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

Central limit theorems for entropy-regularized optimal transport on finite spaces and statistical applications