Optimal Kullback-Leibler Aggregation via Information Bottleneck

TL;DR

This paper introduces an efficient method for reducing Markov chains by approximating the optimal partition using an information bottleneck approach, significantly simplifying the computation while maintaining accuracy.

Contribution

It formulates Markov chain reduction as an information bottleneck problem and proposes a greedy algorithm for near-optimal state space partitioning.

Findings

The method provides tight bounds when the chain is lumpable.

The approach reduces computational complexity compared to exhaustive search.

Application examples demonstrate effectiveness in bio-molecular modeling.

Abstract

In this paper, we present a method for reducing a regular, discrete-time Markov chain (DTMC) to another DTMC with a given, typically much smaller number of states. The cost of reduction is defined as the Kullback-Leibler divergence rate between a projection of the original process through a partition function and a DTMC on the correspondingly partitioned state space. Finding the reduced model with minimal cost is computationally expensive, as it requires an exhaustive search among all state space partitions, and an exact evaluation of the reduction cost for each candidate partition. Our approach deals with the latter problem by minimizing an upper bound on the reduction cost instead of minimizing the exact cost; The proposed upper bound is easy to compute and it is tight if the original chain is lumpable with respect to the partition. Then, we express the problem in the form of information bottleneck optimization, and propose using the agglomerative information bottleneck algorithm for searching a sub-optimal partition greedily, rather than exhaustively. The theory is illustrated with examples and one application scenario in the context of modeling bio-molecular interactions.