Graph-Based Lossless Markov Lumpings

TL;DR

This paper introduces a graph-based approach to identify lossless lumpings of Markov chains using zero-error information theory, enabling information-preserving state space reductions through clique partitioning.

Contribution

It presents a novel method leveraging graph theory and zero-error information theory to find lossless lumpings of Markov chains, with bounds on the trade-off between alphabet size and information loss.

Findings

Lumpings can be characterized by clique partitions of a related graph.

Lossless lumpings are achievable by exploiting the Markov chain's sparse temporal structure.

Bounds are provided on the trade-off between alphabet size and information loss.

Abstract

We use results from zero-error information theory to determine the set of non-injective functions through which a Markov chain can be projected without losing information. These lumping functions can be found by clique partitioning of a graph related to the Markov chain. Lossless lumping is made possible by exploiting the (sufficiently sparse) temporal structure of the Markov chain. Eliminating edges in the transition graph of the Markov chain trades the required output alphabet size versus information loss, for which we present bounds.