Bounds on Lifting Continuous Markov Chains to Speed Up Mixing

TL;DR

This paper extends the evolving set method to derive bounds on the mixing times of lifted Markov chains on continuous state spaces, improving existing bounds and providing insights into speeding up mixing processes.

Contribution

It introduces a refined method for bounding mixing times of lifted Markov chains on continuous spaces, enhancing previous finite-space bounds and addressing continuous cases.

Findings

Derived bounds for continuous state space Markov chains.

Improved existing bounds for finite state space chains.

Provided theoretical tools for analyzing lifted Markov chains.

Abstract

It is often possible to speed up the mixing of a Markov chain $\{ X_{t} \}_{t \in \mathbb{N}}$ on a state space $\Omega$ by \textit{lifting}, that is, running a more efficient Markov chain $\{ \hat{X}_{t} \}_{t \in \mathbb{N}}$ on a larger state space $\hat{\Omega} \supset \Omega$ that projects to $\{ X_{t} \}_{t \in \mathbb{N}}$ in a certain sense. In [CLP99], Chen, Lov{\'a}sz and Pak prove that for Markov chains on finite state spaces, the mixing time of any lift of a Markov chain is at least the square root of the mixing time of the original chain, up to a factor that depends on the stationary measure. Unfortunately, this extra factor makes the bound in [CLP99] very loose for Markov chains on large state spaces and useless for Markov chains on continuous state spaces. In this paper, we develop an extension of the evolving set method that allows us to refine this extra factor and find bounds for Markov chains on continuous state spaces that are analogous to the bounds in [CLP99]. These bounds also allow us to improve on the bounds in [CLP99] for some chains on finite state spaces.