On the Precision of the Spectral Profile Bound for the Mixing Time of Continuous State Markov Chains

TL;DR

This paper examines the accuracy of spectral profile bounds for the mixing times of continuous state Markov chains, showing they are nearly sharp and useful for chain comparison even without spectral bounds.

Contribution

It extends sharpness results of spectral profile bounds from finite to continuous state spaces and demonstrates their application in chain comparison.

Findings

Spectral profile bound is sharp up to a log-log factor.

Extension of sharpness results from finite to continuous state spaces.

Application of bounds for comparing Markov chains.

Abstract

We investigate the sharpness of the spectral profile bound presented by Goel et al. and Chen et al. on the $L^{2}$ mixing time of Markov chains on continuous state spaces. We show that the bound provided by Chen et al. is sharp up to a factor of $\log\log$ of the initial density. This result extends the findings of Kozma, which showed the analogous result for the original spectral profile bound of Goel et al. for Markov chains on finite state spaces. Kozma shows that the spectral profile bound is sharp up to a multiplicative factor of $\log(\log(\pi_{min}))$, where $\pi_{\min}$ is the smallest value of the probability mass function of the stationary distribution. We discuss the application of our primary finding to the comparison of Markov chains. Our main result can be used as a comparison bound, indicating that it is possible to compare chains even when only non-spectral bounds exist for a known chain.