Coarse-graining and reconstruction for Markov matrices

TL;DR

This paper introduces a novel coarse-graining method for Markov matrices that preserves key properties and enables reconstruction, applicable to tensor spaces and fluxes, with connections to functional inequalities.

Contribution

The method is a new, structure-preserving approach for coarse-graining Markov matrices that does not rely on Hilbert space tools and extends to tensor spaces and fluxes.

Findings

Preserves positivity and mass during coarse-graining.

Provides a generalized inverse for reconstruction.

Connects with functional inequalities and Poincaré constants.

Abstract

We present a coarse-graining (or model order reduction) procedure for stochastic matrices by clustering. The method is consistent with the natural structure of Markov theory, preserving positivity and mass, and does not rely on any tools from Hilbert space theory. The reconstruction is provided by a generalized Penrose-Moore inverse of the coarse-graining operator incorporating the inhomogeneous invariant measure of the Markov matrix. As we show, the method provides coarse-graining and reconstruction also on the level of tensor spaces, which is consistent with the notion of an incidence matrix and quotient graphs, and, moreover, allows to coarse-grain and reconstruct fluxes. Furthermore, we investigate the connection with functional inequalities and Poincar\'e-type constants.