Markov Chains on Orbits of Permutation Groups
This work addresses the challenge of slow mixing in MCMC methods for probabilistic graphical models, offering a novel symmetry-based approach that could benefit researchers and practitioners in machine learning and statistics.
The paper tackled the problem of detecting and utilizing symmetries in probabilistic graphical models to improve Markov chain Monte Carlo (MCMC) efficiency, resulting in reduced mixing times and the introduction of the first lifted MCMC algorithm for such models, with analytical and empirical validation.
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.