Expanderizing Higher Order Random Walks

TL;DR

This paper introduces expanderized higher order random walks, which leverage auxiliary expander graphs to improve mixing times and reduce randomness in sampling problems over simplicial complexes.

Contribution

It generalizes higher order random walks using expander graphs, establishing inequalities and analyzing mixing times for sampling colorings and Ising models.

Findings

Expanderized walks satisfy log-Sobolev and Poincaré inequalities with quality depending on the auxiliary graph's expansion.

Achieves $O(n \, \log n)$ mixing time for sampling list colorings in bounded degree graphs.

Provides simplified proofs for log-Sobolev constants and local-to-global entropy contraction in simplicial complexes.

Abstract

We study a variant of the down-up and up-down walks over an $n$-partite simplicial complex, which we call expanderized higher order random walks -- where the sequence of updated coordinates correspond to the sequence of vertices visited by a random walk over an auxiliary expander graph $H$. When $H$ is the clique, this random walk reduces to the usual down-up walk and when $H$ is the directed cycle, this random walk reduces to the well-known systematic scan Glauber dynamics. We show that whenever the usual higher order random walks satisfy a log-Sobolev inequality or a Poincar\'e inequality, the expanderized walks satisfy the same inequalities with a loss of quality related to the two-sided expansion of the auxillary graph $H$. Our construction can be thought as a higher order random walk generalization of the derandomized squaring algorithm of Rozenman and Vadhan. We show that when initiated with an expander graph our expanderized random walks have mixing time $O(n \log n)$ for sampling a uniformly random list colorings of a graph $G$ of maximum degree $\Delta = O(1)$ where each vertex has at least $(11/6 - \epsilon) \Delta$ and at most $O(\Delta)$ colors and $O\left( \frac{n \log n}{(1 - \| J\|)^2}\right)$ for sampling the Ising model with a PSD interaction matrix $J \in R^{n \times n}$ satisfying $\| J \| \le 1$ and the external field $h \in R^n$-- here the $O(\bullet)$ notation hides a constant that depends linearly on the largest entry of $h$. As expander graphs can be very sparse, this decreases the amount of randomness required to simulate the down-up walks by a logarithmic factor. We also prove some simple results which enable us to argue about log-Sobolev constants of higher order random walks and provide a simple and self-contained analysis of local-to-global $\Phi$-entropy contraction in simplicial complexes -- giving simpler proofs for many pre-existing results.