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Tensor Quasi-Random Groups
[article]

2021
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arXiv
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pre-print

Gowers has elegantly characterized the finite groups G in which A_1A_2A_3 = G for any positive density subsets A_1,A_2,A_3. This property, quasi-randomness, holds if and only if G does not admit a nontrivial irreducible representation of constant dimension. We present a dual characterization of tensor quasi-random groups in which multiplication of subsets is replaced by tensor product of representations.

arXiv:2103.11048v1
fatcat:7tjsmxogvjazvecpr523wy44di
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Metrical Service Systems with Transformations
[article]

2020
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arXiv
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pre-print

We consider a generalization of the fundamental online metrical service systems (MSS) problem where the feasible region can be transformed between requests. In this problem, which we call T-MSS, an algorithm maintains a point in a metric space and has to serve a sequence of requests. Each request is a map (transformation) f_t A_t→ B_t between subsets A_t and B_t of the metric space. To serve it, the algorithm has to go to a point a_t∈ A_t, paying the distance from its previous position. Then,

arXiv:2009.08266v1
fatcat:vj55vlcitjcz3htwo2xbzfbune
## more »

... e transformation is applied, modifying the algorithm's state to f_t(a_t). Such transformations can model, e.g., changes to the environment that are outside of an algorithm's control, and we therefore do not charge any additional cost to the algorithm when the transformation is applied. The transformations also allow to model requests occurring in the k-taxi problem. We show that for α-Lipschitz transformations, the competitive ratio is Θ(α)^n-2 on n-point metrics. Here, the upper bound is achieved by a deterministic algorithm and the lower bound holds even for randomized algorithms. For the k-taxi problem, we prove a competitive ratio of Õ((nlog k)^2). For chasing convex bodies, we show that even with contracting transformations no competitive algorithm exists. The problem T-MSS has a striking connection to the following deep mathematical question: Given a finite metric space M, what is the required cardinality of an extension M̂⊇ M where each partial isometry on M extends to an automorphism? We give partial answers for special cases.##
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Online Multiserver Convex Chasing and Optimization
[article]

2020
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arXiv
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pre-print

We introduce the problem of k-chasing of convex functions, a simultaneous generalization of both the famous k-server problem in R^d, and of the problem of chasing convex bodies and functions. Aside from fundamental interest in this general form, it has natural applications to online k-clustering problems with objectives such as k-median or k-means. We show that this problem exhibits a rich landscape of behavior. In general, if both k > 1 and d > 1 there does not exist any online algorithm with

arXiv:2004.07346v1
fatcat:2qb6b3f3djbsta7j3njlovq4ly
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... ounded competitiveness. By contrast, we exhibit a class of nicely behaved functions (which include in particular the above-mentioned clustering problems), for which we show that competitive online algorithms exist, and moreover with dimension-free competitive ratio. We also introduce a parallel question of top-k action regret minimization in the realm of online convex optimization. There, too, a much rougher landscape emerges for k > 1. While it is possible to achieve vanishing regret, unlike the top-one action case the rate of vanishing does not speed up for strongly convex functions. Moreover, vanishing regret necessitates both intractable computations and randomness. Finally we leave open whether almost dimension-free regret is achievable for k > 1 and general convex losses. As evidence that it might be possible, we prove dimension-free regret for linear losses via an information-theoretic argument.##
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Optimization of Mean-field Spin Glasses
[article]

2020
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arXiv
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pre-print

Mean-field spin glasses are families of random energy functions (Hamiltonians) on high-dimensional product spaces. In this paper we consider the case of Ising mixed p-spin models, namely Hamiltonians H_N:Σ_N→R on the Hamming hypercube Σ_N = {± 1}^N, which are defined by the property that {H_N(σ)}_σ∈Σ_N is a centered Gaussian process with covariance E{H_N(σ_1)H_N(σ_2)} depending only on the scalar product 〈σ_1,σ_2〉. The asymptotic value of the optimum max_σ∈Σ_NH_N(σ) was characterized in terms

arXiv:2001.00904v1
fatcat:kzbnx4m57zbgxmf7xpl5sdweri
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... a variational principle known as the Parisi formula, first proved by Talagrand and, in a more general setting, by Panchenko. The structure of superlevel sets is extremely rich and has been studied by a number of authors. Here we ask whether a near optimal configuration σ can be computed in polynomial time. We develop a message passing algorithm whose complexity per-iteration is of the same order as the complexity of evaluating the gradient of H_N, and characterize the typical energy value it achieves. When the p-spin model H_N satisfies a certain no-overlap gap assumption, for any ε>0, the algorithm outputs σ∈Σ_N such that H_N(σ)> (1-ε)max_σ' H_N(σ'), with high probability. The number of iterations is bounded in N and depends uniquely on ε. More generally, regardless of whether the no-overlap gap assumption holds, the energy achieved is given by an extended variational principle, which generalizes the Parisi formula.##
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Competitively Chasing Convex Bodies
[article]

