Approximate and exact solutions of intertwining equations through random spanning forests

TL;DR

This paper introduces a method using random forests to find approximate and exact solutions to intertwining equations in Markov processes, with applications in signal processing and metastability, emphasizing small overlap among probability families.

Contribution

It develops a novel approach employing random forests and a squeezing function to construct and analyze approximate solutions to intertwining equations with bounded errors.

Findings

Bounded expected squeezing and total variation errors in approximate solutions

Method to convert approximate solutions into exact solutions using Laplacian eigenvalues

Applicable to reversible Markov kernels on finite state spaces

Abstract

For different reversible Markov kernels on finite state spaces, we look for families of probability measures for which the time evolution almost remains in their convex hull. Motivated by signal processing problems and metastability studies we are interested in the case when the size of such families is smaller than the size of the state space, and we want such distributions to be with small overlap among them. To this aim we introduce a squeezing function to measure the common overlap of such families, and we use random forests to build random approximate solutions of the associated intertwining equations for which we can bound from above the expected values of both squeezing and total variation errors. We also explain how to modify some of these approximate solutions into exact solutions by using those eigenvalues of the associated Laplacian with the largest absolute values.