Optimal mixing of Glauber dynamics: entropy factorization via high-dimensional expansion

Chen, Zongchen; Liu, Kuikui; Vigoda, Eric

doi:10.1145/3406325.3451035

Cited by 51 publications

(76 citation statements)

References 41 publications

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“…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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Section: Introductionmentioning

confidence: 99%

Spectral Independence via Stability and Applications to Holant-Type Problems

Chen¹,

Liu²,

Vigoda³

2021

Preprint

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“…It follows from Theorem 5.2 that for all λ < λ c and large enough ℓ (depending on λ) the expected run-time of the algorithm is constant per assigned spin. It is worth noting that for the case of a general graph of maximum degree ∆ it was recently shown, by Chen, Liu and Vigoda [CLV20a], that Glauber dynamics mixes in time O(n log n) when λ < λ c . Thus, for general graphs our algorithm is efficient in the same regime as the Glauber dynamics, but we do require the additional condition of subexponential growth.…”

Section: The Hardcore Modelmentioning

confidence: 99%

“…Under the same conditions, there already exists a sampling algorithm, based on Glauber dynamics, that runs in polynomial time [JS89]. Indeed, recent work of Chen et al [CLV20a] shows that Glauber dynamics mixes in time O(n log n). The advantage of the current algorithm is that it is exact and we can sample just a partial configuration on the graph.…”

Section: The Monomer-dimer Modelmentioning

confidence: 99%

“…Many of the most successful algorithms have been based on Markov chain simulation. Particularly prominent is the familiar Glauber dynamics [Gla63] whose mixing time has remained an active area of research, with wonderful new results coming in recent years [CLV20a;CLV20b], prompted by pioneering work on high dimensional expanders [KO20;ALGV18]. Deterministic approaches have also been proposed, based on decay of correlations [Wei06] or Taylor expansion in a zero-free region of the partition function [PR17;BS17].…”

Section: Introductionmentioning

confidence: 99%

“…Various authors have sought to leverage the property of strong spatial mixing to algorithmic ends [FGY19; Spi20; SSY13; SSvY17]. Intriguingly, it has been suggested [CLV20a] that strong spatial mixing may be equivalent to spectral independence, a concept that is currently the subject of intensive study. As for the second condition, subexponential growth currently seems important for perfect sampling, even though some approaches to approximate sampling dispense with it.…”

Section: Introductionmentioning

confidence: 99%

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Perfect Sampling in Infinite Spin Systems via Strong Spatial Mixing

Konrad¹,

Jerrum²

2021

Preprint

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Perfect sampling from spatial mixing

Feng

Guo

Yin

2022

Random Struct Algorithms

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We introduce a new perfect sampling technique that can be applied to general Gibbs distributions and runs in linear time if the correlation decays faster than the neighborhood growth. In particular, in graphs with subexponential neighborhood growth like ℤd, our algorithm achieves linear running time as long as Gibbs sampling is rapidly mixing. As concrete applications, we obtain the currently best perfect samplers for colorings and for monomer‐dimer models in such graphs.

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Optimal mixing of Glauber dynamics: entropy factorization via high-dimensional expansion

Cited by 51 publications

References 41 publications

Spectral Independence via Stability and Applications to Holant-Type Problems

Spectral Independence via Stability and Applications to Holant-Type Problems

Perfect Sampling in Infinite Spin Systems via Strong Spatial Mixing

Perfect sampling from spatial mixing

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