Covariance estimation using Markov chain Monte Carlo

TL;DR

This paper analyzes the sample complexity of covariance matrix estimation using Markov chain Monte Carlo (MCMC) methods, demonstrating conditions under which MCMC achieves similar efficiency to i.i.d. sampling and applying this to convex body sampling.

Contribution

It establishes theoretical guarantees for covariance estimation with MCMC under Poincaré inequality and spectral gap conditions, improving query complexity bounds for specific sampling problems.

Findings

MCMC can match i.i.d. sample complexity for covariance estimation under certain conditions.

Improved query complexity bounds for sampling uniformly on convex bodies.

Guarantees for isotropic rounding procedures in convex geometry.

Abstract

We investigate the complexity of covariance matrix estimation for Gibbs distributions based on dependent samples from a Markov chain. We show that when $\pi$ satisfies a Poincar\'e inequality and the chain possesses a spectral gap, we can achieve similar sample complexity using MCMC as compared to an estimator constructed using i.i.d. samples, with potentially much better query complexity. As an application of our methods, we show improvements for the query complexity in both constrained and unconstrained settings for concrete instances of MCMC. In particular, we provide guarantees regarding isotropic rounding procedures for sampling uniformly on convex bodies.