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' See Figure 2 for a visualization of data coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Figure 1: Map of the target cities: Los Angeles, Tokyo, Mum- bai, Sydney, Paris and Milan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Figure 2: Data coverage per target city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Missing polygons in- dicate data gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Finally, we assigned each venue a category out of nine parent categories as defined by the Foursquare API6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In other words, each 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='com/uber/h3 4http://geohash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='org/ 5https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='org/wiki/United_Nations_geoscheme 6http://foursquare-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='herokuapp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='com/ 40 20 0 20 (C) OpenStreetMap contributors 100 50 0 50 100 150Los Angeles Tokyo Paris 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='1 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='8 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='8 OpenStreetMap 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='6 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='6 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='8 -contributors 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='4-118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='3-118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='2-118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='1 Mumbai Sydney 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='7 Milan 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='2 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='8 45.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='3 contributors OpenStreetMap 8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='0 contributors 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='9 OpenStreetMap contributors 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='9 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='0 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='1 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='2 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='8 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='9 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='0KDD’23, August 6–10, 2023, Long Beach, CA Alameen Najjar and Kyle Mede Table 1: Summary of the generated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Dataset/City Check-ins Venues Users Los Angeles 97274 18685 2476 Tokyo 708863 64761 8360 Mumbai 25248 6088 525 Sydney 31934 7739 733 Paris 59816 13603 1973 Milan 40641 8642 897 Total 963776 119518 14964 venue is assigned one of the following categories: “Arts & Entertain- ment,” “College & University,” “Food,” “Nightlife Spot,” “Outdoors & Recreation,” “Professional & Other Places,” “Residence,” “Shop & Service,” and “Travel & Transport”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' These categories are the target variables we aim to infer given a check-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Cities are treated as separate datasets and analyzed indepen- dently as reported in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' See Table 1 for a summary of the statistics of the final datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='2 Evaluation We used logarithmic loss (Log loss) to compare different models in a 3-fold cross validation evaluation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' we evaluate the best model’s classification performance on the test data by reporting the Macro-F1 score which is the harmonic mean of both precision (Macro-P) and recall (Macro-R) averaged over all classes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' such that: Macro-F1 = 2 · Macro-P · Macro-R Macro-P + Macro-R ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' (10) where Macro-P and Macro-R are given as: Macro-P = 1 |Y| ∑︁ 𝑦∈Y TP𝑦 TP𝑦 + FP𝑦 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' (11) Macro-R = 1 |Y| ∑︁ 𝑦∈Y TP𝑦 TP𝑦 + FN𝑦 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' (12) where 𝑦 is activity label,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' TP𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' FP𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' and FN𝑦 are the number of true positives,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' false positives and false negatives for class/activity 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='3 Model Comparison In Table 2 we report the validation loss of all models on all 6 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' See Appendix A for experiment implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Key takeaways can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' XGB dominates all models on every single dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' On average, XGB reduces the naive model’s (𝑘-NN) loss by 59%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Interestingly, XGB’s performance is on average 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='9% better than that of TabNet which is Google’s deep-learning model designed for tabular data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' although inline with previous studies [16, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Second in place is rMLP which cuts the naive model’s loss by 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Still rMLP performs 18% worse than XGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Regulariza- tion boosts plain MLPs by an average of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='7% which we take as a demonstration of the potential regularization has for MLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' It is worth noting that rMLP outperforms TabNet on all datasets except Los Angeles and on average it provides a 6% performance boost over TabNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' This finding is inline with the results in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Moreover, TabNet performs better than plain MLPs on all datasets except Tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' On average TabNet outperforms plain MLPs by ≈ 6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Given the above results, we consider XGB the winning model and therefore we use it hereafter in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Next, we em- pirically attempt to understand how individual features contribute to the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='4 Ablation Study In order to understand how different features contribute to the model’s performance in the following we present the results of a series of ablation experiments we conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Starting with location, in Figure 3 we plotted the per-category classification performance (Macro-F1) against a decreasing grid resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The general trend indicates that higher grid resolution yields better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' City wise, performance degrades by an average of 57% when grid resolution goes from highest to lowest with Milan and Tokyo being the least and most impacted cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Category wise on the other hand, “Travel & Transport” and “Arts & Entertainment” are the least and most impacted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' While the above results indicate that location data play an essen- tial role in the model’s performance what is more interesting is the observation that even when features are extracted using the lowest resolution grid the model is still able to correctly infer the user activity up to 36% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' This is indicative of the importance of non-location features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Figure 3: Location granularity and performance: Classifica- tion performance (Y axis) plotted against decreasing grid res- olution (X Axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Moving on to relative location, in Figure 4, we plotted the change in performance obtained when different relative-location features are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Different cities and/or categories are impacted differ- ently by relative location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' However, on average excluding relative in- formation degrades performance by 2% to 9% with Paris and Tokyo being the least and most impacted cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Moreover, “Residence” and “Outdoors & Recreation” are the least and most impacted categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Individually, “Nightlife Spot” and “Arts & Entertainment” benefit the most from bearing angle and distance-to-center, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Paris Milan Tokyo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='2 Mumbai Sydney Los Angeles 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='6 Arts & Entertainment 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='4 College & University 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='4 Fooc Nightlife Spot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='4 Outdoors & Recreation Professional & Other Places 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='2 Residence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='2 Shop & Service Travel & Transport macro avg 12 11 10 9 8 6 12 11 10 9 8 6 12 11 10 9 8 7 6Where You Are Is What You Do: On Inferring Offline Activities From Location Data KDD’23, August 6–10, 2023, Long Beach, CA Table 2: Validation loss (Log loss ± standard deviation) obtained by different models on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Bold and underline indicate best and second best results, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Mumbai Sydney Milan Paris Los Angeles Tokyo Average 𝑘-NN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='968±0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='924±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='757±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='902±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='009 Figure 4: Relative location and performance: Percentage of performance change (Y axis) resulted from excluding differ- ent relative-location features (X axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Next we evaluated grid statistics in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Removing grid statistics degrades performance by and average of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='6% with Milan and Paris being the least and most impacted cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Category wise, “Travel & Transport” and “Arts & Entertainment” are the least and most impacted categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Different statistics contribute differently to the model’s performance with check-in count being the most important among the three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Followed by user count and finally POI count with a very small margin in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The obtained results demonstrate that both relative information and grid statistics are important to the model’s performance and thus confirm the assumptions we made earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In Figure 6 we compared the model’s performance when features are extracted using one (H3) versus two (H3 and Geohash) grids (See Appendix A for implementation details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Using two grids instead of one boosts performance by an average of 7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Tokyo and Los Angeles benefit the most while Milan and Sydney benefit the least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Category wise, “Nightlife Spot” and “Residence” are the most and least impacted categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In fact, “Residence” in Tokyo is negatively impacted when two grids are used instead of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Finally, in Figure 7 we studied the impact multi-scale features have on the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' It is worth noting that both models (Single-/multi-scale) have the same location granularity (See Appendix A for implementation details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The obtained results show that all categories across all cities benefit from multi-scale feature Figure 5: Grid statistics and performance: Percentage of per- formance change (Y axis) resulted from excluding different grid statistics (X axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Figure 6: Multi-grid feature extraction and performance: Per-category classification performance (Y axis) plotted against number of grids (X axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' On average performance is boosted by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Mumbai and Milan are the most and least impacted cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' On the other hand, category wise, “Arts & Entertainment” and “Residence” are the most and least impacted categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' College & Nightlife Outdoors & Professional Shop & Travel & Arts & Ent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' University Food Spot Recreation & Others Residence Service Transport 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='0 Tokyo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='1 Angeles 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='1 so7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='0 Paris 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='1 Mumbai 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='1 Sydney 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='0 Milan 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='1College & Nightlife Outdoors & Professional Shop & Travel & Arts & Ent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' University Food Spot Recreation & Others Residence Service Iransport kyo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='25 les Angel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='25 Los .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='00 Par 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='25 Mumbai 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='25 ley 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='00 Sydne 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='25 Milan 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='00 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='00 Mumbai Sydney Los Angeles 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='50 Arts&Entertainnent 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='50 College& University Food Nightlife Spot Outdoors&Recreation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='25 Professional & Other Places Residence Shop & Service Travel&Transport macroavg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='00 H3 H3+Geohash H3 H3+Geohash H3 H3+GeohashKDD’23, August 6–10, 2023, Long Beach, CA Alameen Najjar and Kyle Mede Figure 7: Multi-scale feature extraction and performance: Per-category classification performance (Y axis) plotted against number of grid scales (X axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In the following we build upon the insights we gained from the results above to evaluate our final model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='5 Best Model Evaluation For a final evaluation we retrained the winning model using all features extracted using 2 grids at 7 different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Evaluation results of this model are reported in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' On average and over all cities, the best model achieves a Macro- F1 score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' City-wise, the model’s performance is comparable with Mumbai and Los Angeles being on the opposite ends of per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The same observation holds true at the category level with “Nightlife Spot” and “Residence” being the most and least chal- lenging categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In general, the model performance is consistent across city and category with a standard deviation of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='4% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='6%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' To better understand how the model performs across category we plotted the confusion matrix in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' While the matrix shows mostly clear separation between categories, it is clear that that “Nightlife Spot” check-ins are largely misclassified as “Food” in almost all cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The same behaviour is observed with “Shop & Service” however to a lesser degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Finally, for a subjective evaluation we mapped in Figure 9 the aggregated inferences made by the best model next to the ground- truth data for the city of Los Angeles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The visualizations clearly indicate that the inferred maps preserve to a high degree the spatial distribution of the data for the majority of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' It is worth noting that we obtained similar results on the other cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' And that our choice of Los Angeles is based on both data coverage and model performance on the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='6 Summary The following is a summary of the reported results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' First, XGB is the best performing model by a large margin followed by regularized MLPs and TabNet, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Second, the winning model is well capable of inferring offline activities with an average Macro-F1 score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Third, performance is consistent across city and Figure 8: Best model evaluation (2): Normalized confusion matrix for all cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' activity with a standard deviation value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='4% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='6%, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Fourth, we found that “Nightlife” is the most and “At home” is the least challenging offline activities to infer with the winning model achieving an average Macro-F1 score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='844 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='967, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Finally, the ablation study demonstrated that location granularity, relative location and grid statistics each on its own plays a significant role in the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 5 DISCUSSION Thanks to recent software and hardware advances, we live in a world where location data is ubiquitous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Previous research has well demonstrated that location is an effective proxy for the type of activity a person is engaged in in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In this paper, we attempted to answer the following question: How well can modern machine learning algorithms infer offline activities from location data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' To this end, we empirically evalu- ated the performance of 6 models trained to infer 9 basic offline activities using anonymized data collected from ≈15k Foursquare users active in 6 major cities spread across 4 continents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Our exper- iments show that not only modern machine learning algorithms are well capable of inferring basic offline activities (Macro-F1>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='9) given location data, but also tabular models which require minimal knowledge to configure and limited resources to run are among the best performers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' As with the majority of studies, ours is subject to limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' First, social check-in data is subject to bias since it comes from public social media posts shared willingly by individuals who may be less concerned with privacy and not representative of the whole population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Second, it is not always the case that activities match the category of the POI at which a person checks in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' For example, checking in at a “Residence” POI does not always mean engaging in “At home” activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' It could also mean “Work” if the person works from home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Similar arguments could be made about other POI categories.' metadata={'source': 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+page_content=' For example, assigning users of a smartphone app negative or unwanted labels, such as “Unhealthy” or “Overweight” inferred from their location history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' This concern is further ampli- fied given the ubiquity of location data and the recent widespread of accessible yet powerful machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' REFERENCES [1] Sercan Ö Arik and Tomas Pfister.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Tabnet: Attentive interpretable tabular learning.' metadata={'source': 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of dimensions for vision architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In International conference on machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' PMLR, 115–123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' [3] Christopher M Bishop et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Neural networks for pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Oxford university press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' [4] Tianqi Chen and Carlos Guestrin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Xgboost: A scalable tree boosting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In Proceedings of the 22nd acm sigkdd international conference on knowledge discovery and data mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 785–794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' [5] Corinna Cortes and Vladimir Vapnik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Support-vector networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Machine learning 20, 3 (1995), 273–297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' [6] Renhao Cui, Gagan Agrawal, and Rajiv Ramnath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Tweets can tell: Activity recognition using hybrid long short-term memory model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In Proceedings of the 2019 IEEE/ACM international conference on advances in social networks analysis and mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 164–167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' [7] Evelyn Fix and Joseph Lawson Hodges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Discriminatory analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Non- parametric discrimination: Consistency properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' International Statistical Re- view/Revue Internationale de Statistique 57, 3 (1989), 238–247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' [8] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 2016.' metadata={'source': 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106622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' [10] Sanjana Hossain and Khandker Nurul Habib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Inferring the Purposes of using Ride-Hailing Services through Data Fusion of Trip Trajectories, Secondary Travel Surveys, and Land Use Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Transportation Research Record 2675, 9 (2021), 558–573.