Spectral Independence and Local-to-Global Techniques for Optimal Mixing of Markov Chains

TL;DR

This paper explores spectral independence as a powerful technique to analyze and ensure the rapid mixing of Markov chains, with applications to graphical models and matroids, providing new proofs and extending existing results.

Contribution

It introduces local-to-global theorems for spectral independence, demonstrating their use in proving fast mixing of Markov chains for various models and extending previous results.

Findings

Spectral independence implies fast mixing of Gibbs sampler on graphs.

Polynomial-time mixing established at the critical point of the uniqueness threshold.

Fast mixing of the bases-exchange walk for arbitrary matroids.

Abstract

This monograph is an exposition on an exciting new technique known as spectral independence, which has been instrumental in analyzing the convergence rate of Markov Chain Monte Carlo (MCMC) algorithms. For a high-dimensional distribution defined on labelings of the vertices of an n-vertex graph, the spectral independence condition, introduced by Anari, Liu, and Oveis Gharan (2020), is a bound on the maximum eigenvalue of the nxn influence matrix whose entries capture the influence between pairs of vertices (closely related to the covariance between the variables). In the first part of the monograph, we present results showing that spectral independence (and related techniques) imply fast mixing of simple Markov chains such as the Gibbs sampler. These proofs utilize local-to-global theorems which we will detail in this work. We focus on two applications: the hard-core model on independent sets of a graph (which is a binary graphical model) and random bases of a matroid. We demonstrate several methods for establishing spectral independence, including the correlation decay approach introduced by Weitz (2006) and the analytic approach introduced by Barvinok (2015) using zero-freeness of the associated complex-valued partition function. As a consequence of the techniques presented in this monograph we obtain fast mixing of the Gibbs sampler on general graphs in the so-called tree-uniqueness region, polynomial-time mixing on general graphs at the critical point for the uniqueness threshold, and polynomial-time mixing on random regular graphs beyond the uniqueness threshold. We also utilize the local-to-global framework of spectral independence and the Trickle-Down theorem of Oppenheim (2018) to prove fast mixing of the bases-exchange walk for generating a random basis of an arbitrary matroid, which was established by Anari, Liu, Oveis Gharan and Vinzant (2019).