A rate-distortion framework for MCMC algorithms: geometry and factorization of multivariate Markov chains

We introduce a framework rooted in a rate distortion problem for Markov chains, and show how a suite of commonly used Markov Chain Monte Carlo (MCMC) algorithms are specific instances within it, where the target stationary distribution is controlled by the distortion function. Our approach offers a unified variational view on the optimality of algorithms such as Metropolis-Hastings, Glauber dynamics, the swapping algorithm and Feynman-Kac path models. Along the way, we analyze factorizability and geometry of multivariate Markov chains. Specifically, we demonstrate that induced chains on factors of a product space can be regarded as information projections with respect to a particular divergence. This perspective yields Han--Shearer type inequalities for Markov chains as well as applications in the context of large deviations and mixing time comparison. Finally, to demonstrate the significance of our framework, we propose a new projection sampler based on the swapping algorithm that provably accelerates the mixing time by multiplicative factors related to the number of temperatures and the dimension of the underlying state space.