Geometry and factorization of multivariate Markov chains with applications to the swapping algorithm

This paper analyzes the factorizability and geometry of transition matrices of multivariate Markov chains. Specifically, we demonstrate that the induced chains on factors of a product space can be regarded as information projections with respect to the Kullback-Leibler divergence. This perspective yields Han-Shearer type inequalities and submodularity of the entropy rate of Markov chains, as well as applications in the context of large deviations and mixing time comparison. As a concrete algorithmic application, we introduce a projection sampler based on the swapping algorithm, which resamples the highest-temperature coordinate at stationarity at each step. We prove that such practice accelerates the mixing time by multiplicative factors related to the number of temperatures and the dimension of the underlying state space when compared with the original swapping algorithm.