Spectral asymptotics for Metropolis algorithm on singular domains

TL;DR

This paper analyzes the spectral properties of the Metropolis algorithm on domains with cusp singularities, establishing a spectral gap and exponential convergence to equilibrium as the proposal scale shrinks.

Contribution

It provides the first spectral asymptotics for the Metropolis algorithm on singular domains, including cusp points, as the proposal scale approaches zero.

Findings

Existence of a spectral gap g(h) for small h

Spectral gap g(h) behavior as h approaches zero

Exponential convergence to equilibrium in total variation distance

Abstract

We study the Metropolis algorithm on a bounded connected domain $\Omega$ of the euclidean space with proposal kernel localized at a small scale $h > 0$. We consider the case of a domain $\Omega$ that may have cusp singularities. For small values of the parameter $h$ we prove the existence of a spectral gap $g(h)$ and study the behavior of $g(h)$ when $h$ goes to zero. As a consequence, we obtain exponentially fast return to equilibrium in total variation distance.