Spectral Independence in High-Dimensional Expanders and Applications to the Hardcore Model

Anari, Nima; Liu, Kuikui; Gharan, Shayan Oveis

doi:10.1137/20m1367696

Cited by 36 publications

(102 citation statements)

References 57 publications

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“…Entropic independence is a natural analog for spectral independence, another recently established notion by Anari, Liu, and Oveis Gharan [ALO20], if one replaces variance by entropy.…”

Section: P[s] ∝ µ(S)mentioning

confidence: 99%

“…In [ALO20], spectral independence is defined as an upper bound on the spectral norm of the pairwise correlation matrix of µ (Definition 23), or equivalently, an upper bound on the second largest eigenvalue of the simple (non-lazy) random walk on the 1-skeleton of µ, when viewing µ as a weighted high-dimensional expander [KO18;Opp18]. 1 The simple random walk on the 1-skeleton of µ samples from µD k→1 by transitioning from {i} to {j} with probability proportional to µD k→2 ({i, j}).…”

Section: P[s] ∝ µ(S)mentioning

confidence: 99%

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Entropic Independence I: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Distributions and High-Temperature Ising Models

Anari¹,

Jain²,

Koehler³

et al. 2021

Preprint

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We introduce a notion called entropic independence for distributions µ defined on pure simplicial complexes, i.e., subsets of size k of a ground set of elements. Informally, we call a background measure µ entropically independent if for any (possibly randomly chosen) set S, the relative entropy of an element of S drawn uniformly at random carries at most O(1/k) fraction of the relative entropy of S, a constant multiple of its "share of entropy." Entropic independence is the natural analog of spectral independence, another recently established notion, if one replaces variance by entropy.In our main result, we show that µ is entropically independent exactly when a transformed version of the generating polynomial of µ can be upper bounded by its linear tangent, a property implied by concavity of the said transformation. We further show that this concavity is equivalent to spectral independence under arbitrary external fields, an assumption that also goes by the name of fractional log-concavity. Our result can be seen as a new tool to establish entropy contraction from the much simpler variance contraction inequalities. A key differentiating feature of our result is that we make no assumptions on marginals of µ or the degrees of the underlying graphical model when µ is based on one. We leverage our results to derive tight modified log-Sobolev inequalities for multi-step down-up walks on fractionally log-concave distributions. As our main application, we establish the tight mixing time of O(n log n) for Glauber dynamics on Ising models with interaction matrix of operator norm smaller than 1, improving upon the prior quadratic dependence on n.

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“…Entropic independence is a natural analog for spectral independence, another recently established notion by Anari, Liu, and Oveis Gharan [ALO20], if one replaces variance by entropy.…”

Section: P[s] ∝ µ(S)mentioning

confidence: 99%

Section: P[s] ∝ µ(S)mentioning

confidence: 99%

Entropic Independence I: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Distributions and High-Temperature Ising Models

Anari¹,

Jain²,

Koehler³

et al. 2021

Preprint

Self Cite

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“…v } be the collection of feasible vertex-spin pairs under τ , where Ω τ v represents the set of feasible spins at v conditioned on τ . The following definition is taken from [CG ŠV21]; see also [ALO20,FGYZ21].…”

Section: Spectral Independence Via Stability Of the Partition Functionmentioning

confidence: 99%

“…Anari et al [ALO20] presented a powerful new tool for analyzing MCMC methods known as spectral independence. Spectral independence yields optimal mixing time bounds for the Glauber dynamics (which updates a randomly chosen vertex in each step) [CLV21], and more generally yields optimal mixing time bounds for any block dynamics and the Swendsen-Wang dynamics [BCC + 21].…”

