Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem

We prove several fundamental statistical bounds for entropic OT with the squared Euclidean cost between subgaussian probability measures in arbitrary dimension. First, through a new sample complexity result we establish the rate of convergence of entropic OT for empirical measures. Our analysis improves exponentially on the bound of Genevay et al. (2019) and extends their work to unbounded measures. Second, we establish a central limit theorem for entropic OT, based on techniques developed by Del Barrio and Loubes (2019). Previously, such a result was only known for finite metric spaces. As an application of our results, we develop and analyze a new technique for estimating the entropy of a random variable corrupted by gaussian noise.