2018
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arXiv
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pre-print

Let F be a family of sets in some metric space. In the F-chasing problem, an online algorithm observes a request sequence of sets in F and responds (online) by giving a sequence of points in these sets. The movement cost is the distance between consecutive such points. The competitive ratio is the worst case ratio (over request sequences) between the total movement of the online algorithm and the smallest movement one could have achieved by knowing in advance the request sequence. The family F

arXiv:1811.00887v1
fatcat:q6aonfedmnbylp72c6v56qahwy
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... s said to be chaseable if there exists an online algorithm with finite competitive ratio. In 1991, Linial and Friedman conjectured that the family of convex sets in Euclidean space is chaseable. We prove this conjecture.##
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Algorithmic pure states for the negative spherical perceptron
[article]

2020
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arXiv
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pre-print

We consider the spherical perceptron with Gaussian disorder. This is the set S of points σ∈ℝ^N on the sphere of radius √(N) satisfying ⟨ g_a , σ⟩≥κ√(N) for all 1 ≤ a ≤ M, where (g_a)_a=1^M are independent standard gaussian vectors and κ∈ℝ is fixed. Various characteristics of S such as its surface measure and the largest M for which it is non-empty, were computed heuristically in statistical physics in the asymptotic regime N →∞, M/N →α. The case κ<0 is of special interest as S is conjectured to

arXiv:2010.15811v1
fatcat:denaadcot5ampgxvcmmj3thjgq
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... exhibit a hierarchical tree-like geometry known as "full replica-symmetry breaking" (FRSB) close to the satisfiability threshold α_SAT(κ), and whose characteristics are captured by a Parisi variational principle akin to the one appearing in the Sherrington-Kirkpatrick model. In this paper we design an efficient algorithm which, given oracle access to the solution of the Parisi variational principle, exploits this conjectured FRSB structure for κ<0 and outputs a vector σ̂ satisfying ⟨ g_a , σ̂⟩≥κ√(N) for all 1≤ a ≤ M and lying on a sphere of non-trivial radius √(q̅ N), where q̅∈ (0,1) is the right-end of the support of the associated Parisi measure. We expect σ̂ to be approximately the barycenter of a pure state of the spherical perceptron. Moreover we expect that q̅→ 1 as α→α_SAT(κ), so that ⟨ g_a,σ̂/|σ̂|⟩≥ (κ-o(1))√(N) near criticality.##
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Approximate Ground States of Hypercube Spin Glasses are Near Corners
[article]

2020
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arXiv
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pre-print

We show that with probability exponentially close to 1, all near-maximizers of any mean-field mixed p-spin glass Hamiltonian on the hypercube [-1,1]^N are near a corner. This confirms a recent conjecture of Gamarnik and Jagannath. The proof is elementary and generalizes to arbitrary polytopes with e^o(N^2) faces.

arXiv:2009.09316v1
fatcat:33xy5zaefba5jp6nd5bqzeka6q
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Vertex Sparsifiers for c-Edge Connectivity
[article]

2019
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arXiv
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pre-print

We show the existence of O(f(c)k) sized vertex sparsifiers that preserve all edge-connectivity values up to c between a set of k terminal vertices, where f(c) is a function that only depends on c, the edge-connectivity value. This construction is algorithmic: we also provide an algorithm whose running time depends linearly on k, but exponentially in c. It implies that for constant values of c, an offline sequence of edge insertions/deletions and c-edge-connectivity queries can be answered in

arXiv:1910.10359v1
fatcat:s5zje623vffozkbelss6qjgqn4
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... ylog time per operation. These results are obtained by combining structural results about minimum terminal separating cuts in undirected graphs with recent developments in expander decomposition based methods for finding small vertex/edge cuts in graphs.##
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Chasing Convex Bodies Optimally
[article]

2021
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arXiv
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pre-print

