  The notion of entropy-regularized optimal transport, also known as Sinkhorn
divergence, has recently gained popularity in machine learning and statistics,
as it makes feasible the use of smoothed optimal transportation distances for
data analysis. The Sinkhorn divergence allows the fast computation of an
entropically regularized Wasserstein distance between two probability
distributions supported on a finite metric space of (possibly) high-dimension.
For data sampled from one or two unknown probability distributions, we derive
the distributional limits of the empirical Sinkhorn divergence and its centered
version (Sinkhorn loss). We also propose a bootstrap procedure which allows to
obtain new test statistics for measuring the discrepancies between multivariate
probability distributions. Our work is inspired by the results of Sommerfeld
and Munk (2016) on the asymptotic distribution of empirical Wasserstein
distance on finite space using unregularized transportation costs. Incidentally
we also analyze the asymptotic distribution of entropy-regularized Wasserstein
distances when the regularization parameter tends to zero. Simulated and real
datasets are used to illustrate our approach.
