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

Abstract

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 in [32] 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.

Empirical Regularized Optimal Transport: Statistical Theory and Applications

“…It includes the widely applied entropy ROT in (1.4) and hence limit distributions for the empirical entropy ROT plan and, as a consequence, for the empirical Sinkhorn divergence. For the latter see also Bigot et al (2017) who obtained limit distributions in a similar fashion to (1.8) for the optimal value of (1.3) with a straightforward application of the technique in . Note that the technique we introduce here allows to treat the ROT plan itself also in notable distinction to λ = 0, where such a result as in (1.7) is not known.…”

“…as then ∇S λ (r, r) = 0 (see also Bigot et al (2017)). However, a second order expansion which is based on a perturbation analysis for the dual solutions provides a non degenerate asymptotic limit of nS λ (r n , r).…”

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

“…(ii) A central limit theorem characterizing the fluctuations S(P n , Q n ) − ES(P n , Q n ) when P and Q are subgaussian (Section 3). Such a central limit theorem was previously only known for probability measures supported on a finite number of points (Bigot et al, 2017;Klatt et al, 2018). We use completely different techniques, inspired by recent work of Del Barrio and Loubes (2019), to prove our theorem for general subgaussian distributions.…”

“…In this section, we accomplish this goal by showing a central limit theorem (CLT) for S(P n , Q n ), valid for any subgaussian measures. