Constant curvature metrics for Markov chains

TL;DR

This paper explores the existence and uniqueness of pseudo-metrics with constant curvature for Markov chains, providing conditions for their existence, and applies these results to derive concentration inequalities.

Contribution

It introduces the concept of pseudo-metrics with constant curvature for Markov chains and establishes conditions for their existence and uniqueness.

Findings

Existence of pseudo-metrics with constant curvature on compact Polish spaces.

Uniqueness of such metrics in finite, irreducible Markov chains up to scalar factors.

Application to concentration inequalities for Markov chains.

Abstract

We consider metrics which are preserved under a $p$-Wasserstein transport map, up to a possible contraction. In the case $p=1$ this corresponds to a metric which is uniformly curved in the sense of coarse Ricci curvature. We investigate the existence of such metrics in the more general sense of pseudo-metrics, where the distance between distinct points is allowed to be 0, and show the existence for general Markov chains on compact Polish spaces. Further we discuss a notion of algebraic reducibility and its relation to the existence of multiple true pseudo-metrics with constant curvature. Conversely, when the Markov chain is irreducible and the state space finite we obtain effective uniqueness of a metric with uniform curvature up to scalar multiplication and taking the $p$th root, making this a natural intrinsic distance of the Markov chain. An application is given in the form of concentration inequalities for the Markov chain.