Asymptotics of Entropy Regularized Optimal Transport via Chaos Decomposition
Abstract
Consider the problem of finding the optimal coupling (or matching) between two i.i.d. samples from respective two densities on Euclidean spaces. For both computational efficiency and smoothness, this discrete problem is usually regularized by an entropy term. We introduce a modification to the commonly used discrete entropic regularization [Cuturi (‘13)] so that the optimal coupling of the regularized problem can be viewed as the static Schroedinger bridge given a finite number of particles. We show that this discrete Schroedinger bridge converges weakly to its continuum counterpart as the sample size goes to infinity. The proofs are based on a change of measure argument equipped with a novel contiguity result on the sequence of laws of the pair of empirical measures. We derive a functional CLT and second order Gaussian chaos limits when the limiting Gaussian is degenerate. This is achieved by a new chaos decomposition for paired samples using Markov operators. This idea generalizes the Hoeffding decomposition from the classical U-statistics theory, where the samples are assumed to be independent. This talk is based on joint work with Zaid Harchaoui and Soumik Pal.