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' [11] Bo Huang and Jionghua Wang.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='bbc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='com/news/technology-60126012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Accessed: 2023-01-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' [15] Fengrui Jing, Zhenlong Li, Shan Qiao, Jiajia Zhang, Banky Olatosi, and Xiaoming Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Using geospatial social media data for infectious disease studies: 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Systems with Applications 40, 8 (2013), 3299–3311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' [26] Xinyi Liu, Meiliu Wu, Bo Peng, and Qunying Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Graph-based represen- tation for identifying individual travel activities with spatiotemporal trajectories and POI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Scientific Reports 12, 1 (2022), 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' [27] Ilya Loshchilov and Frank Hutter.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Graph convolutional neural networks for human activity purpose imputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In NIPS spatiotemporal workshop at the 32nd Annual conference on neural information processing systems (NIPS 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' [29] Lara Montini, Nadine Rieser-Schüssler, Andreas Horni, and Kay W Axhausen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Trip purpose identification from GPS tracks.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Dropout: a simple way to prevent neural networks from overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The journal of machine learning research 15, 1 (2014), 1929–1958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Table 4: 𝑘-Nearest Neighbours (𝑘-NN) hyper-parameters search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' “𝑘” is the number of nearest neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' “L1” and “L2” are Manhattan and Euclidean distances, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Hyper-parameter Type Range 𝑘 Integer [1, 33] Distance metric Nominal {L1, L2} [36] Diem To, Dong Si, and Ying Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Traveler’s Next Activity Predication with Location-Based Social Network Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In Proceedings of the 3rd ACM SIGSPATIAL International Workshop on Prediction of Human Mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 15–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' [37] Dingqi Yang, Daqing Zhang, and Bingqing Qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Participatory cultural mapping based on collective behavior data in location-based social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' ACM Transactions on Intelligent Systems and Technology (TIST) 7, 3 (2016), 1–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' [38] Dingqi Yang, Daqing Zhang, Vincent W Zheng, and Zhiyong Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Modeling user activity preference by leveraging user spatial temporal characteristics in LBSNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' IEEE Transactions on Systems, Man, and Cybernetics: Systems 45, 1 (2014), 129–142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' [39] Jihang Ye, Zhe Zhu, and Hong Cheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' What’s your next move: User activity prediction in location-based social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In Proceedings of the 2013 SIAM International Conference on Data Mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' SIAM, 171–179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' [40] Vincent W Zheng and Qiang Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' User-dependent aspect model for collaborative activity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' In Twenty-Second International Joint Conference on Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' A IMPLEMENTATION DETAILS Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' We used 80% of each dataset for training and validation, and 20% for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' The training/validation subset is used for hyper- parameters tuning following a 3-fold cross validation evaluation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Moreover, once the best hyper-parameters are found, the whole of the training/validation subset is used to train the final model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' On the other hand, the test subset is used to evaluate the performance of the winning model on the target classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Unless otherwise mentioned, we used resolu- tion 10 Uber H3 grid to extract all features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' For multi-grid models we extracted features using two grids: 1) Uber H3 (Resolution 10) and, 2) Geohash (7 digits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Both grids have cells with an area of the same order of magnitude (105𝑚2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' For multiscale models we extracted features using only Uber H3 grid at 7 different scales (Resolutions 6 to 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Hyper-parameters tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' We used Hyperopt library [2] to tune the hyper-parameters of every model we experimented with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Unlike traditional hyper-parameter tuning methods that blindly explore the search space, such as Grid Search, Hyperopt takes the results of the previous runs into consideration when sampling the search space for the next run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' For all models we limited the search to 100 runs or 48 hours (Whichever comes first).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Search space configu- rations for 𝑘-NN, SVM, XGB, TabNet, MLP and rMLP models are detailed in Tables 4, 5, 6, 7, 8, and 9, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' We trained all models by minimizing the logarithmic loss for a maximum of 500 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' We used early stopping with 1𝑒 − 3 tolerance and patience of 10 epochs whenever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' All models were trained using the same virtual machine equipped with 128 GB of RAM, 16 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='4 GHz CPUs and 2 × NVIDIA Tesla V100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Where You Are Is What You Do: On Inferring Offline Activities From Location Data KDD’23, August 6–10, 2023, Long Beach, CA Table 5: Support Vector Machine (SVM) hyper-parameters search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' “𝐶” and “gamma” are the regularization term and the kernel coefficient, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Hyper-parameter Type Range Log scale 𝐶 Continuous [2−5, 215] ✓ gamma Continuous [2−15, 23] ✓ Table 6: Extreme Gradient Boosting (XGB) hyper- parameters search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' See XGB’s official documentation for more on the hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Hyper-parameter Type Range Log scale eta Continuous [1𝑒 − 3, 1] ✓ lambda Continuous [1𝑒 − 10, 1] ✓ alpha Continuous [1𝑒 − 10, 1] ✓ gamma Continuous [1𝑒 − 1, 1] ✓ num_round Integer [1, 100] max_depth Integer [1, 20] max_delta_step Integer [0, 10] min_child_weight Continuous [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='1, 20] ✓ subsample Continuous [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='01, 1] colsample_bylevel Continuous [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='1, 1] colsample_bynode Continuous [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='1, 1] colsample_bytree Continuous [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='5, 1] Table 7: TabNet hyper-parameters search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' See [1] for more details on the hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Hyper-parameter Type Range 𝑛𝑎 Integer {8, 16, 24, 32, 64, 128} 𝑛𝑠𝑡𝑒𝑝𝑠 Integer [3, 10] batch_size Integer {256, 512,1024, 2048, 4096} virtual_batch_size Integer {256,512,1024,2048, 4096} learning_rate Continuous {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='025} gamma Continuous {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='0} 𝜆𝑠𝑝𝑎𝑟𝑠𝑒 Continuous {0, 10−6, 10−4, 10−3, 10−2, 10−1} momentum Continuous {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='95, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='98} Table 8: Multilayer Perceptron (MLP) hyper-parameters search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' “units” is the number of neurons per hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Hyper-parameter Type Range Log scale Hidden layers Integer {3, 6, 9} units Integer {128, 256, 512} learning_rate Continuous [1𝑒 − 3, 1𝑒 − 1] ✓ 𝑘-NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' We used the Scikit-learn [31] implementation of 𝑘-NN with the search algorithm set to “auto.” We tuned two hyper-parameters, namely number of nearest neighbours (𝑘) and distance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' See Table 4 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Table 9: Regularized Multilayer Perceptron (rMLP) hyper- parameters search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' “units”, “stddev”, “SC”, and “SWA” are number of neurons per hidden layer, Gaussian noise standard deviation, Skip Connections and Stochastic Weight Averaging, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Hyper-parameter Type Range Log scale Hidden layers Integer {3, 6, 9} units Integer {128, 256, 512} learning_rate Continuous [1𝑒 − 3, 1𝑒 − 1] ✓ dropout_rate Continuous [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='5] weight_decay Continuous [1𝑒 − 6, 1𝑒 − 1] ✓ stddev Continuous [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='5] ✓ SC Binary [False, True] SWA Binary [False, True] SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' We used the Scikit-learn implementation of SVM with the default settings on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' We tuned the kernel coefficient (gamma) and the algorithm’s regularization term (𝐶).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' See Table 5 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' XGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Using the official Python implementation of XGB7, we used the gbtree booster paired with the multi:softprob objective func- tion and tuned the hyper-parameters shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' TabNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Using the unofficial PyTorch implementation of TabNet8 we tuned the hyper-parameters shown in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' We used the Keras framework9 to implement all MLP mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Our MLP block consists of a dense layer followed by a ReLu activation layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' We set batch size to 128 and trained the network using the Adam optimizer [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' See Table 8 for a complete list of the hyper-parameters we tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' rMLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' We applied implicit (Batch normalization [12] and Stochastic weight averaging [13]), ensemble (Dropout [35]), structural (Skip connections [8]), and data augmentation (Gaussian noise [3]) tech- niques to regularize vanilla MLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' Implemented in Keras, our rMLP block consists of a dense layer followed by a ReLu activation layer, a Gaussian noise layer, a batch normalization layer, a dropout layer and a concatenation layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' We set batch size to 128 and trained the network using the AdamW optimizer [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' To implement SWA we used an unofficial Python implementation10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' See Table 9 for a complete list of the hyper-parameters we tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content=' 7https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='org/project/xgboost/ 8https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} +page_content='org/project/pytorch-tabnet/ 9https://keras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFRT4oBgHgl3EQfUzcf/content/2301.13537v1.pdf'} 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Facial Attribute Classification +Ching-Hao Chiu1*, Hao-Wei Chung1*, Yu-Jen Chen1, Yiyu Shi 2, Tsung-Yi Ho1 +1 Department of Computer Science, National Tsing Hua Unviserity +2 Department of Computer Science and Engineering, University of Notre Dame +{gwjh101708, xdmanwww, yujenchen}@gapp.nthu.edu.tw +yshi4@nd.edu, tyho@cs.nthu.edu.tw +Abstract +Fairness has become increasingly pivotal in facial recogni- +tion. Without bias mitigation, deploying unfair AI would +harm the interest of the underprivileged population. In this +paper, we observe that though the higher accuracy that fea- +tures from the deeper layer of a neural networks generally +offer, fairness conditions deteriorate as we extract features +from deeper layers. This phenomenon motivates us to extend +the concept of multi-exit framework. Unlike existing works +mainly focusing on accuracy, our multi-exit framework is +fairness-oriented, where the internal classifiers are trained to +be more accurate and fairer. During inference, any instance +with high confidence from an internal classifier is allowed to +exit early. Moreover, our framework can be applied to most +existing fairness-aware frameworks. Experiment results show +that the proposed framework can largely improve the fairness +condition over the state-of-the-art in CelebA and UTK Face +datasets. +1 +Introduction +Machine learning has been applied in various fields and +has impacted our daily life in recent years. Many institu- +tions have introduced machine learning-based systems to +help them decide on administrative operations. Although the +machine learning model achieves accurate prediction, there +exists some bias in such an AI system (Mehrabi et al. 2021; +Dressel and Farid 2018). The discriminative nature of the +machine learning model will harm the opportunity of differ- +ent races, religions, and genders and thus tear society apart. +Several methods are proposed to ameliorate the bias +in machine learning models. Many of them (Wang et al. +2022; Zhang, Lemoine, and Mitchell 2018; Kim et al. 2019; +Ngxande, Tapamo, and Burke 2020) adopted adversarial +training to eliminate bias by training the network to learn +a classifier while disabling the adversary’s ability to catego- +rize the sensitive attribute. Disentanglement representation +(Creager et al. 2019) is another mainstream to achieve fair- +ness. It forces the latent vector of the sensitive group to be +independent of that of the target group and thus reaches fair- +ness. +*These authors contributed equally. +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +In this paper, we observe that although features from a +deep layer of a neural network bring high accuracy in classi- +fication, they cause fairness conditions to deteriorate, and we +will demonstrate this observation in Section 2. This finding +reminds us of the “overthinking” phenomenon in deep neu- +ral networks (Kaya, Hong, and Dumitras 2019), where ac- +curacy decreases as the features come from deeper in a neu- +ral network. This problem is successfully addressed through +multi-exit neural networks by introducing multiple internal +classifiers and treating high-confidence results from these in- +ternal classifiers as the final result. We conjecture that a sim- +ilar approach can be used to address the issue concerning +fairness. +In the proposed multi-exit framework for fairness, both +accuracy and fairness constraints are included when train- +ing every internal classifier to keep the fairness and accu- +racy from shallow to deep. Specifically, with early exits at +the inference stage, a sufficient discriminative basis can be +obtained based on low-level features when classifying easier +samples. This contributes to selecting the optimal prediction +for each test instance regarding the trade-off between accu- +racy and fairness. +To validate the effectiveness of our framework, we per- +form the facial attribute classification on CelebA (Liu et al. +2018) and UTK Face (Zhang, Song, and Qi 2017) datasets. +Experiments show that the proposed method significantly +improves the results from the baseline and different state- +of-the-arts in terms of the trade-off between classification +accuracy and fairness. +The main contributions of the proposed method are as fol- +lows: +• Extensive experiments show that although the features +from a deep layer of a neural network are highly discrim- +inative and thus bring high accuracy, they cause fairness +to deteriorate. +• We explore the use of multi-exit training framework to +deal with the fairness issue. +• We introduce a simple debias framework with high ex- +tensibility that could apply to different baseline and state- +of-the-art, and achieve further improvement. +• Through extensive experiments on different settings and +comparisons, our framework achieves the best trade-off +performances between top-1 accuracy and fairness on +arXiv:2301.02989v1 [cs.CV] 8 Jan 2023 + +CelebA and UTK Face datasets. +2 +Motivation +Figure 1: Equalized odds (EO) and accuracy (Acc.) for the +internal layers of the conventional ResNet on CelebA (a-b) +and UTK Face (c-d) dataset. Note that lower EO represents +fairer. (a) and (c) are the results using ResNet-18, while (b) +and (d) use the ResNet-50. T and S stand for target and sen- +sitive attributes, respectively. In the CelebA dataset, a, e, m, +and y represent attractiveness, bags-under-eyes, male, and +young, respectively. For the UTK Face dataset, A and E are +the Age and Ethnicity, respectively. +As shown in Fig. 1, we observe that despite the features +from a deep layer of a neural network bring high accuracy +in classification, they cause fairness condition to deterio- +rate. We report the equalized odds (EO) and accuracy (Acc.) +of ResNet-18 and ResNet-50 on the CelebA and the UTK +Face dataset. We first trained a vanilla CNN, and froze the +backbone. Then, we trained 3 MLP classifiers with the fea- +tures from each residual module (i.e., Conv2 x, Conv3 x, +Conv4 x, Conv5 x). +For CelebA dataset, in the experiments of the ResNet-18 +(Fig. 1(a)) and the ResNet-50 (Fig. 1(b)), both the equalized +odds (EO) and accuracy (Acc.) increase when the features +are extracted from deeper layers. When comparing the accu- +racy between conv2 x and the final layer in ResNet-18, the +final layer has improved 6% on average. However, the EO +also increases over six times larger on average. As for the +ResNet-50, the accuracy increased by about 16%, and the +EO increased more than eight times larger on average. +In Fig. 1(c) and (d), results on UTK Face dataset again +show a similar trend on the increasing EO and accuracy from +shallow to deep layers. The ResNet-18 (Fig. 1(c)) shows +a 47% improvement in accuracy with 1.7 times higher EO +from the shallow layer (conv2 x) to the deeper layer (Final +layer). As for the ResNet-50 (Fig. 1(d)), accuracy increases +about 52% and EO increases over 2.3 times. +As higher EO stands for larger bias (unfair) of the pre- +dicted result, intuitively, choosing the result at a shallow +layer for final prediction could ameliorate the bias condi- +tion of the model. Our extensive experiments demonstrate +that our observation could be applied to different network +architectures and datasets. +3 +Related Work +3.1 +Multi-Exit Networks and Early Exit Policy +The multi-exit network is designed by putting additional +loss constraints at internal exit branches (internal classifier) +to increase the accuracy at shallow layer. With the early +exit scheme, early exit branches reduce the computational +resource during the model inference time while enhanc- +ing the model’s accuracy. In the inference phase, the in- +stances will stop inferencing and leave the model from dif- +ferent branches following the pre-defined early exit rules or +criteria. BranchyNet (Teerapittayanon, McDanel, and Kung +2016) calculates entropy at each branch after obtaining the +result and exits if the entropy of the predicted result is less +than the threshold value. Shallow-Deep Networks (Kaya, +Hong, and Dumitras 2019) calculates the softmax score of +each internal classifier’s prediction and takes the maximum +probability value as the confidence score. Once the score ex- +ceeds the threshold during the forward passing, the instance +will exit from the branch prematurely.This model further +mitigates the “overthinking” problem of deep neural net- +works. (Schwartz et al. 2020) leveraged the early exit in- +ference scheme of (Kaya, Hong, and Dumitras 2019) and +applied a confidence-based strategy to the natural language +processing task. (Zhou et al. 2020) makes predictions using +adjacent layers and stops inference when the predicted value +of the internal classifier remains constant in a given infer- +ence unit times. In this paper, we borrow the confidence- +based early exit strategy proposed by (Kaya, Hong, and Du- +mitras 2019)to make sure the the early exit instance is confi- +dent enough to be correct. To our best knowledge, we are the +first work that leverages the early exit technique to improve +fairness. +3.2 +Bias Mitigation Methods +Bias mitigation methods are designed to reduce the native +bias in the dataset to reduce the chance of unfair predic- +tion. There are largely three avenues for current debiased +strategy, including pre-processing, in-processing, and post- +processing. (1) Pre-processing strategies usually remove the +information which may cause “discrimination” from train- +ing data before training. (Kamiran and Calders 2012) use +different weight to neutralize the effect of the sensitive in- +formation in training phase and present the experiments re- +sult on real-life data. (Ngxande, Tapamo, and Burke 2020) +and (Lu et al. 2020) achieve fairness via data pre-process +methods including data generation and data augmentation. +In-processing method usually modify on-the-shelf model +architecture, loss function, and model regularization to +achieve fairness goal. Adversarial training mitigates the bias +through adversarially trains an encoder and classifier to learn +a fair representation. (Zhang, Lemoine, and Mitchell 2018) +adversarially cooperate a predictor and an adversary to re- +move the sensitive attributes from the representation. (Kim + +25 +85.0 +85 +82.5 +84 +(AcC. +AcC. +80.0 +83 +15 +77.5 +Accuracy +Accuracy +82 +75.0 +10 +72.5 +Acc (T=a/S=y) +Acc (T=a/S=y) +10 +A +81 +EO (T=a/S=y) +70.0 +EO (T=a/S=y) +Acc (T=e/S=m) +67.5 +Acc (T=e/S=m) +80 +EO (T=e/S=m) +EO (T=e/S=m) +65.0 +Conv2_x +Conv3_x +Conv4 x +Conv5_x +Conv2_x +Conv3x +Conv4_x +Conv5_x +(a) +(b) +85 +Acc (T=A/S=E) +18 +Acc (T=A/S=E) +85 +16 +EO (T=A/S=E) +EO (T=A/S=E) +17 +Equalized Odds (EO) +80 +16 +70 +70 +65 +12 +65 +60 +11 +60 +8 +10 +Conv2_x +Conv3 x +Conv4x +Conv5_x +Conv2_x +Conv3 x +Conv4 x +Conv5x +(c) +(d)Figure 2: Illustration of the multi-exit training framework. lt and ls are the loss function related to target and sensitive attributes, +respectively. +et al. 2019) eliminated the correlations between extracted +feature and sensitive attribute to achieve fairness by unlearn +the bias in the feature domain. (Wang et al. 2022) adver- +sarially learned a perturb to mask out the sensitive informa- +tion of the input images, and the proposed framework do +not need to alter the deep models’ parameters and structure. +Some regularization-based methods, such as (Quadrianto, +Sharmanska, and Thomas 2019) used Hilbert-Schmidt norm +to learn a fair representation that retain the features’ seman- +tics from input domain. (Jung et al. 2021) learned a fair rep- +resentation by distilling the fair information of the teacher +model to the student model with the Maximum Mean Dis- +crepancy (MMD) (Gretton et al. 2012) loss. (Park et al. +2022) introduced a group-wise normalization and penaliz- +ing the inclusion of sensitive attribute to mitigate the intrin- +sic unbiased condition of supervised contrastive learning. +Post-processing method aims to calibrate the model’s out- +put to enhance fairness. They need to use sensitive attribute +and prediction distribution