Markov Chains on Orbits of Permutation Groups

TL;DR

This paper introduces orbital Markov chains that exploit symmetries in probabilistic graphical models to improve sampling efficiency, providing the first lifted MCMC algorithm with proven rapid mixing.

Contribution

It presents a scalable method for computing permutation group symmetries and introduces orbital Markov chains that leverage these symmetries to enhance mixing times.

Findings

Efficient computation of permutation group generators.

Orbital Markov chains achieve faster mixing.

Empirical results confirm improved performance.

Abstract

We present a novel approach to detecting and utilizing symmetries in probabilistic graphical models with two main contributions. First, we present a scalable approach to computing generating sets of permutation groups representing the symmetries of graphical models. Second, we introduce orbital Markov chains, a novel family of Markov chains leveraging model symmetries to reduce mixing times. We establish an insightful connection between model symmetries and rapid mixing of orbital Markov chains. Thus, we present the first lifted MCMC algorithm for probabilistic graphical models. Both analytical and empirical results demonstrate the effectiveness and efficiency of the approach.