Covariance estimation using Markov chain Monte Carlo
This addresses covariance estimation problems in statistical inference and sampling, with applications to isotropic rounding for convex bodies, though it appears incremental as it builds on existing MCMC theory.
The paper tackles covariance matrix estimation for Gibbs distributions using dependent samples from Markov chains, showing that under Poincaré inequality and spectral gap conditions, MCMC achieves similar sample complexity to i.i.d. estimators with potentially better query complexity.
We investigate the complexity of covariance matrix estimation for Gibbs distributions based on dependent samples from a Markov chain. We show that when $π$ satisfies a Poincaré 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.