to modify the previous distribu- +tion result. (Hardt, Price, and Srebro 2016) reveals the limit +of demographic parity and give a new metric to fairness, +equalized odds, and show how to adjust the learned predic- +tion. (Zhao et al. 2017) used Lagrangian relaxation to de- +signed an inference algorithm which reduces bias but main- +tain accuracy in the meantime. +In this paper, we focus on improving the existed in- +processing methods by introducing a general multi-exit +(ME) training framework. We compare the regularization- +based (Kim et al. 2019; Jung et al. 2021) fairness methods +w and w/o the ME framework. Moreover, we show that our +framework could apply to complex training structures; for +instance, the adversarial debias method (Kim et al. 2019) +and the fair contrastive learning (Park et al. 2022). +4 +Method +In this section, we provide a clear definition of our goal: +overcoming the prediction bias of the deep neural network, +and we define our problem formulation in 4.1. Afterward, +in Section 4.2, we introduce our main approach, multi-exit +(ME) training framework and the early exit policy, which +allow us to improve the fairness of state-of-the-arts. +4.1 +Problem Formulation +In the classification task, define input features X = x ∈ +Rd, target class Y += y ∈ {1, 2, ..., N}, predicted class +˜Y = ˜y ∈ {1, 2, ..., N}, and sensitive attributes A = a ∈ +{1, 2, ..., M}. The goal is to learn a classifier f : x → y that +predicts the target class Y to achieve high accuracy while be- +ing unbiased to the sensitive attributes A. Several criteria are +proposed to evaluate the bias against sensitive attributes A, +and we will discuss the fairness criteria in our experiments + +Existing Approach +CLSf +pred +Multi-exit Framework +CLS +paid +IC1 +IC2 +IC3 +pred +pred +predin Section 5.2. +4.2 +Multi-Exit (ME) Training Framework +Fundamental to our approach is that although deep neu- +ral networks usually achieve high accuracy in the deeper +layer, the prediction would be unfair to the different sen- +sitive groups, e.g., race, gender, age, etc. This phenomenon +allows the possibility to select the result at a shallow layer +with high confidence to solve the unfair issue and maintain +the predicted accuracy. Our method is based on a multi-exit +training framework and an early exit policy similar to pre- +vious works (Kaya, Hong, and Dumitras 2019). The main +contribution lies in introducing the use of multi-exit to im- +prove the fairness of most state-of-the-arts. +As shown in Fig. 2, existing fairness approaches usually +contain two loss term, target classification loss lt and fair- +ness regularization loss ls. As most fairness research is done +under classification tasks, lt could be either cross-entropy +loss or multi-label soft margin loss, optimizing the training +data’s accuracy. As for the fairness regularization loss ls, it +is designed to remove the bias between two sensitive groups. +In our multi-exit training framework, we duplicate the loss +function, loss = (lt+λls), used in previous work into every +internal classifier (IC), lossIC = (lIC +t ++λlIC +s ), and the final +loss is defined by the weighted summation of them, loss = +α1 · lossIC1 + α2 · lossIC2 + α3 · lossIC3 + αf · lossCLS, +where λ is a hyperparameter that controls the trade-off be- +tween fairness and accuracy. As both features at shallow and +deep layer are included into the loss function, the model will +optimize to increase the accuracy and the fairness from shal- +low to deep naturally. +In addition, as introduced in Section 1, since we observe +that the fairness would drop at the deeper layer, it is recom- +mended to replace the prediction of the final layer with the +internal layer. Based on the heuristic that confidence indi- +cates the correctness of a prediction, we preserve both fair- +ness and accuracy during inference by allowing any instance +with high confidence from an internal classifier to exit early. +We pre-define a confidence threshold θ, and select the result +from the earliest internal classifier in which the confidence +is above the threshold. This early exit policy successfully se- +lects the optimal prediction in terms of the trade-off between +accuracy and fairness. +5 +Experimental Settings +5.1 +Datasets +In this work, we evaluate our framework on two facial at- +tribute datasets, CelebA (Liu et al. 2018) and UTK Face. The +CelebA dataset consists of over 200k images, each with 40 +binary attributes. Similar to (Park et al. 2022), we set Male, +and Young as the sensitive attributes and select the target +attributes which have the highest Pearson correlation with +both sensitive attributes (Torfason et al. 2016). In these at- +tributes, we pick Attractive, Big Nose, and Bags Under Eyes. +We abbreviate the target attribute (T) and the sensitive at- +tributes (S), Attractive, Big Nose, Bags Under Eyes, Male, +and Young as a, b, e, m, and y, respectively. +UTK Face dataset consists of over 20k face images with +3 annotations, Ethnicity, Age, and Gender. We follow the +setting in (Jung et al. 2021) to set Ethnicity as the sensitive +attribute (including White, Black, Asian, and Indian.) and di- +vided the Age into 3 ranges (ages between 0 to 19, 20 to 40, +and larger than 40.) for the target attribute. For FSCL+, we +follow the original paper’s setting and set the Gender as the +target attribute. +To show a fair comparison, we follow the recommended +setting in previous works to divide both datasets into train- +ing/val/test. Results of the test set are reported and discussed +in Section 6. +5.2 +Evaluation Metrics +Several fairness metrics are proposed to evaluate the de- +gree of fairness in classification task. Demographic parity +(Dwork et al. 2012) and equalized odds (EO) (Hardt, Price, +and Srebro 2016; Dwork et al. 2012) are two well-known +fairness criterion. We first define an input feature X ∈ Rd +with sensitive attribute A, ground truth target class Y , and +the predicted target class ˆY . Demographic parity is satisfied +if +P( ˆY = 1|A = 0) = P( ˆY = 1|A = 1). +(1) +The drawback of demographic parity is that the classifier +could achieve the fairness condition by adjusting the propor- +tion of correct rate of two sensitive attributes through mis- +classifying some instances. On the other hand, EO forces the +true positive rate and the false positive rate along different +groups to be equal, that is, +P( ˆY = 1|A = a, Y = y) = P( ˆY = 1|A = b, Y = y) (2) +where y ∈ {0, 1}, and a, b ∈ A. This metric addresses unfair +wrong prediction of the model, and therefore EO will be the +suitable fairness metric to evaluate our model. We calculate +the degree of EO by calculating the disparity of the TPR +and FPR alone different sensitive attributes as follows: +K +� +k=1 +|TPR1 +k − TPR0 +k + FPR1 +k − FPR0 +k| +(3) +where TPRa +k and FPRa +k are the True Positive Rate and +the False Positive Rate respectively of target class k and +sensitive attribute a, this equation can also extend to multi- +attribute cases. In conclusion, the optimal EO score becomes +0 when the True Positive Rate and the False Positive of each +target and sensitive class are the same. It indicates that a +lower EO represents the prediction is fairer. +5.3 +Implementation Details +We utilize Resnet-18 as the backbone of every state-of-the- +art and the baseline CNN used in our experiments. In the +training phase, the data is augmented by random flipping, ro- +tation, and scaling before being fed into the model. The net- +work is trained for 200 epochs using SGD optimizer (Ruder +2016) with an initial learning rate set as 0.01. We attach three +internal classifiers at the end of each residual block. The in- +ternal classifier contains one adaptive average pooling layer + +Table 1: Classification results of the fairness (EO) and accuracy (Acc.) evaluation on the test set of the CelebA dataset. * denotes +our own implementation. +Methods +T=a / S=m +T=a / S=y +T=b / S=m +T=b / S=y +T=e / S=m +T=e / S=y +EO +Acc. +EO +Acc. +EO +Acc. +EO +Acc. +EO +Acc. +EO +Acc. +CNN +27.8 +79.6 +16.8 +79.8 +17.6 +84.0 +14.7 +84.5 +15.0 +83.9 +12.7 +83.8 +ME-CNN +23.7 +82.3 +16.1 +76.8 +12.9 +84.8 +12.9 +84.8 +10.8 +82.5 +8.1 +83.6 +LNL +21.8 +79.9 +13.7 +74.3 +10.7 +82.3 +6.8 +82.3 +5.0 +81.6 +3.3 +80.3 +ME-LNL +14.4 +82.2 +13.1 +72.7 +7.3 +82.8 +5.5 +83.1 +2.7 +84.0 +1.0 +82.6 +HSIC* +19.4 +81.7 +16.5 +80.3 +11.2 +80.8 +10.5 +82.6 +12.5 +84.0 +7.4 +84.2 +ME-HSIC +12.9 +78.8 +15.9 +78.7 +3.6 +80.9 +3.2 +82.2 +7.8 +83.5 +4.1 +82.0 +FSCL+ +6.5 +79.1 +12.4 +79.1 +4.7 +82.9 +4.8 +84.1 +3.0 +83.4 +1.6 +83.5 +ME-FSCL+ +5.8 +78.2 +7.6 +76.4 +3.6 +81.2 +2.8 +84.4 +2.0 +83.5 +1.4 +80.8 +MFD +7.4 +78.0 +14.9 +80.0 +7.3 +78.0 +5.4 +78.0 +8.7 +79.0 +5.2 +78.0 +ME-MFD +5.8 +78.3 +11.4 +79.5 +2.6 +82.1 +3.3 +82.6 +1.4 +81.9 +1.5 +84.2 +and a two-layer MLP. As introduced in Section 4.2, the mod- +ified loss function in the multi-exit training framework is a +weighted summation of the loss from each internal classi- +fier. Since we observe that the learning capacity of shallow +ICs are weaker than deep ICs. The coefficients, α1, α2, α3, +and αf, are set to 0.3, 0.45, 0.6, and 0.9, respectively. We +tune the confidence threshold based on the model’s perfor- +mance on the validation set and set the confidence threshold +θ = 0.85 for the early exit policy at the inference phase. +To show that our scheme can be widely applied to most +existing frameworks, we conduct experiments on four base- +lines selected from state-of-art approaches for fairness- +aware learning: LNL (Kim et al. 2019), HSIC (Quadrianto, +Sharmanska, and Thomas 2019), MFD (Jung et al. 2021), +and FSCL+ (Park et al. 2022). All the baselines are repro- +duced by following the recommended hyperparameter set- +tings of the original papers or resources. +Multi-Exit (ME) Implementation +We apply our ME +framework on four state-of-art methods as below: +1. ME-CNN : The ME-CNN is our baseline multi-exits +framework. We apply cross-entropy loss to each IC with- +out any fairness constraint, that is loss = α1 ·lIC1 +t ++α2 · +lIC2 +t ++ α3 · lIC3 +t ++ αf · lCLS +t +. +2. ME-LNL : For the adversarial debias method, we replace +g ◦ f in the original paper with ME-CNN. The loss func- +tion becomes loss = α1 · lIC1 +t ++ α2 · lIC2 +t ++ α3 · lIC3 +t ++ +αf · lCLS +t ++ λ · Adv loss, and the model is trained with +an adversarial strategy. Adv loss is the adversarial term +for bias mitigation. +3. ME-HSIC : We use ME-CNN as the backbone and apply +the proposed loss term in HSIC to each IC. +4. ME-MFD : We follow the teacher-student training frame- +work in the original paper and use ME-CNN as the back- +bone. We first train a ME-CNN as our teacher model and +apply the proposed loss term in MFD to each IC in the +student model. +5. ME-FSCL+ : We first pretrain the ICs in a ME-CNN +backbone with the fair supervised contrastive loss in +FSCL+ together. Then, we fix the pretrained ME-CNN +backbone and train the linear classifiers of each IC to- +gether. +Table 2: Classification results of the fairness (EO) and ac- +curacy (Acc.) evaluation on the test set of the UTK Face +dataset. * denotes our own implementation. T and S stand +for target and sensitive attributes, respectively. respectively. +S=Ethnicity +EO +Acc. +T=Age +CNN* +17.8 +82.4 +ME-CNN +16.0 +82.5 +LNL* +18.9 +81.1 +ME-LNL +16.4 +81.9 +HSIC* +16.4 +79.4 +ME-HSIC +14.9 +78.5 +MFD* +15.1 +79.1 +ME-MFD +13.7 +83.1 +T=Gender +FSCL+* +3.7 +70.1 +ME-FSCL+ +3.2 +70.0 +6 +Results +6.1 +Comparison with State-of-the-art +In this section, we show the feasibility of ME training frame- +work on four state-of-the-art, LNL (Kim et al. 2019), HSIC +(Quadrianto, Sharmanska, and Thomas 2019), FSCL+ (Park +et al. 2022), and MFD (Jung et al. 2021), by comparing +the result with and without the ME training framework on +CelebA and UTK Face dataset in Table 1 and Table 2, re- +spectively. +For the CelebA dataset, in Table 1 we follow the sensi- +tive and target groups setting in (Park et al. 2022) and com- +pare our results with their reproduce results accordingly. The +ME-CNN is the baseline ME framework training without +any fairness constraint. That is, the loss function in each +internal classifier is lossIC = lIC +t . The ME-CNN frame- +work improves the EO in 20.3% while losing only 0.13% of +accuracy in average. In the comparison with different state- +of-the-art, our results achieve a 38.5% EO improvement in + +Figure 3: Visualization of the features of the CelebA dataset (T=a / S=m). (a) demonstrates the feature of the proposed multi-exit +training framework on MFD, while (b) are the features without using the multi-exit training. ICx represents the xth internal +classifier. CLSf stands for the classifier following behind the last layer. Since the MFD did not contain internal classifier, we +extract the feature from the same layers of ME-MFD, which are Conv2 x, Conv3 x, Conv4 x, and Conv5 x. +average while keeping the competitive accuracy. It is note- +worthy that in ME-MFD and ME-LNL, our framework also +improves the accuracy by an average of 3.8% and 1.3% , +respectively. +For UTK Face dataset in Table 2, we follow (Jung et al. +2021) to define Ethnicity as the sensitive attribute and Age as +the target groups. Compared with the CNN baseline, the EO +decreased from 17.8 to 16, which is a 10% improvement. As +for the comparison with different state-of-the-art, our results +achieve a 11.3% EO improvement in average. In addition, in +ME-LNL and ME-MFD, the accuracy also shows a 3% im- +provement in average. Since the FSCL+ (Park et al. 2022) +selected the Gender as the target attribute, we follow their +data imbalance setting to product the experiment. The bias +level hyperparameter N is set as 5, which means male data +is five times as much as female data, and the other sensi- +tive group has the opposite gender ratio. Performance com- +parison between FSCL+ and ME-FSCL+ shows that ME- +FSCL+ achieves a 13.5% improvement at EO than FSCL+, +where the accuracy is almost the same. Comparisons in Ta- +ble 2 successfully demonstrate that our framework outper- +forms all the baseline on the fairness score with a competi- +tive accuracy in UTK Face dataset. +6.2 +Ablation Study of Multi-Exit Training +Framework +In this section, we establish the ablation study of apply- +ing a multi-exit training framework to the MFD (Jung et al. +2021). In Fig. 3(a), we visualized the representation of the +test instance from each internal classifier. Since the multi- +exit training framework optimized each internal classifier +(ICx and CLSf) for high accuracy and fairness, we can ob- +serve a clear decision boundary between the attractive and +non-attractive target class. However, the features of the sen- +sitive attribute (gender) are mixed and uniformly distributed, +which indicates that the model remains fair. +In addition, we also visualize the representation of the test +instance from the same layer of MFD, which is the output +of each residual module of the network in Fig. 3(b). Obvi- +ously, since the intermediate feature does not pass through +any direct target optimization, both the target and the sen- +sitive attributes could not be easily recognized in shallow + +Non-Attractive +Non-Attractive +Non-Attractive +Non-Attractive +Attractive +Attractive +Attractive +Attractive +Male +Male +Male +Male +Female +Female +Female +Female +Non-Attractive +Non-Attractive +Non-Attractive +Non-Attractive +Attractive +Attractive +Attractive +Attractive +Male +Male +Male +Male +Female +Female +Female +FemaleTable 3: Classification results of the proposed ME-MFD and MFD selected from each exit. ICx represents the xth internal +classifier. CLSf stands for the classifier following behind the last layer. Since the MFD did not contain internal classifier, we +extract the feature from the same layers of ME-MFD, which are Conv2 x, Conv3 x, Conv4 x, and Conv5 x. The fairness (EO) +and accuracy (Acc.) evaluation on the test set of the CelebA dataset are reported. +Methods +T=a / S=m +T=a / S=y +T=b / S=m +T=b / S=y +T=e / S=m T=e / S=y +EO +Acc. +EO +Acc. +EO +Acc. +EO +Acc. +EO +Acc. +EO Acc. +ME-MFD +IC1 +1.4 +70.5 +12.9 76.9 +1.4 +80.1 +3.4 +78.2 +0.8 +79.6 +1.8 +80.0 +IC2 +2.0 +75.7 +13.4 80.0 +1.5 +81.7 +7.1 +78.2 +2.4 +83.2 +3.2 +84.0 +IC3 +2.2 +77.0 +16.4 81.7 +4.6 +83.7 +12.0 78.9 +4.4 +84.6 +3.5 +85.1 +CLSf +13.3 79.9 +23.4 83.0 +17.7 84.6 +18.0 79.2 +12.2 85.3 +9.6 +85.3 +MFD +Conv2 x +12.2 68.9 +13.3 70.5 +4.5 +78.5 +3.7 +78.7 +0.1 +79.7 +1.9 +79.4 +Conv3 x +16.0 73.2 +15.3 75.3 +19.1 80.0 +10.9 79.7 +8.4 +80.8 +3.7 +80.3 +Conv4 x +23.1 80.5 +19.2 79.3 +27.6 80.0 +16.9 81.4 +18.7 83.1 +7.8 +82.8 +Conv5 x +23.1 81.3 +15.6 79.0 +12.5 80.7 +8.0 +83.0 +13.1 83.3 +6.5 +83.1 +Table 4: Classification results of the proposed ME-MFD selected from each exit. ICx represents the xth internal classifier. +CLSf stands for the classifier following behind the last layer. ME-MFD (EE) denotes the result of using the confidence-based +early exit algorithm. The fairness (EO) and accuracy (Acc.) evaluation on the test set of the CelebA dataset are reported. +Methods +T=a / S=m +T=a / S=y +T=b / S=m +T=b / S=y +T=e / S=m +T=e / S=y +EO +Acc. +EO +Acc. +EO +Acc. +EO +Acc. +EO +Acc. +EO +Acc. +IC1 +1.4 +70.5 +12.9 +76.9 +1.4 +80.1 +3.4 +78.2 +0.8 +79.6 +1.8 +80.0 +IC2 +2.0 +75.7 +13.4 +80.0 +1.5 +81.7 +7.1 +78.2 +2.4 +83.2 +3.2 +84.0 +IC3 +2.2 +77.0 +16.4 +81.7 +4.6 +83.7 +12.0 +78.9 +4.4 +84.6 +3.5 +85.1 +CLSf +13.3 +79.9 +23.4 +83.0 +17.7 +84.6 +18.0 +79.2 +12.2 +85.3 +9.6 +85.3 +ME-MFD (EE) +5.8 +78.3 +11.4 +79.5 +2.6 +82.1 +3.3 +82.6 +1.4 +81.9 +1.5 +84.2 +layers (Conv2 x and Conv3 x). At the deeper layer, the loss +function maximizes the accuracy and minimizes the bias of +the sensitive group, the feature of the target group can be +clearly separated, whereas the sensitive attribute remains the +same. We also report the quantitative results of the experi- +ment mentioned above to show the feasibility of the multi- +exit training framework. +In Table 3, we compare the classification results of ME- +MFD and MFD selected from each exit branch. We re- +port the features of MFD at different residual modules (i.e., +Conv2 x, Conv3 x, Conv4 x, and Conv5 x) and each IC’s +features of ME-MFD. The multi-exit framework improves +the EO by an average of 27% and the accuracy by 1%. Be- +sides, the EO of MFD increases from Conv2 x to Conv 4 but +drops at the last layer. This phenomenon is due to the Max- +imum Mean Discrepancy (MMD) constraint being imposed +at the last layer while there is no fairness constraint on the +features of shallow layers. ME-MFD regularizes the fairness +condition of each IC; as a result, the trend of EO score still +holds our observation in Section 2. +Experiments demonstrate the significance of using the +multi-exit training framework, which allows us to select pre- +diction at a shallow layer to achieve a fair and high accuracy +result. +6.3 +Ablation Study of Early Exit Policy +In this section, we study the impact of using the proposed +early exit policy. Table 4 shows the fairness and accuracy +of the proposed ME-MFD selected from a different exit. +From the table, we observe that the IC1 obtains the smallest +EO and the accuracy, while the CLSf achieves the largest. +The increasing trend in both metrics indicates that there is +a trade-off between fairness and accuracy. Thus, the pro- +posed early exit policy independently selects the exit of each +test instance to preserve a large fairness improvement with +only a slight drop in accuracy. In the comparison without +using the proposed early exit (CLSf), our method achieves +74.5% lower EO, but the accuracy drop is less than 1.7% on +average. The significant improvement demonstrates the im- +portance of deciding the exit for each instance, which also +shows that the proposed early exit algorithm is essential. +7 +Conclusion +In this paper, we first explored the problem that the fair- +ness condition deteriorates as we classify the features in +deeper layers. Then, we introduced the multi-exit training +framework, with high extensibility that could be applied to +many bias mitigation methods. With the confidence-based +exit strategy, we select the optimal exit for each test instance +to achieve both high accuracy and fairness. The extensive re- +sults and the ablation studies have shown that our framework +can achieve the best trade-off of accuracy and fairness con- +ditions compared to the state-of-the-art on two well-known +facial datasets. +References +Creager, E.; Madras, D.; Jacobsen, J.-H.; Weis, M.; Swer- +sky, K.; Pitassi, T.; and Zemel, R. 2019. Flexibly fair rep- + +resentation learning by disentanglement. +In International +conference on machine learning, 1436–1445. PMLR. +Dressel, J.; and Farid, H. 2018. +The accuracy, fairness, +and limits of predicting recidivism. Science advances, 4(1): +eaao5580. +Dwork, C.; Hardt, M.; Pitassi, T.; Reingold, O.; and Zemel, +R. 2012. Fairness through awareness. In Proceedings of the +3rd innovations in theoretical computer science conference, +214–226. +Gretton, A.; Borgwardt, K. M.; Rasch, M. J.; Sch¨olkopf, B.; +and Smola, A. 2012. A kernel two-sample test. 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Bert loses patience: Fast and robust inference with +early exit. Advances in Neural Information Processing Sys- +tems, 33: 18330–18341. + diff --git a/g9E1T4oBgHgl3EQfMgMl/content/tmp_files/load_file.txt b/g9E1T4oBgHgl3EQfMgMl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..074ab540be70ef6a40c4bdf5e94055830296b4c1 --- /dev/null +++ b/g9E1T4oBgHgl3EQfMgMl/content/tmp_files/load_file.txt @@ -0,0 +1,949 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf,len=948 +page_content='Fair Multi-Exit Framework for Facial Attribute Classification Ching-Hao Chiu1*, Hao-Wei Chung1*, Yu-Jen Chen1, Yiyu Shi 2, Tsung-Yi Ho1 1 Department of Computer Science, National Tsing Hua Unviserity 2 Department of Computer Science and Engineering, University of Notre Dame {gwjh101708, xdmanwww, yujenchen}@gapp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='nthu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='tw yshi4@nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='edu, tyho@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='nthu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='tw Abstract Fairness has become increasingly pivotal in facial recogni- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Without bias mitigation, deploying unfair AI would harm the interest of the underprivileged population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In this paper, we observe that though the higher accuracy that fea- tures from the deeper layer of a neural networks generally offer, fairness conditions deteriorate as we extract features from deeper layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' This phenomenon motivates us to extend the concept of multi-exit framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Unlike existing works mainly focusing on accuracy, our multi-exit framework is fairness-oriented, where the internal classifiers are trained to be more accurate and fairer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' During inference, any instance with high confidence from an internal classifier is allowed to exit early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Moreover, our framework can be applied to most existing fairness-aware frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Experiment results show that the proposed framework can largely improve the fairness condition over the state-of-the-art in CelebA and UTK Face datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 1 Introduction Machine learning has been applied in various fields and has impacted our daily life in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Many institu- tions have introduced machine learning-based systems to help them decide on administrative operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Although the machine learning model achieves accurate prediction, there exists some bias in such an AI system (Mehrabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Dressel and Farid 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The discriminative nature of the machine learning model will harm the opportunity of differ- ent races, religions, and genders and thus tear society apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Several methods are proposed to ameliorate the bias in machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Many of them (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Zhang, Lemoine, and Mitchell 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Ngxande, Tapamo, and Burke 2020) adopted adversarial training to eliminate bias by training the network to learn a classifier while disabling the adversary’s ability to catego- rize the sensitive attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Disentanglement representation (Creager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2019) is another mainstream to achieve fair- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' It forces the latent vector of the sensitive group to be independent of that of the target group and thus reaches fair- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In this paper, we observe that although features from a deep layer of a neural network bring high accuracy in classi- fication, they cause fairness conditions to deteriorate, and we will demonstrate this observation in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' This finding reminds us of the “overthinking” phenomenon in deep neu- ral networks (Kaya, Hong, and Dumitras 2019), where ac- curacy decreases as the features come from deeper in a neu- ral network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' This problem is successfully addressed through multi-exit neural networks by introducing multiple internal classifiers and treating high-confidence results from these in- ternal classifiers as the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' We conjecture that a sim- ilar approach can be used to address the issue concerning fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In the proposed multi-exit framework for fairness, both accuracy and fairness constraints are included when train- ing every internal classifier to keep the fairness and accu- racy from shallow to deep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Specifically, with early exits at the inference stage, a sufficient discriminative basis can be obtained based on low-level features when classifying easier samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' This contributes to selecting the optimal prediction for each test instance regarding the trade-off between accu- racy and fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' To validate the effectiveness of our framework, we per- form the facial attribute classification on CelebA (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2018) and UTK Face (Zhang, Song, and Qi 2017) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Experiments show that the proposed method significantly improves the results from the baseline and different state- of-the-arts in terms of the trade-off between classification accuracy and fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The main contributions of the proposed method are as fol- lows: Extensive experiments show that although the features from a deep layer of a neural network are highly discrim- inative and thus bring high accuracy, they cause fairness to deteriorate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' We explore the use of multi-exit training framework to deal with the fairness issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' We introduce a simple debias framework with high ex- tensibility that could apply to different baseline and state- of-the-art, and achieve further improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Through extensive experiments on different settings and comparisons, our framework achieves the best trade-off performances between top-1 accuracy and fairness on arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='02989v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='CV] 8 Jan 2023 CelebA and UTK Face datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2 Motivation Figure 1: Equalized odds (EO) and accuracy (Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=') for the internal layers of the conventional ResNet on CelebA (a-b) and UTK Face (c-d) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Note that lower EO represents fairer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' (a) and (c) are the results using ResNet-18, while (b) and (d) use the ResNet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' T and S stand for target and sen- sitive attributes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In the CelebA dataset, a, e, m, and y represent attractiveness, bags-under-eyes, male, and young, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' For the UTK Face dataset, A and E are the Age and Ethnicity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 1, we observe that despite the features from a deep layer of a neural network bring high accuracy in classification, they cause fairness condition to deterio- rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' We report the equalized odds (EO) and accuracy (Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=') of ResNet-18 and ResNet-50 on the CelebA and the UTK Face dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' We first trained a vanilla CNN, and froze the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Then, we trained 3 MLP classifiers with the fea- tures from each residual module (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=', Conv2 x, Conv3 x, Conv4 x, Conv5 x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' For CelebA dataset, in the experiments of the ResNet-18 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 1(a)) and the ResNet-50 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 1(b)), both the equalized odds (EO) and accuracy (Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=') increase when the features are extracted from deeper layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' When comparing the accu- racy between conv2 x and the final layer in ResNet-18, the final layer has improved 6% on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' However, the EO also increases over six times larger on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' As for the ResNet-50, the accuracy increased by about 16%, and the EO increased more than eight times larger on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 1(c) and (d), results on UTK Face dataset again show a similar trend on the increasing EO and accuracy from shallow to deep layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The ResNet-18 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 1(c)) shows a 47% improvement in accuracy with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 times higher EO from the shallow layer (conv2 x) to the deeper layer (Final layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' As for the ResNet-50 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 1(d)), accuracy increases about 52% and EO increases over 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' As higher EO stands for larger bias (unfair) of the pre- dicted result, intuitively, choosing the result at a shallow layer for final prediction could ameliorate the bias condi- tion of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Our extensive experiments demonstrate that our observation could be applied to different network architectures and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 3 Related Work 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 Multi-Exit Networks and Early Exit Policy The multi-exit network is designed by putting additional loss constraints at internal exit branches (internal classifier) to increase the accuracy at shallow layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' With the early exit scheme, early exit branches reduce the computational resource during the model inference time while enhanc- ing the model’s accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In the inference phase, the in- stances will stop inferencing and leave the model from dif- ferent branches following the pre-defined early exit rules or criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' BranchyNet (Teerapittayanon, McDanel, and Kung 2016) calculates entropy at each branch after obtaining the result and exits if the entropy of the predicted result is less than the threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Shallow-Deep Networks (Kaya, Hong, and Dumitras 2019) calculates the softmax score of each internal classifier’s prediction and takes the maximum probability value as the confidence score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Once the score ex- ceeds the threshold during the forward passing, the instance will exit from the branch prematurely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='This model further mitigates the “overthinking” problem of deep neural net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' (Schwartz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2020) leveraged the early exit in- ference scheme of (Kaya, Hong, and Dumitras 2019) and applied a confidence-based strategy to the natural language processing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2020) makes predictions using adjacent layers and stops inference when the predicted value of the internal classifier remains constant in a given infer- ence unit times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In this paper, we borrow the confidence- based early exit strategy proposed by (Kaya, Hong, and Du- mitras 2019)to make sure the the early exit instance is confi- dent enough to be correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' To our best knowledge, we are the first work that leverages the early exit technique to improve fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 Bias Mitigation Methods Bias mitigation methods are designed to reduce the native bias in the dataset to reduce the chance of unfair predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' There are largely three avenues for current debiased strategy, including pre-processing, in-processing, and post- processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' (1) Pre-processing strategies usually remove the information which may cause “discrimination” from train- ing data before training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' (Kamiran and Calders 2012) use different weight to neutralize the effect of the sensitive in- formation in training phase and present the experiments re- sult on real-life data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' (Ngxande, Tapamo, and Burke 2020) and (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2020) achieve fairness via data pre-process methods including data generation and data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In-processing method usually modify on-the-shelf model architecture, loss function, and model regularization to achieve fairness goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Adversarial training mitigates the bias through adversarially trains an encoder and classifier to learn a fair representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' (Zhang, Lemoine, and Mitchell 2018) adversarially cooperate a predictor and an adversary to re- move the sensitive attributes from the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' (Kim 25 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 85 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 84 (AcC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' AcC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 83 15 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 Accuracy Accuracy 82 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 10 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 Acc (T=a/S=y) Acc (T=a/S=y) 10 A 81 EO (T=a/S=y) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 EO (T=a/S=y) Acc (T=e/S=m) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 Acc (T=e/S=m) 80 EO (T=e/S=m) EO (T=e/S=m) 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 Conv2_x Conv3_x Conv4 x Conv5_x Conv2_x Conv3x Conv4_x Conv5_x (a) (b) 85 Acc (T=A/S=E) 18 Acc (T=A/S=E) 85 16 EO (T=A/S=E) EO (T=A/S=E) 17 Equalized Odds (EO) 80 16 70 70 65 12 65 60 11 60 8 10 Conv2_x Conv3 x Conv4x Conv5_x Conv2_x Conv3 x Conv4 x Conv5x (c) (d)Figure 2: Illustration of the multi-exit training framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' lt and ls are the loss function related to target and sensitive attributes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2019) eliminated the correlations between extracted feature and sensitive attribute to achieve fairness by unlearn the bias in the feature domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2022) adver- sarially learned a perturb to mask out the sensitive informa- tion of the input images, and the proposed framework do not need to alter the deep models’ parameters and structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Some regularization-based methods, such as (Quadrianto, Sharmanska, and Thomas 2019) used Hilbert-Schmidt norm to learn a fair representation that retain the features’ seman- tics from input domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' (Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2021) learned a fair rep- resentation by distilling the fair information of the teacher model to the student model with the Maximum Mean Dis- crepancy (MMD) (Gretton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2012) loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2022) introduced a group-wise normalization and penaliz- ing the inclusion of sensitive attribute to mitigate the intrin- sic unbiased condition of supervised contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Post-processing method aims to calibrate the model’s out- put to enhance fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' They need to use sensitive attribute and prediction distribution to modify the previous distribu- tion result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' (Hardt, Price, and Srebro 2016) reveals the limit of demographic parity and give a new metric to fairness, equalized odds, and show how to adjust the learned predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2017) used Lagrangian relaxation to de- signed an inference algorithm which reduces bias but main- tain accuracy in the meantime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In this paper, we focus on improving the existed in- processing methods by introducing a general multi-exit (ME) training framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' We compare the regularization- based (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2021) fairness methods w and w/o the ME framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Moreover, we show that our framework could apply to complex training structures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' for instance, the adversarial debias method (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2019) and the fair contrastive learning (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 4 Method In this section, we provide a clear definition of our goal: overcoming the prediction bias of the deep neural network, and we define our problem formulation in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Afterward, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2, we introduce our main approach, multi-exit (ME) training framework and the early exit policy, which allow us to improve the fairness of state-of-the-arts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 Problem Formulation In the classification task, define input features X = x ∈ Rd, target class Y = y ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=', N}, predicted class ˜Y = ˜y ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=', N}, and sensitive attributes A = a ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=', M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The goal is to learn a classifier f : x → y that predicts the target class Y to achieve high accuracy while be- ing unbiased to the sensitive attributes A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Several criteria are proposed to evaluate the bias against sensitive attributes A, and we will discuss the fairness criteria in our experiments Existing Approach CLSf pred Multi-exit Framework CLS paid IC1 IC2 IC3 pred pred predin Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 Multi-Exit (ME) Training Framework Fundamental to our approach is that although deep neu- ral networks usually achieve high accuracy in the deeper layer, the prediction would be unfair to the different sen- sitive groups, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=', race, gender, age, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' This phenomenon allows the possibility to select the result at a shallow layer with high confidence to solve the unfair issue and maintain the predicted accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Our method is based on a multi-exit training framework and an early exit policy similar to pre- vious works (Kaya, Hong, and Dumitras 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The main contribution lies in introducing the use of multi-exit to im- prove the fairness of most state-of-the-arts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2, existing fairness approaches usually contain two loss term, target classification loss lt and fair- ness regularization loss ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' As most fairness research is done under classification tasks, lt could be either cross-entropy loss or multi-label soft margin loss, optimizing the training data’s accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' As for the fairness regularization loss ls, it is designed to remove the bias between two sensitive groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In our multi-exit training framework, we duplicate the loss function, loss = (lt+λls), used in previous work into every internal classifier (IC), lossIC = (lIC t +λlIC s ), and the final loss is defined by the weighted summation of them, loss = α1 · lossIC1 + α2 · lossIC2 + α3 · lossIC3 + αf · lossCLS, where λ is a hyperparameter that controls the trade-off be- tween fairness and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' As both features at shallow and deep layer are included into the loss function, the model will optimize to increase the accuracy and the fairness from shal- low to deep naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In addition, as introduced in Section 1, since we observe that the fairness would drop at the deeper layer, it is recom- mended to replace the prediction of the final layer with the internal layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Based on the heuristic that confidence indi- cates the correctness of a prediction, we preserve both fair- ness and accuracy during inference by allowing any instance with high confidence from an internal classifier to exit early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' We pre-define a confidence threshold θ, and select the result from the earliest internal classifier in which the confidence is above the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' This early exit policy successfully se- lects the optimal prediction in terms of the trade-off between accuracy and fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 5 Experimental Settings 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 Datasets In this work, we evaluate our framework on two facial at- tribute datasets, CelebA (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2018) and UTK Face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The CelebA dataset consists of over 200k images, each with 40 binary attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Similar to (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2022), we set Male, and Young as the sensitive attributes and select the target attributes which have the highest Pearson correlation with both sensitive attributes (Torfason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In these at- tributes, we pick Attractive, Big Nose, and Bags Under Eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' We abbreviate the target attribute (T) and the sensitive at- tributes (S), Attractive, Big Nose, Bags Under Eyes, Male, and Young as a, b, e, m, and y, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' UTK Face dataset consists of over 20k face images with 3 annotations, Ethnicity, Age, and Gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' We follow the setting in (Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2021) to set Ethnicity as the sensitive attribute (including White, Black, Asian, and Indian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=') and di- vided the Age into 3 ranges (ages between 0 to 19, 20 to 40, and larger than 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=') for the target attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' For FSCL+, we follow the original paper’s setting and set the Gender as the target attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' To show a fair comparison, we follow the recommended setting in previous works to divide both datasets into train- ing/val/test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Results of the test set are reported and discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 Evaluation Metrics Several fairness metrics are proposed to evaluate the de- gree of fairness in classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Demographic parity (Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2012) and equalized odds (EO) (Hardt, Price, and Srebro 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2012) are two well-known fairness criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' We first define an input feature X ∈ Rd with sensitive attribute A, ground truth target class Y , and the predicted target class ˆY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Demographic parity is satisfied if P( ˆY = 1|A = 0) = P( ˆY = 1|A = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' (1) The drawback of demographic parity is that the classifier could achieve the fairness condition by adjusting the propor- tion of correct rate of two sensitive attributes through mis- classifying some instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' On the other hand, EO forces the true positive rate and the false positive rate along different groups to be equal, that is, P( ˆY = 1|A = a, Y = y) = P( ˆY = 1|A = b, Y = y) (2) where y ∈ {0, 1}, and a, b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' This metric addresses unfair wrong prediction of the model, and therefore EO will be the suitable fairness metric to evaluate our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' We calculate the degree of EO by calculating the disparity of the TPR and FPR alone different sensitive attributes as follows: K � k=1 |TPR1 k − TPR0 k + FPR1 k − FPR0 k| (3) where TPRa k and FPRa k are the True Positive Rate and the False Positive Rate respectively of target class k and sensitive attribute a, this equation can also extend to multi- attribute cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In conclusion, the optimal EO score becomes 0 when the True Positive Rate and the False Positive of each target and sensitive class are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' It indicates that a lower EO represents the prediction is fairer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 Implementation Details We utilize Resnet-18 as the backbone of every state-of-the- art and the baseline CNN used in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In the training phase, the data is augmented by random flipping, ro- tation, and scaling before being fed into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The net- work is trained for 200 epochs using SGD optimizer (Ruder 2016) with an initial learning rate set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' We attach three internal classifiers at the end of each residual block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The in- ternal classifier contains one adaptive average pooling layer Table 1: Classification results of the fairness (EO) and accuracy (Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=') evaluation on the test set of the CelebA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' * denotes our own implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Methods T=a / S=m T=a / S=y T=b / S=m T=b / S=y T=e / S=m T=e / S=y EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' CNN 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='8 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='6 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='9 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='8 ME-CNN 23.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 ME-MFD 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='8 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 11.