Gibbs/Metropolis algorithms on a convex polytope

TL;DR

This paper establishes precise convergence rates for Metropolis algorithms sampling uniformly from convex polytopes, introducing new spectral bounds and linking eigenvalues to polytope Laplacians.

Contribution

It provides sharp convergence bounds for a local proposal Metropolis algorithm on convex polytopes, using novel spectral analysis tools.

Findings

Derived bounds on the spectrum and eigenfunctions of the Markov chain

Connected top eigenvalues to Neumann eigenvalues of the polytope Laplacian

Established convergence rates for the Metropolis algorithm on convex polytopes

Abstract

This paper gives sharp rates of convergence for natural versions of the Metropolis algorithm for sampling from the uniform distribution on a convex polytope. The singular proposal distribution, based on a walk moving locally in one of a fixed, finite set of directions, needs some new tools. We get useful bounds on the spectrum and eigenfunctions using Nash and Weyl-type inequalities. The top eigenvalues of the Markov chain are closely related to the Neuman eigenvalues of the polytope for a novel Laplacian.