🤖 AI Summary
This work investigates the decomposability and geometric structure of transition matrices for multivariate Markov chains to accelerate mixing. We model factor chains as information projections onto submanifolds under the KL divergence, establishing—for the first time—a rigorous correspondence between factorization structure and information-geometric projection. This framework unifies Han–Shearer-type inequalities and submodularity of entropy rate, while revealing intrinsic connections among entropy-rate submodularity, large-deviation rate functions, and mixing times. Building on this insight, we design a projection sampler parameterized by temperature coordinates—a geometric refinement of the exchange algorithm. Theoretical analysis shows that its mixing time improves over the classical exchange algorithm by a factor proportional to the product of the number of temperatures and the dimension of the state space, substantially enhancing sampling efficiency for high-dimensional multivariate chains.
📝 Abstract
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.