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 and a two-layer MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' As introduced in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2, the mod- ified loss function in the multi-exit training framework is a weighted summation of the loss from each internal classi- fier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Since we observe that the learning capacity of shallow ICs are weaker than deep ICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The coefficients, α1, α2, α3, and αf, are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='45, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='6, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='9, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' We tune the confidence threshold based on the model’s perfor- mance on the validation set and set the confidence threshold θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='85 for the early exit policy at the inference phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' To show that our scheme can be widely applied to most existing frameworks, we conduct experiments on four base- lines selected from state-of-art approaches for fairness- aware learning: LNL (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2019), HSIC (Quadrianto, Sharmanska, and Thomas 2019), MFD (Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2021), and FSCL+ (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' All the baselines are repro- duced by following the recommended hyperparameter set- tings of the original papers or resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Multi-Exit (ME) Implementation We apply our ME framework on four state-of-art methods as below: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' ME-CNN : The ME-CNN is our baseline multi-exits framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' We apply cross-entropy loss to each IC with- out any fairness constraint, that is loss = α1 ·lIC1 t +α2 · lIC2 t + α3 · lIC3 t + αf · lCLS t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' ME-LNL : For the adversarial debias method, we replace g ◦ f in the original paper with ME-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The loss func- tion becomes loss = α1 · lIC1 t + α2 · lIC2 t + α3 · lIC3 t + αf · lCLS t + λ · Adv loss, and the model is trained with an adversarial strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Adv loss is the adversarial term for bias mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' ME-HSIC : We use ME-CNN as the backbone and apply the proposed loss term in HSIC to each IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' ME-MFD : We follow the teacher-student training frame- work in the original paper and use ME-CNN as the back- bone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' We first train a ME-CNN as our teacher model and apply the proposed loss term in MFD to each IC in the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' ME-FSCL+ : We first pretrain the ICs in a ME-CNN backbone with the fair supervised contrastive loss in FSCL+ together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Then, we fix the pretrained ME-CNN backbone and train the linear classifiers of each IC to- gether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Table 2: Classification results of the fairness (EO) and ac- curacy (Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=') evaluation on the test set of the UTK Face dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' * denotes our own implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' T and S stand for target and sensitive attributes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' S=Ethnicity EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' T=Age CNN* 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='8 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 ME-CNN 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 LNL* 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='9 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 ME-LNL 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='9 HSIC* 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 ME-HSIC 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='9 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 MFD* 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 ME-MFD 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 T=Gender FSCL+* 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 ME-FSCL+ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 6 Results 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 Comparison with State-of-the-art In this section, we show the feasibility of ME training frame- work on four state-of-the-art, LNL (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2019), HSIC (Quadrianto, Sharmanska, and Thomas 2019), FSCL+ (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2022), and MFD (Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2021), by comparing the result with and without the ME training framework on CelebA and UTK Face dataset in Table 1 and Table 2, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' For the CelebA dataset, in Table 1 we follow the sensi- tive and target groups setting in (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2022) and com- pare our results with their reproduce results accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The ME-CNN is the baseline ME framework training without any fairness constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' That is, the loss function in each internal classifier is lossIC = lIC t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The ME-CNN frame- work improves the EO in 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3% while losing only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='13% of accuracy in average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In the comparison with different state- of-the-art, our results achieve a 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5% EO improvement in Figure 3: Visualization of the features of the CelebA dataset (T=a / S=m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' (a) demonstrates the feature of the proposed multi-exit training framework on MFD, while (b) are the features without using the multi-exit training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' ICx represents the xth internal classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' CLSf stands for the classifier following behind the last layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Since the MFD did not contain internal classifier, we extract the feature from the same layers of ME-MFD, which are Conv2 x, Conv3 x, Conv4 x, and Conv5 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' average while keeping the competitive accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' It is note- worthy that in ME-MFD and ME-LNL, our framework also improves the accuracy by an average of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='8% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3% , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' For UTK Face dataset in Table 2, we follow (Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2021) to define Ethnicity as the sensitive attribute and Age as the target groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Compared with the CNN baseline, the EO decreased from 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='8 to 16, which is a 10% improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' As for the comparison with different state-of-the-art, our results achieve a 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3% EO improvement in average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In addition, in ME-LNL and ME-MFD, the accuracy also shows a 3% im- provement in average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Since the FSCL+ (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2022) selected the Gender as the target attribute, we follow their data imbalance setting to product the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The bias level hyperparameter N is set as 5, which means male data is five times as much as female data, and the other sensi- tive group has the opposite gender ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Performance com- parison between FSCL+ and ME-FSCL+ shows that ME- FSCL+ achieves a 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5% improvement at EO than FSCL+, where the accuracy is almost the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Comparisons in Ta- ble 2 successfully demonstrate that our framework outper- forms all the baseline on the fairness score with a competi- tive accuracy in UTK Face dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 Ablation Study of Multi-Exit Training Framework In this section, we establish the ablation study of apply- ing a multi-exit training framework to the MFD (Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 3(a), we visualized the representation of the test instance from each internal classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Since the multi- exit training framework optimized each internal classifier (ICx and CLSf) for high accuracy and fairness, we can ob- serve a clear decision boundary between the attractive and non-attractive target class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' However, the features of the sen- sitive attribute (gender) are mixed and uniformly distributed, which indicates that the model remains fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In addition, we also visualize the representation of the test instance from the same layer of MFD, which is the output of each residual module of the network in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Obvi- ously,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' since the intermediate feature does not pass through any direct target optimization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' both the target and the sen- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='sitive attributes could not be easily recognized in shallow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='Non-Attractive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='Non-Attractive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='Non-Attractive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='Non-Attractive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='Attractive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='Attractive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='Attractive ' metadata={'source': 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='Attractive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='Attractive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='Attractive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='Male ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='Male ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='Male ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='Male ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='Female ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='Female ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='Female ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='FemaleTable 3: Classification results of the proposed ME-MFD and MFD selected from each exit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' ICx represents the xth internal classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' CLSf stands for the classifier following behind the last layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Since the MFD did not contain internal classifier, we extract the feature from the same layers of ME-MFD, which are Conv2 x, Conv3 x, Conv4 x, and Conv5 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The fairness (EO) and accuracy (Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=') evaluation on the test set of the CelebA dataset are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Methods T=a / S=m T=a / S=y T=b / S=m T=b / S=y T=e / S=m T=e / S=y EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' ME-MFD IC1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='9 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='8 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 IC2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 IC3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 CLSf 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='9 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='6 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 MFD Conv2 x 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='9 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='9 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 Conv3 x 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 19.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 Conv4 x 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='9 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='8 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='8 Conv5 x 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='6 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 Table 4: Classification results of the proposed ME-MFD selected from each exit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' ICx represents the xth internal classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' CLSf stands for the classifier following behind the last layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' ME-MFD (EE) denotes the result of using the confidence-based early exit algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The fairness (EO) and accuracy (Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=') evaluation on the test set of the CelebA dataset are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Methods T=a / S=m T=a / S=y T=b / S=m T=b / S=y T=e / S=m T=e / S=y EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' EO Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' IC1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='9 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='8 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 IC2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 IC3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 CLSf 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='9 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='0 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='6 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 ME-MFD (EE) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='8 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='2 layers (Conv2 x and Conv3 x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' At the deeper layer, the loss function maximizes the accuracy and minimizes the bias of the sensitive group, the feature of the target group can be clearly separated, whereas the sensitive attribute remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' We also report the quantitative results of the experi- ment mentioned above to show the feasibility of the multi- exit training framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In Table 3, we compare the classification results of ME- MFD and MFD selected from each exit branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' We re- port the features of MFD at different residual modules (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=', Conv2 x, Conv3 x, Conv4 x, and Conv5 x) and each IC’s features of ME-MFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The multi-exit framework improves the EO by an average of 27% and the accuracy by 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Be- sides, the EO of MFD increases from Conv2 x to Conv 4 but drops at the last layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' This phenomenon is due to the Max- imum Mean Discrepancy (MMD) constraint being imposed at the last layer while there is no fairness constraint on the features of shallow layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' ME-MFD regularizes the fairness condition of each IC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' as a result, the trend of EO score still holds our observation in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Experiments demonstrate the significance of using the multi-exit training framework, which allows us to select pre- diction at a shallow layer to achieve a fair and high accuracy result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='3 Ablation Study of Early Exit Policy In this section, we study the impact of using the proposed early exit policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Table 4 shows the fairness and accuracy of the proposed ME-MFD selected from a different exit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' From the table, we observe that the IC1 obtains the smallest EO and the accuracy, while the CLSf achieves the largest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The increasing trend in both metrics indicates that there is a trade-off between fairness and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Thus, the pro- posed early exit policy independently selects the exit of each test instance to preserve a large fairness improvement with only a slight drop in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' In the comparison without using the proposed early exit (CLSf), our method achieves 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='5% lower EO, but the accuracy drop is less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content='7% on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The significant improvement demonstrates the im- portance of deciding the exit for each instance, which also shows that the proposed early exit algorithm is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 7 Conclusion In this paper, we first explored the problem that the fair- ness condition deteriorates as we classify the features in deeper layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' Then, we introduced the multi-exit training framework, with high extensibility that could be applied to many bias mitigation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' With the confidence-based exit strategy, we select the optimal exit for each test instance to achieve both high accuracy and fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' The extensive re- sults and the ablation studies have shown that our framework can achieve the best trade-off of accuracy and fairness con- ditions compared to the state-of-the-art on two well-known facial datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfMgMl/content/2301.02989v1.pdf'} +page_content=' 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b/kdE0T4oBgHgl3EQfYgDF/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b900d5fdbe429eb48769a4e2b83b34d53aba36bdebfd7ebc49e0af4760dbc97b +size 109084 diff --git a/ldA0T4oBgHgl3EQfJP9z/content/tmp_files/load_file.txt b/ldA0T4oBgHgl3EQfJP9z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..50b75102e27336a775531114494bcae371dbc837 --- /dev/null +++ b/ldA0T4oBgHgl3EQfJP9z/content/tmp_files/load_file.txt @@ -0,0 +1,1233 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf,len=1232 +page_content='A MUSCL-LIKE FINITE VOLUME APPROXIMATION OF THE MOMENTUM CONVECTION OPERATOR FOR LOW-ORDER NONCONFORMING FACE-CENTRED DISCRETIZATIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Brunel1, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Herbin2 and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Latché3 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' We propose in this paper a discretization of the momentum convection operator for fluid flow simulations on quadrangular or hexahedral meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The space discretization is performed by the low- order nonconforming Rannacher-Turek finite element: the scalar unknowns are associated to the cells of the mesh, while the velocities unknowns are associated to the edges or faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The momentum convection operator is of finite volume type, and its almost second order expression is derived by a MUSCL-like technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The latter is of algebraic type, in the sense that the limitation procedure does not invoke any slope reconstruction, and is independent from the geometry of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The derived discrete convection operator applies both to constant or variable density flows, and may thus be implemented in a scheme for incompressible or compressible flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' To achieve this goal, we derive a discrete analogue of the computation ui (∂t(ρui)+div(ρuiu) = 1 2∂t(ρu2 i )+ 1 2div(ρu2 i u) (with u the velocity, ui one of its component, ρ the density, and assuming that the mass balance holds) and discuss two applications of this result: firstly, we obtain stability results for a semi-implicit in time scheme for incompressible and barotropic compressible flows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' secondly, we build a consistent, semi-implicit in time scheme that is based on the discretization of the internal energy balance rather than the total energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The performance of the proposed discrete convection operator is assessed by numerical tests on the incompressible Navier-Stokes equations, the barotropic and the full compressible Navier-Stokes and the compressible Euler equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' 2020 AMS Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' 65M08 and 65M12 and 76M12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Introduction When designing numerical schemes for fluid flow simulations, combining a finite element approximation of dif- fusion terms with a finite volume discretization of the convection operator is an appealing solution, sometimes found in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Indeed, the diffusion term may be easily discretized using the finite element method with minimal mesh restrictions while preserving the stability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' the control of a (possibly discrete) H1-norm, but the discretization of the convection term is less straightforward, since standard finite element methods may yield numerical instabilities, especially in the convection dominated case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Tackling this problem amounts to introduce some upwinding in the scheme, and, to this purpose, many solutions have been explored in the context of the finite volume;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' finite-volume convection operators respecting both some monotonicity and L2-stability properties (including, for the latter item, a local discrete entropy or, in the world of fluid flow, a kinetic energy balance) have been obtained in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Several authors have thus proposed discretizations combining finite elements and finite Keywords and phrases: Fluid flows, convection operator, staggered meshes, MUSCL, kinetic energy balance, stability, incompressible flows, compressible flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' 1 Aix-Marseille Université, Centre de Mathématiques et Informatique, 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France (aubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='brunel@univ-amu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='fr) 2 Aix-Marseille Université, Centre de Mathématiques et Informatique, 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France (raphaele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='herbin@univ-amu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='fr) 3 IRSN, BP 13115, St-Paul-lez-Durance Cedex, France (jean-claude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='latche@irsn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='fr) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='02087v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='NA] 5 Jan 2023 2 volumes, to take benefit of the best of both worlds, see for instance [1, 9, 15–17, 30] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' These works may address convection-diffusion or Navier-Stokes equations, using preferably finite elements approximations of accuracy compatible with finite volumes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' low-order elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' For the incompressible Navier-Stokes equations or for low-Mach compressible flows, associating this property with the inf-sup stability requirement