An approximation theory for Markov chain compression

TL;DR

This paper introduces a rigorous framework for compressing reversible Markov chains with error bounds, proposing two schemes and validating their effectiveness through numerical experiments.

Contribution

It develops a novel approximation framework for Markov chain compression, including two new schemes and error control methods based on Nyström approximation.

Findings

Error bounds are established for the compression schemes.

Numerical experiments confirm the scalability and accuracy of the methods.

The approach effectively reduces Markov chain complexity while maintaining fidelity.

Abstract

We develop a framework for the compression of reversible Markov chains with rigorous error control. Given a subset of selected states, we construct reduced dynamics that can be lifted to an approximation of the full dynamics, and we prove simple spectral and nuclear norm bounds on the recovery error in terms of a suitably interpreted Nystr\"{o}m approximation error. We introduce two compression schemes: a projective compression based on committor functions and a structure-preserving compression defined in terms of an induced Markov chain over the selected states. The Nystr\"{o}m error appearing in our bounds can be controlled using recent results on column subset selection by nuclear maximization. Numerical experiments validate our theory and demonstrate the scalability of our approach.