In the chasing convex bodies problem, an online player receives a request sequence of N convex sets K_1,..., K_N contained in a normed space ℝ^d. The player starts at x_0∈ℝ^d, and after observing each K_n picks a new point x_n∈ K_n. At each step the player pays a movement cost of ||x_n-x_n-1||. The player aims to maintain a constant competitive ratio against the minimum cost possible in hindsight, i.e. knowing all requests in advance. The existence of a finite competitive ratio for convex body

arXiv:1905.11968v3
fatcat:fs55kpbd2fdedg4z3zv4djp7bq
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... hasing was first conjectured in 1991 by Friedman and Linial. This conjecture was recently resolved with an exponential 2^O(d) upper bound on the competitive ratio. We give an improved algorithm achieving competitive ratio d in any normed space, which is exactly tight for ℓ^∞. In Euclidean space, our algorithm also achieves competitive ratio O(√(dlog N)), nearly matching a √(d) lower bound when N is subexponential in d. The approach extends our prior work for nested convex bodies, which is based on the classical Steiner point of a convex body. We define the functional Steiner point of a convex function and apply it to the associated work function.##
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Tight Lipschitz Hardness for Optimizing Mean Field Spin Glasses
[article]

2021
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arXiv
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pre-print

*Mark*

*Sellke*was supported by an NSF graduate research fellowship, the William R. and Sara Hart Kimball Stanford graduate fellowship, and NSF award CCF-2006489. ...

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Vertex Sparsification for Edge Connectivity
[article]

2020
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arXiv
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pre-print

We now

arXiv:2007.07862v1
fatcat:7hdvksnutzg2lcw6en5kxmyhka
*mark*all the removed edges as essential, delete them, and*mark*their endpoints as additional terminals. ... In this way, a violating cut corresponds to the "small cut" in G whose edges we*mark*as essential. ...##
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Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localization
[article]

2022
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arXiv
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pre-print

We consider the Sherrington-Kirkpatrick model of spin glasses at high-temperature and no external field, and study the problem of sampling from the Gibbs distribution μ in polynomial time. We prove that, for any inverse temperature β<1/2, there exists an algorithm with complexity O(n^2) that samples from a distribution μ^alg which is close in normalized Wasserstein distance to μ. Namely, there exists a coupling of μ and μ^alg such that if (x,x^alg)∈{-1,+1}^n×{-1,+1}^n is a pair drawn from this

arXiv:2203.05093v1
fatcat:2bo6vpxe4rabzdjwdgq6z4v42y
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... oupling, then n^-1𝔼{||x-x^alg||_2^2}=o_n(1). The best previous results, by Bauerschmidt and Bodineau and by Eldan, Koehler, and Zeitouni, implied efficient algorithms to approximately sample (under a stronger metric) for β<1/4. We complement this result with a negative one, by introducing a suitable "stability" property for sampling algorithms, which is verified by many standard techniques. We prove that no stable algorithm can approximately sample for β>1, even under the normalized Wasserstein metric. Our sampling method is based on an algorithmic implementation of stochastic localization, which progressively tilts the measure μ towards a single configuration, together with an approximate message passing algorithm that is used to approximate the mean of the tilted measure.##
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Tensor quasi-random groups

2022
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Proceedings of the American Mathematical Society, Series B
*

*MARK*

*SELLKE*We induct again from N to G, obtaining G-representations W θ := Ind G K θ = Ind G N V θ . ...

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Approximating Continuous Functions by ReLU Nets of Minimal Width
[article]

2018
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arXiv
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pre-print

This article concerns the expressive power of depth in deep feed-forward neural nets with ReLU activations. Specifically, we answer the following question: for a fixed d_in≥ 1, what is the minimal width w so that neural nets with ReLU activations, input dimension d_in, hidden layer widths at most w, and arbitrary depth can approximate any continuous, real-valued function of d_in variables arbitrarily well? It turns out that this minimal width is exactly equal to d_in+1. That is, if all the

arXiv:1710.11278v2
fatcat:aoluxiwi5jgzzc7w5fv42jpqj4
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... n layer widths are bounded by d_in, then even in the infinite depth limit, ReLU nets can only express a very limited class of functions, and, on the other hand, any continuous function on the d_in-dimensional unit cube can be approximated to arbitrary precision by ReLU nets in which all hidden layers have width exactly d_in+1. Our construction in fact shows that any continuous function f:[0,1]^d_in→ R^d_out can be approximated by a net of width d_in+d_out. We obtain quantitative depth estimates for such an approximation in terms of the modulus of continuity of f.##
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The Saxl Conjecture for Fourth Powers via the Semigroup Property
[article]

2016
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arXiv
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pre-print

The tensor square conjecture states that for n ≥ 10, there is an irreducible representation V of the symmetric group S_n such that V ⊗ V contains every irreducible representation of S_n. Our main result is that for large enough n, there exists an irreducible representation V such that V^⊗ 4 contains every irreducible representation. We also show that tensor squares of certain irreducible representations contain (1-o(1))-fraction of irreducible representations with respect to two natural

arXiv:1511.02387v2
fatcat:bphpqbqypbad5efsv6gq24blte
## more »

... ity distributions. Our main tool is the semigroup property, which allows us to break partitions down into smaller ones.
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