Spectral Telescope: Convergence Rate Bounds for Random-Scan Gibbs Samplers Based on a Hierarchical Structure

TL;DR

This paper introduces new spectral gap bounds for random-scan Gibbs samplers by exploiting their hierarchical structure, extending spectral independence techniques to general state spaces, and demonstrating applications on high-dimensional uniform distributions.

Contribution

It develops three novel bounds on spectral gaps of Gibbs samplers using hierarchical and correlation structures, generalizing spectral independence to broader domains.

Findings

Derived three new spectral gap bounds for Gibbs samplers.

Extended spectral independence techniques to general state spaces.

Applied bounds to high-dimensional uniform distributions on the cube corner.

Abstract

Random-scan Gibbs samplers possess a natural hierarchical structure. The structure connects Gibbs samplers targeting higher dimensional distributions to those targeting lower dimensional ones. This leads to a quasi-telescoping property of their spectral gaps. Based on this property, we derive three new bounds on the spectral gaps and convergence rates of Gibbs samplers on general domains. The three bounds relate a chain's spectral gap to, respectively, the correlation structure of the target distribution, a class of random walk chains, and a collection of influence matrices. Notably, one of our results generalizes the technique of spectral independence, which has received considerable attention for its success on finite domains, to general state spaces. We illustrate our methods through a sampler targeting the uniform distribution on a corner of an $n$-cube.