Section: Introductionmentioning

confidence: 99%

Spectral Independence via Stability and Applications to Holant-Type Problems

Chen¹,

Liu²,

Vigoda³

2021

Preprint

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This paper formalizes connections between stability of polynomials and convergence rates of Markov Chain Monte Carlo (MCMC) algorithms. We prove that if a (multivariate) partition function is nonzero in a region around a real point λ then spectral independence holds at λ. As a consequence, for Holant-type problems (e.g., spin systems) on bounded-degree graphs, we obtain optimal O(n log n) mixing time bounds for the single-site update Markov chain known as the Glauber dynamics. Our result significantly improves the running time guarantees obtained via the polynomial interpolation method of Barvinok (2017), refined by Patel and Regts (2017).There are a variety of applications of our results. In this paper, we focus on Holant-type (i.e., edge-coloring) problems, including weighted edge covers and weighted even subgraphs. For the weighted edge cover problem (and several natural generalizations) we obtain an O(n log n) sampling algorithm on bounded-degree graphs. The even subgraphs problem corresponds to the high-temperature expansion of the ferromagnetic Ising model. We obtain an O(n log n) sampling algorithm for the ferromagnetic Ising model with a nonzero external field on bounded-degree graphs, which improves upon the classical result of Jerrum and Sinclair (1993) for this class of graphs. We obtain further applications to antiferromagnetic two-spin models on line graphs, weighted graph homomorphisms, tensor networks, and more.

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“…The past few years have witnessed the emergence of an attractive method for bounding the mixing time, based on local-to-global arguments for highdimensional expanders [ALOG20, DK17, KO18, AL20, Opp18]. Of direct relevance to us is the work of Anari, Liu, and Oveis Gharan [ALOG20], who introduced the notion of spectral independence (see Section 2.2 for an introduction) as a way of proving that the Glauber dynamics mixes rapidly. This notion, introduced in [ALOG20] for Boolean spin systems, was further developed in the works [FGYZ21,CGŠV21].…”

Section: Introductionmentioning

confidence: 99%

Spectral independence, coupling with the stationary distribution, and the spectral gap of the Glauber dynamics

Jain,

Pham,

Vuong

2021

Preprint

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We present a new lower bound on the spectral gap of the Glauber dynamics for the Gibbs distribution of a spectrally independent q-spin system on a graph G = (V, E) with maximum degree ∆. Notably, for several interesting examples, our bound covers the entire regime of ∆ excluded by arguments based on coupling with the stationary distribution. As concrete applications, by combining our new lower bound with known spectral independence computations and known coupling arguments:• We show that for a triangle-free graph G = (V, E) with maximum degree ∆ ≥ 3, the Glauber dynamics for the uniform distribution on proper k-colorings with k ≥ (1.763 • • • + δ)∆ colors has spectral gap Ωδ (|V | −1 ). Previously, such a result was known either if the girth of G is at least 5 [Dyer et. al, FOCS 2004], or under restrictions on ∆ [Chen et. al, STOC 2021; Hayes-Vigoda, FOCS 2003].• We show that for a regular graph G = (V, E) with degree ∆ ≥ 3 and girth at least 6, and for any ε, δ > 0, the partition function of the hardcore model with fugacity λ ≤ (1 − δ)λ c (∆) may be approximated within a (1 + ε)-multiplicative factor in time Õδ (n 2 ε −2 ). Previously, such a result was known if the girth is at least 7 [Efthymiou et. al, SICOMP 2019].• We show for the binomial random graph G(n, d/n) with d = O(1), with high probability, an approximately uniformly random matching may be sampled in time O d (n 2+o(1) ). This improves the corresponding running time of Õd (n 3 ) due to [Jerrum-Sinclair, SICOMP 1989;Jerrum, 2003].

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Spectral Independence in High-Dimensional Expanders and Applications to the Hardcore Model

Cited by 36 publications

References 57 publications

Entropic Independence I: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Distributions and High-Temperature Ising Models

Entropic Independence I: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Distributions and High-Temperature Ising Models

Spectral Independence via Stability and Applications to Holant-Type Problems

Spectral independence, coupling with the stationary distribution, and the spectral gap of the Glauber dynamics

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