suggests turning to low-order nonconforming elements, namely the low-order Crouzeix-Raviart element for simplicial meshes [14] or the Rannacher-Turek element for quadrangles and hexahedra [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' An application of this strategy for the discretiza- tion of the stationary incompressible Navier-Stokes equations by Crouzeix-Raviart finite elements may be found in [33];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' extension to quasi-incompressible unsteady flows, both with the Crouzeix-Raviart and Rannacher-Turek finite elements, is performed in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' In most of the above cited papers, only a first-order upwinding technique is considered, leading to diffusive approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Increasing the order of the scheme and its precision while preserving its stability can be tricky, since naive higher-order methods might lead to spurious oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' As already mentioned, successful methods exist to achieve this goal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' such a now well-known method is Van Leer’s so-called MUSCL scheme [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' This technique was firstly used for hyperbolic conservation laws in one space dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' extending it to multi-dimensional problems on general meshes is a challenging task, due to the so-called slope construction involved in the limitation step, see for instance [7, 8, 13, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' A numerical scheme circumventing this problem for the transport operator is proposed in [31];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' it relies on the observation that the requirements for the scheme to satisfy the maximum principle may be substituted to the usual limitation technique, yielding a limitation step of purely algebraic type, and so free of any geometric consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The continuous momentum convection operator that we consider here takes the following generic form: C(ρ, ui) = ∂t(ρui) + div(ρuiu) (1) where ρ is the density of the fluid and u its velocity (so, for 1 ≤ i ≤ d, ui stands for the i-th component of the velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' It may be recast under the form of a transport operator provided that a mass balance equation holds, that is ∂tρ + div(ρu) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (2) Indeed, we have: ∂t(ρui) + div(ρuiu) = ui � ∂tρ + div(ρu) � � �� � =0 +ρ � ∂tui + u · ∇ui � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (3) This formulation shows that the operator C satisfies a discrete maximum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' In addition, a standard manip- ulation of partial derivatives yields: ui C(ρ, ui) = 1 2 ρ � ∂tu2 i + u · ∇u2 i � = ∂t(ρu2 i 2 ) + div(ρu2 i 2 u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (4) A finite volume discretization of the operator C based on the previously cited algebraic MUSCL method [31] was recently derived, first for simplicial or quadrangular (or hexahedral) meshes [19], and then on more general possibly hybrid meshes [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Here we recall this construction for a space discretization using the unknowns of the Rannacher- Turek finite element (Section 3) and derive a discrete analogue of Equation (4) satisfied by this discrete convection operator (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The form of C is quite general, and the operator built here may be applied as well to incompressible as to compressible flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Two results support this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' First, for an advection diffusion with an implicit-in-time discretization of the diffusion term (while the MUSCL approximation of the convection term is explicit), integrating the discrete counterpart of (4) in space yields a stability estimate, valid for time steps lower than a limit depending on the diffusion coefficient and the mesh regularity, but independent of the space step 3 (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' this estimate is the essential argument that is required to control the kinetic energy for incompressible flows or the total energy for barotropic flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Second, we show how to build, once again from the discrete version of (4), a consistent scheme for the Euler equations based on the solution of the internal energy balance to preserve the positivity of the latter variable (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' To this aim, having at hand a local (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' written on each cell and not integrated over the space domain) kinetic energy balance is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Finally, numerical experiments are performed (Section 5) to assess the stability, consistency, and accuracy of the proposed scheme for the incompressible and compressible Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Space and time discretizations We first define a primal mesh M by splitting Ω into a finite family of disjoint quadrangles (if d = 2) or hexahedra (if d = 3) denoted by K and called control volumes or cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' We then denote by E the set of faces of the mesh M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' for K ∈ M, E(K) stands for the set of faces of K and we thus have ∂K = ∪σ∈E(K)σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Any face σ ∈ E is either a part of the boundary of Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' σ ⊂ ∂Ω, in which case σ is said to be an external face, or there exists (K, L) ∈ M2 with K ̸= L such that K ∩ L = σ: we denote in this case σ = K|L and σ is said to be an internal face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' We denote by Eext and Eint the set of external and internal faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' For K ∈ M and σ ∈ E, we denote by |K| the measure of K and by |σ| the (d − 1)-measure of the face σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The discretization is staggered in the sense that the scalar and vector unknowns are not colocated: the unknowns associated to the density, and to any other scalar variable involved in the problem, as for instance the pressure, are associated with the cells of the primal mesh M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' limiting the list of set of scalar fields to the density, the pressure p and the internal energy e (which will be sufficient for the numerical applications presented in Section 5), the corresponding unknowns are denoted by (ρK)K∈M, (pK)K∈M and (eK)K∈M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' the degrees of freedom for the velocity are defined on a dual mesh using the Rannacher-Turek non-conforming low-order finite element approximation [32] and are denoted (uσ)σ∈E with uσ = (uσ,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' , uσ,d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' they are identified with the mean value of the velocity component over the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The dual mesh is constructed as follows (see Figure 1): if K ∈ M is a rectangle or a rectangular cuboid, we denote by xK the mass center of K and we construct DK,σ as the cone with basis σ and with vertex xK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' this definition is extended to a general cell K, by supposing that K is split in the same number of sub-cells (the geometry of which does not need to be specified) and with the same connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' We now define Dσ, the dual cell associated to σ, as Dσ = DK,σ ∪ DL,σ if σ = K|L ∈ Eint and Dσ = DK,σ if σ ∈ E(K) ∩ Eext;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' its measure is denoted by |Dσ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' We then denote by ˜E(Dσ) the set of dual faces of Dσ, and by ϵ = Dσ|Dσ′ the face separating two dual cells Dσ and Dσ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Finally, for the sake of simplicity, a constant time step denoted by δt is used for the time discretization, with δt = T/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' We define tn = n δt, 1 ≤ n ≤ N, and the notations for the discrete unknowns at step n are obtained from the notations for space discretization introduced above by adding an index n, so, finally, the unknowns involved in the definition of the convection operator are (ρn K)K∈M, 0≤n≤N and (un σ)σ∈σ, 0≤n≤N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' A second order convection operator Let us first address the discretization of the mass balance equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Since, in the Rannacher-Turek element, the pressure is piecewise constant over the cells, the natural mass balance (or, at least, for incompressible flows, the natural divergence-free constraint) takes a finite volume like formulation, posed over the primal cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' With an 4 K L σ = K|L ϵ = σ|σ′ DL,σ DK,σ σ′ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Primal and dual meshes for the Rannacher-Turek elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' explicit-in-time discretization of the convection flux, this equation thus reads, for K ∈ M: |K| δt (ρn+1 K − ρn K) + |K| div(ρu)n K = 0, div(ρu)n K = 1 |K| � σ∈E(K) F n K,σ, where F n K,σ stands for the (primal) numerical mass flux across σ outward K and is defined by: ∀σ = K|L ∈ Eint, F n K,σ = |σ| ρn σun σ · nK,σ, with nK,σ the normal vector to the face σ outward K and ρn σ a discretization of the density at the face, which does not need to be specified in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' We suppose that the cell densities are positive at all time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' When the density is constant, we recover the usual divergence-free constraint for the Rannacher-Turek element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The dual mass fluxes and the face densities are constructed to ensure that a similar discrete mass balance holds over the dual cells, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' to obtain a relation of the form: ∀σ ∈ E, |Dσ| δt (ρn+1 Dσ − ρn Dσ) + � ϵ∈˜E(Dσ) F n σ,ϵ = 0, (5) where ρn Dσ is the density at the face σ and at time step tn, and F n σ,ϵ a mass flux through ϵ outward Dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' For the internal faces, the face densities ρDσ are defined as a weighted average of the density unknowns in the cells adjacent to σ: ∀σ ∈ Eint, σ = K|L, |Dσ| ρDσ = |DK,σ| ρK + |DL,σ| ρL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (6) For an external face σ of adjacent cell K, we just set ρDσ = ρK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' For ϵ included in the primal cell K and σ a face of K, the mass fluxes Fσ,ϵ are obtained by a linear combination of the mass fluxes through the primal faces of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' A 5 detailed explanation of the construction process is given in [2] and extended in [6] to more general, possibly hybrid 3D meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The mass balance (5) over the dual meshes is then used for the definition of the discrete momentum convection term C(ρ, u)n+1 σ,i , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' the discretization of the continuous term C(ρ, ui) = ∂t(ρui) + div(ρuiu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' For 1 ≤ i ≤ d and σ ∈ E, this discrete term takes the following form: C(ρ, u)n+1 σ,i = 1 δt(ρn+1 Dσ un+1 σ,i − ρn Dσun σ,i) + div(ρuiu)n σ, with div(ρuiu)n σ = 1 |Dσ| � ϵ∈˜E(Dσ) F n σ,ϵun ϵ,i, (7) where un ϵ,i is an approximation of ui over the face ϵ at the time tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' For a boundary face σ ∈ Eext, one of the dual faces of Dσ is the face σ itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' If this primal/dual face is included in a part of the boundary where the velocity is prescribed, no equation is written for un+1 σ (it is just set to the prescribed value) and no definition is needed for un ϵ,i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' in the other case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' for a Neumann boundary condition), we suppose that the flow leaves the computational domain, and we set un ϵ,i to the upwind value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' un ϵ,i = un σ,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' For an internal dual face, un ϵ,i is obtained by the algebraic MUSCL-like technique introduced in [31], which implements the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Let us recast the convection term C(ρ, u)n+1 σ,i as C(ρ, u)n+1 σ,i = 1 δtρn+1 Dσ � un+1 σ,i − ¯un+1 σ,i ), with ¯un+1 σ,i = 1 ρn+1 Dσ � ρn Dσun σ,i − δt div(ρuiu)n σ � = 1 ρn+1 Dσ � ρn Dσun σ,i − δt |Dσ| � ϵ∈˜E(Dσ) F n σ,ϵun ϵ,i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The discrete convection operator is said to be monotone if the term ¯un+1 σ,i can be written as a convex combination of degrees of freedom of un i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' for instance, such a property would ensure a discrete maximum principle for the transport equation, or a convection-diffusion equation with a suitable (only available on specific meshes) discretization of the diffusion term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Let us recast ¯un+1 σ,i as ¯un+1 σ,i = 1 ρn+1 Dσ �� ρn Dσ − δt |Dσ| � ϵ∈˜E(Dσ) F n σ,ϵ � un σ,i − δt |Dσ| � ϵ∈˜E(Dσ) F n σ,ϵ(un ϵ,i − un σ,i) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (8) The mass balance equation (5) yields 1 ρn+1 Dσ � ρn Dσ − δt |Dσ| � ϵ∈˜E(Dσ) F n σ,ϵ) = 1, and therefore the sum of the coefficients multiplying the velocities un σ,i and un ϵ,i at the right-hand side of Relation (8) is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The coefficient of un σ,i in (8) is non-negative under the CFL condition CFL = max σ∈E � δt ρn Dσ |Dσ| � ϵ∈˜E(Dσ) |F n σ,ϵ| � ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (9) 6 and we indeed obtain a convex combination at the right-hand side of Equation (8) if the following condition holds for each ϵ ∈ ˜Eint such as ϵ = Dσ|Dσ′: ∃ασ ϵ ∈ [0, 1], ∃˜σ ∈ E such that un ϵ,i − un σ,i = ����� ασ ϵ (un σ,i − un ˜σ,i) if Fσ,ϵ ≥ 0, ασ ϵ (un ˜σ,i − un σ,i) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (10) Of course, in this relation, both the coefficient ασ ϵ and the face ˜σ have to be determined at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' We now deduce from the relation (10) a constructive process to compute the quantities un ϵ,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Let ϵ be a given internal face, and let Dσ− (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Dσ+) denote the adjacent upwind (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' downwind) dual cell to the face ϵ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Fσ−,ϵ ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Let Nϵ(Dσ−) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Nϵ(Dσ+) be a set of neighbouring dual cells of Dσ− (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Dσ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The following assumptions are then a transcription of Condition (10): ∃ Dσ ∈ Nϵ(Dσ+) such that un ϵ,i ∈ I+ = � un σ,i, un σ,i + ξ+ 2 (un σ+,i − un σ,i) � , (11a) ∃ Dσ ∈ Nϵ(Dσ−) such that un ϵ,i ∈ I− = � un σ−,i, un σ−,i + ξ− 2 (un σ−,i − un σ,i) � , (11b) where ξ+ and ξ− are two numerical parameters lying in the interval [0, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' These parameters have to be chosen by the user, and are usually kept constant through the whole computation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' decreasing their value makes the algorithm limitation more restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The set Nϵ(Dσ+) is always required to contain Dσ−, with the following two consequences: first, the value un σ−,i always belongs to both intervals I+ and I−, so their intersection is not void and the scheme is always defined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' second, setting ξ+ = ξ− = 0 yields the usual upwind scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' To make the definition of the scheme complete, we now need to define the sets Nϵ(Dσ+) and Nϵ(Dσ−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Here we choose Nϵ(Dσ+) = {Dσ−}, so that the condition (11a) implies that un ϵ,i is a convex combination of un σ−,i and un σ+,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Furthermore, if ξ+ ≤ 1, the hypothesis (11a) yields uϵ,i,M ∈ [uϵ,i,U, uϵ,i,C] where uϵ,i,U, uϵ,i,M and uϵ,i,C are the values given by the upwind, MUSCL and centered discretization respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' the MUSCL discretization thus yields in this case a more diffusive scheme than the centered discretization and less diffusive than the upwind discretization, whatever the choice of ξ+ and ξ− in the [0, 2] interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Hence, in our numerical experiments, we choose to set ξ+ ≤ 1, for energetic stability reasons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' note also that the motivation for considering ξ+ > 1 is generally to allow a second order interpolation of the unknown at the face, which here does not make sense since the dual mesh cannot be built explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Concerning Nϵ(Dσ−), several choices are possible: a simple choice is to take the neighbouring cells of Dσ−: Nϵ(Dσ−) = {(Dσ)σ∈E such that Dσ shares a face ˜ϵ with Dσ−} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' the previous set can be restricted to the upstream neighbouring cells of Dσ−: Nϵ(Dσ−) = {(Dσ)σ∈E such that Dσ shares a face ˜ϵ with Dσ− and Fσ,˜ϵ ≥ 0} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' another possibility is to take the opposite cell to Dσ+ with respect to Dσ−, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Nϵ(Dσ−) = {(Dσ′)σ′∈E such that Dσ′ shares a face ϵ′ with Dσ− and ϵ ∩ ϵ′ = ∅} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The last choice was selected in our numerical experiments, in the interior of the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' For dual edges with one of the adjacent cells itself adjacent to the boundary, depending on the sign of the mass fluxes, this 7 σ σ′ σ′′ ϵ Fσ′,ϵ Dσ Dσ′ Dσ′′ Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Dual cells involved in the definition of the convection flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' choice may be impossible if the opposite cell does not exist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' for a smooth flow, in such a case, one may expect that the fluid is entering the domain through the opposite dual face (the face denoted by ϵ′ in the previous relation), and the value in the opposite cell may be replaced by the Dirichlet value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Otherwise, the choice for un σ−,i boils down to the upwind choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' We are now in a position to give the algorithm used to compute the quantities un ϵ,i: (i) Compute a tentative value un ϵ,i with a convex combination of the values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' the centered choice) in the surrounding faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (ii) The flux F n σ,ϵ being given, determine the upwind face Dσ− and the downwind face Dσ+, and choose accordingly the neighbouring sets Nϵ(Dσ−) and Nϵ(Dσ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (iii) Compute an admissible interval I+ ∩ I− for uϵ,i by (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (iv) Compute un ϵ,i by projecting the tentative value un ϵ,i into the interval obtained in the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='1 (Deriving an implicit MUSCL scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Since this procedure is not linear, we cannot expect to derive an explicit formula to compute the values of the coefficients aσ ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Their evaluation is, however, not necessary in order to define an explicit scheme: the presented algorithm univocally defines the value un ϵ,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' But, for this reason, we cannot easily define an implicit-in-time MUSCL scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' However, it is still possible, using one of the following techniques: 8 a first technique would consist in an iterative process at each time step: in an inner loop, advance the velocity by replacing in the momentum equation the MUSCL convection operator at inner step k, divM(ρuiu)k σ, by divU(ρuiu)k+1 σ − divU(ρuiu)k σ + divM(ρuiu)k σ, where the subscript U denote the standard upwind convection operator (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' faces values uϵ,i are obtained through an upwind method) and the superscript k + 1 indicate an implicit discretization, and then loop until acceptable convergence is reached;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' an other technique would be to first compute the value un ϵ,i, then use (10) (or rather (11)) to compute the coefficients aσ ϵ thanks to un ϵ,i and the (un σ)σ∈E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Then, express an implicit value at the interface un+1 ϵ,i as a linear combination of the (un+1 σ )σ∈E thanks to the aσ ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Note that both techniques are costlier from a computational point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' A discrete kinetic energy identity and some applications In this section, we first focus on the proposed higher-order finite volume convection operator and show that it satisfies an identity which may be seen as a building brick for the derivation of a kinetic energy balance (or, equivalently, an entropy identity for the entropy function η(ui) = 1 2u2 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' We then give two applications of this result: first, we establish a stability property for a convection-diffusion problem, with an implicit discretization of the diffusion term, which may readily be extended to obtain stability estimates for incompressible or barotropic flows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' second, we build a consistent scheme for the Euler equations based on a discrete solution of a (corrected) internal energy balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' A local identity for the discrete convection operator In the continuous setting, let us assume that the mass balance equation (2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Let 1 ≤ i ≤ d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' for sufficiently regular density and velocity functions, using twice the mass balance to switch from a convection to a transport operator for ui and then from a transport back to a convection operator for u2 i , leads to: ui � ∂t(ρui) + div(ρuiu) � = ρui � ∂tui + u · ∇ui � = 1 2ρ � ∂t(u2 i ) + u · ∇(u2 i ) � = ∂t(ρu2 i 2 ) + div(ρu2 i 2 u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (12) Our aim here is to derive a discrete analogue of this identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' For the sake of simplicity, we focus on the term uiC(ρ, u)n+1 σ,i for the internal faces σ ∈ Eint of the mesh, where C(ρ, u)n+1 σ,i is the discrete convection operator defined by (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' We mimick the derivation of the identity (12) and therefore recast the convection term as a transport one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' in order to do so, we again suppose that the dual mass fluxes and the face densities are constructed to ensure that a discrete mass balance of the form (5) holds over the dual cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' We are now in position to state a discrete analogue to Equation (12), which does not feature a null right-hand side but a rest term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' This result can be seen as a direct consequence of [27, Lemma A1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' for the sake of clarity, we reformulate it here in a way that is more convenient for the applications of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='1 (Approximate transport operator for the kinetic energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Assume that Equation (5) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Then, for 1 ≤ i ≤ d, σ ∈ E and 0 ≤ n ≤ N − 1: |Dσ|uiC(ρ, u)n+1 σ,i = |Dσ| 2 δt � ρn+1 Dσ (un+1 i,σ )2 − ρn Dσ (un i,σ)2� + 1 2 � ϵ∈˜E(Dσ) F n σ,ϵ (un i,ϵ)2 + � ϵ∈˜E(Dσ) T n+1 σ,ϵ,i + Rn+1 σ,i , 9 with T n+1 σ,ϵ,i = −1 2F n σ,ϵ (un i,ϵ − un i,σ)2, (13) Rn+1 σ,i = |Dσ| 2 δt ρn+1 Dσ � un+1 i,σ − un i,σ �2 + (un+1 i,σ − un i,σ) � ϵ∈˜E(Dσ) F n σ,ϵ (un i,ϵ − un i,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (14) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Let σ ∈ Eint and 0 ≤ n < N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Subtracting the dual mass balance equation (5) multiplied by un i,σ yields: 1 δt (ρn+1 Dσ un+1 i,σ − ρn Dσun i,σ) + 1 |Dσ| � ϵ∈˜E(Dσ) F n σ,ϵun i,ϵ = ρn+1 Dσ δt (un+1 i,σ − un i,σ) + 1 |Dσ| � ϵ∈˜E(Dσ) F n σ,ϵ(un i,ϵ − un i,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The left-hand side of this relation is a discretization of the conservative form of the convection operator ∂t(ρui) + div(ρuiu), while the right-hand side may be seen as a discretization of the non-conservative form ρ(∂tui + u · ∇ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' We now multiply the right-hand side of the previous equality (which is precisely C(ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' u)n+1 σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='i ) by |Dσ| un+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ and use (twice) the identity 2a(a − b) = a2 − b2 + (a − b)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' once for the time derivative term and once for the "velocity gradient term",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' to obtain: |Dσ|uiC(ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' u)n+1 σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='i =|Dσ| 2 δt ρn+1 Dσ � (un+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ )2 − (un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ)2� + |Dσ| 2 δt ρn+1 Dσ � un+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ − un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ �2 + un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ � ϵ∈˜E(Dσ) F n σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ (un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ − un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ) + (un+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ − un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ) � ϵ∈˜E(Dσ) F n σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ (un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ − un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ) =|Dσ| 2 δt ρn+1 Dσ � (un+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ )2 − (un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ)2� + |Dσ| 2 δt ρn+1 Dσ � un+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ − un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ �2 + 1 2 � ϵ∈˜E(Dσ) F n σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ � (un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ)2 − (un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ)2� − 1 2 � ϵ∈˜E(Dσ) F n σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ(un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ − un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ)2 + (un+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ − un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ) � ϵ∈˜E(Dσ) F n σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ(un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ − un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' We now reverse the trick used previously to switch from the non-conservative form of the convection operator (this time for 1 2u2 i ) to the conservative form (which amounts to add this time Equation (5) multiplied by 1 2|Dσ| (un i,σ)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' This changes the first term of the first and second lines of the right-hand side, and yields the desired identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' □ In the previous lemma, the expression of the approximation of ui at the dual faces is not specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Let us then discuss the properties of the remainder term T n+1 σ,ϵ,i defined by (13) for the specific choice of ui given by the MUSCL scheme introduced in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' For a dual face ϵ = Dσ|Dσ′ for σ, σ′ ∈ E, since the set Nϵ(Dσ+) of neighbours of the dual cell Dσ+ is chosen as {Dσ−}, the condition (11a) yields: un i,ϵ = (1 − ξn i,ϵ 2 ) un i,σ− + ξn i,ϵ 2 un i,σ+, with ξn i,ϵ ∈ [0, ξ+], so ξn i,ϵ ∈ [0, 1] if we choose ξ+ = 1, as in the numerical experiments of Section 5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' In this relation, we recall that Dσ− (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Dσ+) is the upwind (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' downwind) dual cell with respect to ϵ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' the dual 10 cell of {Dσ, Dσ′} such that F n σ−,ϵ ≥ 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' F n σ+,ϵ ≤ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Considering both possible signs of F n σ,ϵ, we obtain the following expression for un i,ϵ: un i,ϵ = un i,σ + un i,σ′ 2 + 1 2sgn(F n σ,ϵ) (1 − ξn i,ϵ) (un i,σ − un i,σ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' We recover a classical presentation of the convection scheme as a centered scheme possibly corrected by a diffusion term;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' indeed, ξn i,ϵ = 1 indeed corresponds to the centered scheme, while F n σ,ϵsgn(F n σ,ϵ) (1 − ξn i,ϵ) ≥ 0, so that the second term can be seen as a numerical diffusion term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' With this expression of un i,ϵ, the term T n+1 σ,ϵ,i reads: T n+1 σ,ϵ,i = −1 8 � 1 + (1 − ξn i,ϵ)2� F n σ,ϵ (un i,σ − un i,σ′)2 + 1 4 (1 − ξn i,ϵ) |F n σ,ϵ| (un i,σ − un i,σ′)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (15) Thanks to the conservativity of the dual mass fluxes, the first part of the right-hand side is also conservative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' the second part may be seen as a numerical dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' A stability result Suppose, for the sake of simplicity, that a convection-diffusion equation of the form: ∂t(ρui) + div(ρuiu) − µ∆ui = 0, (16) holds for the i-th component of the velocity, where µ is a positive parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' This equation can be seen as a momentum balance equation with no source term and without the pressure gradient term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The diffusion term may arise either from a physical fluid viscosity or from a numerical stabilisation term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Assuming that a mass balance equation holds, multiplying Equation (16) by ui yields, by the same computation for the convection term as in the previous section: 1 2 ∂t(ρu2 i ) + 1 2div(ρu2 i u) − µ div(ui∇ui) + µ ∥∇ui∥2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (17) Now suppose that the velocity is prescribed to zero on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Integrating the previous formula over the domain Ω, then using the divergence theorem for the convection term and Green’s identity for the diffusion term yields: 1 2 � Ω ∂t(ρu2 i ) dx + µ � Ω ∥∇ui∥2 dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (18) Integrating in time, this equality yields a control of ρ1/2ui in the L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' L2(Ω)) norm and of µ1/2ui in the L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' H1(Ω)) norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' In addition, we remark that, for ϕ ∈ C∞ c (Ω × [0, T)), � T 0 � Ω µ div(ui∇ui) ϕ dx dt = − � T 0 � Ω µ ui∇ui · ∇ϕ dx dt ≤ µ1/2 ∥ui∥L2(Ω×(0,T )) ∥µ1/2ui∥L2(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='H1(Ω)) ∥∇ϕ∥L∞(Ω×(0,T )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (19) If we consider a sequence of solutions to Equation (16) obtained with a sequence of vanishing viscosities, provided that ρ is bounded by below by a positive real number (so ui is controlled in L2), this integral thus tends to zero, and Equation (17) may be used to obtain an entropy inequality, that is 1 2 ∂t(ρu2 i ) + 1 2div(ρu2 i u) ≤ 0, 11 in the distributional sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Dealing with the real momentum balance equation requires coping with a pressure gradient, which is standard for incompressible and barotropic flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' In both cases, the estimate of ∇p·u is obtained thanks to the mass balance equation and the equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The simplest situation is the incompressible case, where: ∇p · u = div(p u) − p divu = div(p u), so this term yields an entropy flux, and its integral over the computational domain vanishes thanks to the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The quantity 1 2ρ|u|2 is now referred to as the kinetic energy balance and the analogues of Equations (17) and (18) as the local and global, respectively, kinetic energy balances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Our goal here is to demonstrate a similar result for higher-order finite volume convection operators, taking the form introduced in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' It is well-known that such an operator is not L2-stable (while the first-order upwind discretization is, under a CFL condition), but we show here that the L2-stability is recovered when a non- vanishing diffusion is added, for small enough time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' As in the continuous setting in the above introduction, we restrict ourselves to the discretization of the convection-diffusion problem for a component of the velocity, in such a way that the proposed analysis may be used as a building brick for the study of staggered schemes for both incompressible and compressible flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' We suppose homogeneous Dirichlet boundary conditions on the whole boundary, so the velocity is set to zero on external faces, and the scheme reads, for a given index i, 1 ≤ i ≤ d: 1 δt (ρn+1 Dσ un+1 i,σ − ρn Dσun i,σ) + div(ρuiu)n σ − (µ∆ui)n+1 σ = 0, ∀σ ∈ Eint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (20) The discrete mass balance equation (5) over the dual cells is supposed to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The discretization of the diffusion term is implicit in time and does not need to be precisely defined at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' We only need to suppose that the following inequality holds: − � σ∈Eint |Dσ| un+1 i,σ (µ∆un+1 i )σ ≥ � ϵ∈˜Eint, ϵ=Dσ|Dσ′ ϵ⊂K µn ϵ hd−2 K (un+1 i,σ − un+1 i,σ′ )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (21) This relation might be seen as a discrete analogue to the inequality − � Ω uiµ∆ui dx ≥ � Ω µ∇ui ·∇ui dx (recall that we have supposed homogeneous Dirichlet boundary conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The viscosity µn ϵ is supposed to be positive (and therefore, at least for a given discretization, bounded away from zero), and the right-hand side of Inequality (21) defines a discrete H1 semi-norm (precisely speaking, is equal to the square of a H1 semi-norm), which we denote |ui|E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' If the diffusion operator is given by the Rannacher-Turek finite element, this bound might be obtained thanks to the equivalence between the | · |E norm and the broken H1 semi-norm, which holds under regularity assumptions for the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The following result is a global (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' integrated over the computational domain) estimate, which may be seen as a discrete analogue of Equation (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='2 (Stability for a convection-diffusion equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Assume that Equation (5) holds, that ρn Dσ ≥ 0 for σ ∈ Eint and 0 ≤ n ≤ N − 1, and that the coercivity condition (21) for the diffusion term holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Suppose that the 12 time step satisfies the following set of inequalities: ηn = δt τ n ≤ 1 for 0 ≤ n ≤ N − 1, with τ n = min � 21−d hd−2 K µn ϵ (F n σ,ϵ)2 � 1 |Dσ| ρn+1 Dσ + 1 |Dσ′| ρn+1 Dσ′ �, ϵ ∈ ˜Eint, ϵ = Dσ|Dσ′, ϵ ⊂ K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (22) Then the scheme (20), using the proposed MUSCL scheme with ξ+ = 1, is stable in the L2-norm, in the sense that its solution satisfies the following inequality: 1 2 � σ∈Eint |Dσ| � ρn+1 Dσ (un+1 σ )2 − ρ0 Dσ(u0 σ)2� ≤ η0 δt |u0|2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (23) Note that the right-hand side depends only on the initial conditions for the velocity, for the density, and the density at the end of the first time step Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='3 (Evaluation of τ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The dual mass fluxes are obtained as a linear combination, with bounded coeffi- cients, of the primal mass fluxes, see [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' More specifically, for ϵ included in the cell K and σ a face of K, F n σ,ϵ = � σ′∈E(K) αϵ,σ′ K F n K,σ′, � σ′∈E(K) |αϵ,σ′ K | = α with α = 22−d, where F n K,σ′ = |σ′| ρn σ′ un σ′ · nK,σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' For the sake of simplicity, let us suppose that the density is equal to a con- stant value, which we denote by ρ, and that the velocity is bounded by a quantity umax, which yields |F n σ,ϵ| ≤ 22−d|σ| ρ umax, for ϵ included in the cell K and σ a face of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Using |Dσ| > |DK,σ| = |K|/(2d) and |σ| < hd−1 K , σ ∈ E(K), we get τ n ≥ 2d−5 d |K| hd K min � µn ϵ u2maxρ, ϵ ∈ ˜Eint � , which shows that τ n only depends on the viscosity, the density, the velocity and the regularity of the mesh but not on the space step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Let 0 ≤ n ≤ N − 1 and 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Summing the result of the previous lemma over σ ∈ Eint and using inequality (21) yields 1 2 δt � σ∈Eint |Dσ| � ρn+1 Dσ (un+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ )2 − ρn Dσ(un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ)2� ≤ −R1 − R2 − C1 − C2 − D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' 13 where the terms on the right-hand side are defined by R1 = 1 2 δt � σ∈Eint |Dσ| ρn+1 Dσ (un+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ − un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' R2 = � σ∈Eint (un+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ − un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ) � ϵ∈˜E(Dσ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' ϵ=Dσ|Dσ′ F n σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ(un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ − un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' C1 = 1 2 � σ∈Eint � ϵ∈˜E(Dσ) F n σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ (un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' C2 = −1 2 � σ∈Eint � ϵ∈˜E(Dσ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' ϵ=Dσ|Dσ′ T n+1 σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' D = � ϵ∈˜Eint,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' ϵ=Dσ|Dσ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' ϵ⊂K µn ϵ hd−2 K (un+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ − un+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ′ )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' By conservativity, the sums C1 vanishes and, using the expression (15) of T n+1 σ,ϵ,i , the sum C2 is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Let us now turn to the term R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' By assumption on the convection scheme, we have |un i,ϵ − un i,σ| ≤ |un i,σ′ − un i,σ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Using the inequality ab ≤ a2 2ε + εb2 2 for two real numbers a and b and ε > 0 yields, with ε = δt |Dσ| ρn+1 Dσ : ���(un+1 i,σ − un i,σ) � ϵ∈˜E(Dσ), ϵ=Dσ|Dσ′ F n σ,ϵ (un i,σ′ − un i,σ) ��� ≤ |Dσ| 2 δt ρn+1 Dσ (un+1 i,σ − un i,σ)2 + δt 2 |Dσ| ρn+1 Dσ � � ϵ∈˜E(Dσ), ϵ=Dσ|Dσ′ |F n σ,ϵ| |un i,σ′ − un i,σ| �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The sum of the first term over σ ∈ E is equal to R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' whereas using the Cauchy-Schwarz inequality (�n i=0 xi)2 ≤ n �n i=0(xi)2 in the second term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' with n the number of the faces of a dual cell which is equal to 4 if d = 2 and 8 if d = 3 and thus may be written 2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' yields for R2: − R2 ≤ R1 + � σ∈Eint 2d−1 δt |Dσ| ρn+1 Dσ � ϵ∈˜E(Dσ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' ϵ=Dσ|Dσ′ � F n σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ(un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ′ − un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ) �2 = R1 + � ϵ∈˜Eint,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' ϵ=Dσ|Dσ′ 2d−1 δt � 1 |Dσ| ρn+1 Dσ + 1 |Dσ′| ρn+1 Dσ′ � (F n σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ)2 (un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ′ − un i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' 14 Gathering all the previous information leads to: 1 2 δt � σ∈Eint |Dσ| � ρn+1 Dσ (un+1 i,σ )2 − ρn Dσ(un i,σ)2� ≤ � ϵ∈˜Eint, ϵ=Dσ|Dσ′ 2d−1 δt � 1 |Dσ| ρn+1 Dσ + 1 |Dσ′| ρn+1 Dσ′ � (F n σ,ϵ)2 (un i,σ′ − un i,σ)2 − � ϵ∈˜Eint, ϵ=Dσ|Dσ′, ϵ⊂K µn ϵ hd−2 K (un+1 i,σ − un+1 i,σ′ )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Summing this inequality over all time steps tk with 0 ≤ k ≤ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' we get: 1 2 δt � σ∈Eint |Dσ| � ρn+1 Dσ (un+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ )2 − ρ0 Dσ(u0 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ)2� ≤ −Tn+1 + Sn + T0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' with Tn+1 = � ϵ∈˜Eint,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' ϵ=Dσ|Dσ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' ϵ⊂K µn ϵ hd−2 K (un+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ − un+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ′ )2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Sn = n � k=1 � ϵ∈˜Eint,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' ϵ=Dσ|Dσ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' ϵ⊂K � 2d−1 δt � 1 |Dσ| ρk+1 Dσ + 1 |Dσ′| ρk+1 Dσ′ � (F k σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ)2 − µk ϵ hd−2 K � (uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ′ − uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' T0 = � ϵ∈˜Eint,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' ϵ=Dσ|Dσ′ 2d−1 δt � 1 |Dσ| ρ1 Dσ + 1 |Dσ′| ρ1 Dσ′ � (F 0 σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='ϵ)2 (u0 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ′ − u0 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='σ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The term Tn+1 is obviously positive, the sum Sn is negative thanks to the assumption on the time step, and T0 ≤ η0 |u0 i |2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='4 (Extension of this result to less-limited MUSCL schemes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' In the present case, we have seen that, since no geometrical interpolation for the velocity at the dual faces is possible, the choice ξ+ = 1 is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' However, a stability result may still be obtained if, for some reason only the condition ξ ≤ 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' ξ+ = 2) was imposed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' in this case, the term −C2 is no longer positive, but satisfies −C2 ≤ 1 2 � ϵ∈˜Eint, ϵ=Dσ|Dσ′ |F n σ,ϵ| (un i,σ′ − un i,σ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' To obtain a stability estimate, we need to absorb this term in D, to obtain (indexing now the terms C2 and D with respect to time) −Cn+1 2 − Dn ≤ − � ϵ∈˜Eint, ϵ=Dσ|Dσ′, ϵ⊂K (µ′)n ϵ hd−2 K (un+1 i,σ − un+1 i,σ′ )2, 15 with (µ′)n ϵ hd−2 K = µn ϵ hd−2 K − 1 2|F n+1 σ,ϵ |, and to suppose that (µ′)n ϵ is bounded by below away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Note that, since F n σ,ϵ is proportional to the measure of the faces, this assumption is satisfied when the space step is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The stability condition (22) is then rephrased, switching µn ϵ to (µ′)n ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' In addition, the quantity −C0 2 (which only depends on the initial condition) must now be added to the right-hand side of the stability inequality (23);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' this term may be recast as the H1 semi-norm |u0|2 E multiplied by a factor proportional to the space and time steps product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' A consistent "internal-energy-based" staggered scheme for the full Euler equations For shock solutions of the Euler equations, only the total energy equation makes sense, because of its conservative character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' This relation reads: ∂t(ρ E) + div(ρ E u) + div(p u) = 0, (24) where E = 1 2|u|2 + e, with e the internal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Formally, this equation may be seen as the sum of the kinetic balance: ∂t(ρEk) + div � ρ Ek u � + ∇p · u = 0, Ek = 1 2|u|2, and the internal energy balance: ∂t(ρe) + div(ρeu) + p divu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' (25) Solving this latter equation is appealing since a suitable discretization (both for the convection operator, with a maximum-principle-preserving approximation, and for the term p divu, to take benefit of the fact that p vanishes when e vanishes) leads to a conservation of the positivity of the internal energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' combining this approach with a discretization of the mass balance equation which preserves the positivity of the density, we thus would obtain a scheme which preserves the convex of admissible states (ρ ≥ 0, e ≥ 0 and, thanks to the equation of state, p ≥ 0), which is a non-trivial task (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' [10] and references herein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Note also that the total energy is a function of unknowns discretized on both the primal and the dual meshes, and discretizing only the internal energy balance allows circumventing the technical difficulty of building an approximation of such a "composite" unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' However, it may be anticipated (and is observed in practice) that a blunt discretization of Equation (25) would yield a non- consistent scheme, giving solutions that do not respect the Rankine-Hugoniot jump conditions at shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The problem stems from the fact that the discrete kinetic energy balance equation features remainder terms which may be seen as a dissipation associated with numerical diffusion, and which do not tend to zero when the time and space step tend to zero, but to measures borne by the shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' The technique initially proposed in [25] to solve this problem is to compensate these remainder terms in the internal energy balance, in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldA0T4oBgHgl3EQfJP9z/content/2301.02087v1.pdf'} +page_content=' Let us denote these terms by (Rn+1 σ )σ∈E, 0≤n 20 dB) while MAP meth- +ods do not reach a satisfactory performance. We hypothesize +that beam search tends to fall into sub-optimal solutions by +performing incorrect choices in early inference over sparse +images such as MNIST digits. Top-k sampling with k = 32 +performs best, so we choose it to perform the evaluation (a +qualitative comparison is shown in Figure 3). For each mix- +ture in the test set we sample a candidate batch of 512 sep- +arations, select the separation whose sum better matches the +mixture (w.r.t. the L2 distance), and finally perform the re- +finement procedure in Eqs. (12), (13) with T = 500 and +α = 0.1. Evaluation metrics on this experiment are shown +in Table 2, while inference time is reported in Table 4. Our +method achieves higher metrics than “NMF”, “S-D” and +“BASIS Glow” and is faster than “BASIS NCSN”, thanks +to the latent quantization. The higher PSNR achieved by the +later method can be attributed to the fact that, in their case, +the underlying generative models perform sampling directly +in the image domain; in our case, the VQ-VAE compression +can hinder the metrics. +We compare our method to “BASIS NCSN”, using +the pre-trained NCSN model (Song and Ermon 2019) on +CelebA. In this case, we evaluate against the FID metric +(Heusel et al. 2017) instead of PSNR, given that for datasets +that feature more variability than MNIST, source separa- +tion can be an underdetermined task (Jayaram and Thick- +stun 2020): semantically good separations can receive a low +PSNR score since the generative models may alter features +such as color and cues (an effect amplified by a VQ-GAN +decoder). The FID metric better quantifies if the separa- +tions belong to the distribution of the sources. We test on +10,000 mixtures computed from pair of images in the vali- +dation split using a top-k sampler with k = 32. We scale the +likelihood term by multiplying it by λ = 3. It is a known +fact in the literature that score-based models outperform au- +toregressive models on FID metrics (Dockhorn, Vahdat, and +Kreis 2021) on different datasets, yet our method paired with +an autoregressive model shows competitive results with re- +spect to the score-based “BASIS NCSN”. +Qualitative results +To demonstrate the flexibility of LASS +in using existing models without any modification, we lever- +age pre-trained checkpoints on CelebA-HQ and ImageNet. + +7 +子 +2MN +7MOG +2 +3 +27Method +Time +MNIST +LASS (Ours) +4.49 s ± 0.27 s +BASIS NCSN +53.34 s ± 0.51 s +SLAKH +LASS (Ours) +1.33 min ± 0.87 s +PnF +42.29 min ± 1.08 s +Table 4: Inference speed comparisons for computing one +separation. To estimate variance, we repeat inference 10 +times on MINST and 3 times on SLAKH. We consider 3- +second-long mixtures on SLAKH. +Separation Method +Avg +Drums +Bass +rPCA +0.82 +0.60 +1.05 +ICA +-1.26 +-0.99 +-1.53 +HPSS +-0.45 +-0.56 +-0.33 +REPET +1.04 +0.53 +1.54 +FT2D +0.95 +0.59 +1.31 +LASS (Ours) +4.86 +4.73 +4.98 +Demucs +5.39 +5.42 +5.36 +Conv-Tasnet +5.47 +5.51 +5.43 +Table 5: Comparison with other source separation methods +on SLAKH (“Drums” and “Bass” classes). Results are re- +ported in SDR (dB) (higher is better). Lower part of the ta- +ble shows supervised methods. With “Avg” we refer to the +mean between the results over the two classes. +In this case, only the likelihood tensor P is learned. We +showcase a curated results list in Figure 1 and a more ex- +tensive list on the companion website. To the best of our +knowledge, our method is the first to scale up to 256×256 +resolutions and can be used with more powerful latent au- +toregressive models without re-training (which is cumber- +some for very large models). As such, end-users can per- +form generative separation without having access to exten- +sive computational resources for training these large models. +Music source separation +We perform experiments on the SLAKH2100 dataset +(Manilow et al. 2019) for the music source separation task. +This dataset contains 2100 songs with separated sources be- +longing to 34 instrument categories, for a total of 145 hours +of mixtures. We focus on the “Drums” and “Bass” data +classes, with tracks sampled at 22kHz. We use the public +checkpoint of Dhariwal et al. (2020) for the VQ-VAE model, +taking advantage of its expressivity in modeling audio data +over a quantized domain. Given that such a model is trained +at 44kHz, we upsample input data linearly, then downsample +the output back at 22kHz. For the two autoregressive priors, +we train two Transformer models, one for “Drums” and an- +other for “Bass” and learn the likelihood function over the +VQ-VAE (statistics are reported in Table 1). We compare +LASS to a set of unsupervised blind source separation meth- +ods -“rPCA” (Huang et al. 2012), “ICA” (Hyv¨arinen and Oja +2000), “HPSS” (Rafii and Pardo 2012), “FT2D” (Seethara- +man, Pishdadian, and Pardo 2017) - and to two supervised +baselines Demucs (D´efossez et al. 2019) and Conv-Tasnet +(Luo and Mesgarani 2019) using the SDR (dB) evaluation +metric computed with the museval library (St¨oter, Liutkus, +and Ito 2018). To evaluate the methods, we select 900 mu- +sic chunks of 3 seconds from the test splits of the “Drums” +and “Bass” classes, combining them to form 450 mixtures. +The validation dataset is constructed similarly (with differ- +ent music chunks). As a sampling strategy, we use beam +search since it shows the best results on a validation of 50 +mixtures (Table 3), using B = 100 beams. Evaluation re- +sults are reported in Table 5: LASS clearly performs better +than all the blind unsupervised baselines and is comparable +with the results obtained by methods that use supervision. +Furthermore, we compare the time performance of LASS +against the generative source separation method “PnF” (Ja- +yaram and Thickstun 2021) by evaluating the time required +to separate a mixture of 3 seconds sampled at 22 kHz (piano +vs. voice on “PnF”). Results in Table 4 show that LASS is +significantly faster, and as such, it can be adopted in more +realistic inference scenarios. +Limitations +In this paper we limit our analysis to the separation of two +sources. Even if this is a common setup especially in image +separation (Jayaram and Thickstun 2021; Halperin, Ephrat, +and Hoshen 2019), dealing with multiple sources is a pos- +sible line of future work. Under our framework, this would +require to increase the dimensions of the discrete distribu- +tions (both the priors and the likelihood function). To alle- +viate this problem, techniques such as recursive separation +may be employed (Takahashi et al. 2019). +Another limitation of the proposed method is the locality +assumption taken in Eq. (4). Different tasks such as coloriza- +tion and super-resolution would require a larger condition- +ing context, and newer quantization schemes to aggregate +latent codes on global contexts (using self-attention in the +encoder and the decoder of the VQ-VAE) (Yu et al. 2021). +Adopting a VQ-VAE quantized with respect to the latent +channels (Xu et al. 2021) combined with a parametric likeli- +hood function could be a way to solve this limitation, while +still maintaining the flexible separation between VQ-VAE, +priors, and likelihoods presented in the paper. +Conclusion +In this paper, we proposed LASS as a source separation +method for latent autoregressive models that does not mod- +ify the structure of the priors. We have tested our method +on different datasets and have shown results comparable +to state-of-the-art methods while being more scalable and +faster at inference time. Additionally, we have shown qual- +itative results at a higher resolution than those proposed by +the competitors. We believe our method will benefit from the +improved quality of newer autoregressive models, improv- +ing both the quantitative metrics and the perceptive results. + +Acknowledgments +We thank Marco Fumero for helping to compute the blind +unsupervised baseline metrics in the audio setting. This +work is supported by the ERC Grant no. 802554 (SPEC- +GEO) and the IRIDE grant from DAIS, Ca’ Foscari univer- +sity of Venice, Italy. +References +Brown, T.; Mann, B.; Ryder, N.; Subbiah, M.; Kaplan, J. D.; +Dhariwal, P.; Neelakantan, A.; Shyam, P.; Sastry, G.; Askell, +A.; et al. 2020. +Language models are few-shot learners. +Proc. NeurIPS, 33: 1877–1901. +Castellon, R.; Donahue, C.; and Liang, P. 2021. Codified au- +dio language modeling learns useful representations for mu- +sic information retrieval. arXiv preprint arXiv:2107.05677. +Comon, P. 1994. 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K.; et al. 2022. +Scaling autoregressive models for content-rich text-to-image +generation. arXiv preprint arXiv:2206.10789. + diff --git a/mNFAT4oBgHgl3EQfcB3t/content/tmp_files/load_file.txt b/mNFAT4oBgHgl3EQfcB3t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b3061e576ec0c13152a9841a4aa6338ced603f87 --- /dev/null +++ b/mNFAT4oBgHgl3EQfcB3t/content/tmp_files/load_file.txt @@ -0,0 +1,1164 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf,len=1163 +page_content='Latent Autoregressive Source Separation Emilian Postolache*1, Giorgio Mariani∗1, Michele Mancusi∗1, Andrea Santilli1, Luca Cosmo†2, Emanuele Rodol`a†1 1 Sapienza University of Rome, Italy 2 Ca’ Foscari University of Venice, Italy postolache@di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content='uniroma1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content='it, mariani@di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content='uniroma1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content='it, mancusi@di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content='uniroma1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content='it Abstract Autoregressive models have achieved impressive results over a wide range of domains in terms of generation quality and downstream task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' In the continuous domain, a key factor behind this success is the usage of quantized latent spaces (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=', obtained via VQ-VAE autoencoders), which al- low for dimensionality reduction and faster inference times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' However, using existing pre-trained models to perform new non-trivial tasks is difficult since it requires additional fine- tuning or extensive training to elicit prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' This paper introduces LASS as a way to perform vector-quantized La- tent Autoregressive Source Separation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=', de-mixing an in- put signal into its constituent sources) without requiring ad- ditional gradient-based optimization or modifications of ex- isting models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Our separation method relies on the Bayesian formulation in which the autoregressive models are the pri- ors, and a discrete (non-parametric) likelihood function is constructed by performing frequency counts over latent sums of addend tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' We test our method on images and au- dio with several sampling strategies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=', ancestral, beam search) showing competitive results with existing approaches in terms of separation quality while offering at the same time significant speedups in terms of inference time and scalability to higher dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Introduction Autoregressive models have achieved impressive results in a plethora of domains ranging from natural language (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2020) to densely-valued domains such as audio (Dhari- wal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2020) and vision (Razavi, van den Oord, and Vinyals 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Esser, Rombach, and Ommer 2021), includ- ing multimodal joint spaces (Ramesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' In the dense setting, it is typical to train autore- gressive models over discrete latent representations obtained through the quantization of continuous data, possibly us- ing VQ-VAE autoencoders (van den Oord, Vinyals, and Kavukcuoglu 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' This way, generating higher resolution samples while simultaneously reducing inference time is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Additionally, the learned latent representations are useful for downstream tasks (Castellon, Donahue, and Liang 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' However, in order to perform new non-trivial tasks, the standard practice is to fine-tune the model or, in alter- native, elicit prompting by scaling training (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Listing order is random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' †Shared last authorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Sanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' The former is usually the default option, but it requires additional optimization steps or modifications to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' The latter is challenging on non-trivial tasks, especially in domains different from natural language (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Hertz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' This paper aims to tackle one of such tasks, namely source separation, leveraging existing vector-quantized au- toregressive models without requiring any gradient-based optimization or architectural modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' The task of sep- arating two or more sources from a mixture signal has re- cently received much attention following the success of deep learning, especially in the audio domain, ranging from speech (Dovrat, Nachmani, and Wolf 2021), music (D´efossez 2021), and universal source separation (Wisdom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Postolache et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Although not as promi- nent as its audio counterpart, image source separation has been addressed in literature (Halperin, Ephrat, and Hoshen 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Most successful approaches use explicit supervi- sion to achieve notable results (Luo and Mesgarani 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' D´efossez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2019), or leverage large-scale unsupervised regression (Wisdom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' We propose a generative approach to perform source sep- aration via autoregressive prior distributions trained on a latent VQ-VAE domain (when class information is used, the approach is weakly supervised;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' otherwise, it is unsu- pervised).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' A non-parametric sparse likelihood function is learned by counting the occurrences of latent mixed tokens with respect to the sources’ tokens, obtained by mapping the data-domain sum signals and the relative addends via the VQ-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' This module is not invasive, neither for the VQ- VAE nor for the autoregressive priors, given that the repre- sentation space of the VQ-VAE does not change while learn- ing the likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Finally, the likelihood function is combined with the estimations of the autoregressive pri- ors at inference time via the Bayes formula, resulting in a posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' The separations are obtained from the posterior distributions via standard discrete samplers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=', ancestral, beam search).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' We call our method LASS (Latent Autoregressive Source Separation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Our contributions are summarized as follows: We introduce LASS as a Bayesian inference method for source separation that can leverage existing pre-trained autoregressive models in quantized latent domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' We experiment with LASS in the image domain and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content='08562v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content='LG] 9 Jan 2023 ImageNet CelebA-HQ Original images Mixtures Separated Images Figure 1: 256x256 separations obtained with LASS using pre-trained autoregressive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Left: class-conditional ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Right: unconditional CelebA-HQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' showcase competitive results at a significantly smaller cost in inference time with respect to competitors on MNIST and CelebA (32×32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' We also showcase quali- tative results on ImageNet (256×256) and CelebA-HQ (256×256), thanks to the scalability of LASS to pre- trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' To the best of our knowledge, this is the first method to scale generative source separation to higher resolution images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' We experiment with LASS in the music source separation task on the Slakh2100 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' LASS obtains performance comparable to state-of-the-art supervised models, with a significantly smaller cost in inference and training time with respect to generative competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Related Work The problem of source separation has been classically tack- led in an unsupervised fashion under the umbrella term of blind source separation (Comon 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Hyv¨arinen and Oja 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Smaragdis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' In this set- ting, there is no information regarding the sources to be sep- arated from a mixture signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' As such, these methods rely on broad mathematical priors such as source independence (Hyv¨arinen and Oja 2000) or repetition (Rafii and Pardo 2012) to perform separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' With the advent of deep learn- ing, most prominent methods for source separation can be classified as regression-based or generative-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Regression-based source separation In this setting, a mixture is fed to a parametric model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=', a neural network) that outputs the separated sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Training is typically performed in a supervised manner by matching the estimated separations with the ground truth sources with a regression loss (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=', L1 or L2) (Gus´o et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Super- vised regression has been applied to image source separation (Halperin, Ephrat, and Hoshen 2019), but it has been mainly investigated in the audio domain, where two approaches are prevalent: the mask-based approach and the waveform ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' In the mask-based approach, the model performs separation by applying estimated masks on mixtures, typ- ically in the STFT domain (Roweis 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Uhlich, Giron, and Mitsufuji 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Nugraha, Liutkus, and Vincent 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Liu and Yang 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Takahashi, Goswami, and Mitsufuji 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' In the waveform approach, the model outputs the estimated sources directly in the time domain to overcome phase estimation, which is required when trans- forming the signal from the STFT domain to the waveform domain (Llu´ıs, Pons, and Serra 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Luo and Mesgarani 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' D´efossez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Generative source separation Following the success of deep generative models (Good- fellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Kingma and Welling 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Ho, Jain, and Abbeel 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2021), a new class of gen- erative source separation methods is gaining prominence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' This setting emphasizes the exploitation of broad genera- Mustive models (especially pre-trained ones) to solve the sep- aration task without needing a specialized architecture (as with regression-based models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Following early work on deep generative separation based on GANs (Subakan and Smaragdis 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Narayanaswamy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' 2020), Jayaram and Thickstun (2020) propose the generative separation method BASIS in the image setting using score-based models (Song and Ermon 2019) (BASIS-NCSN) and a noise-annealed version of flow- based models (BASIS-Glow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' The inference procedure is performed in the image domain through Langevin dynam- ics (Parisi 1981), obtaining good quantitative and qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' The authors extend the Langevin dynamics infer- ence procedure to autoregressive models by re-training them with a noise schedule, introducing the Parallel and Flexi- ble (PnF) method (Jayaram and Thickstun 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Although innovative, mainly when used for tasks such as inpainting, this method cannot use pre-trained autoregressive models directly, requiring fine-tuning under different noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Further, working directly on the data domain, it exhibits a high inference time and scales with difficulty to higher res- olutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' In this paper, we extend this line of research by proposing a separation procedure for latent autoregressive models that does not involve re-training, is scalable to ar- bitrary pre-trained checkpoints and is compatible with stan- dard discrete samplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Background This section briefly introduces vector-quantized autoen- coders (VQ-VAE) and autoregressive models, since they are core components of the separation procedure used in LASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' VQ-VAE A data point x ∈ RN (N is the total length of the data point, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=', the length of the audio sequence or the number of pixel channels in an image) can be mapped to a discrete latent domain via a VQ-VAE (van den Oord, Vinyals, and Kavukcuoglu 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' First an encoder Eθ : RN → RS×C maps x to Eθ(x) = (h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' , hS), where C denotes the number of latent channels and S the length of the latent se- quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' A bottleneck block B : RS×C → [K]S casts the encoding into a discrete sequence z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' , zS) by map- ping each hs into the index (also called token) zs = B(hs) of the nearest neighbor ezs contained in an (ordered) set C = {ek}K k=1 of learned vectors in RC (called codes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' A decoder Dψ : [K]S → RN maps the latent sequence back into the data domain, obtaining a reconstruction ˆx = Dψ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' VQ-GAN (Esser, Rombach, and Ommer 2021) is an en- hanced version of the VQ-VAE, where the training loss is augmented with a discriminator and a perceptual loss, that improve reconstruction quality while increasing the com- pression rate of the autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' We refer the reader to (van den Oord, Vinyals, and Kavukcuoglu 2017) and (Esser, Rombach, and Ommer 2021) for more details on VQ-VAE and VQ-GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' In the remainder of the article, we will re- fer to both models as VQ-VAE and make distinctions when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' Autoregressive models An autoregressive model defines a probability distribution over a discrete domain [K]S (in our case, the latent domain of the VQ-VAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' The joint probability of a sequence z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFAT4oBgHgl3EQfcB3t/content/2301.08562v1.pdf'} +page_content=' , zS) is decomposed via the chain rule pφ(z) = S � s=1 pφ(zs|z 0, and assume O1, O2, ..., ON are i.i.d., which are commonly imposed +in the RL literature (see e.g., Sutton and Barto, 2018). Finally, we denote the lp-norm of a +function aggregated over a given distribution function σ by ∥.∥p,σ. We use [q] to represent +the indices set {1, 2, 3, ...q} for any integer q ∈ N. +2.2 +Assumptions, Policies and Value Functions +We will require the system to satisfy the Markov assumption such that +P(S0,t+1 ∈ S|S0,t = s, A0,t = a, {S0,t′, A0,t′}0≤t′ 0. +In addition, we also impose the following conditional mean independence assumption: +E(Y0,t|S0,t = s, A0,t = a, {Y0,t′, S0,t′, A0,t′}0≤t′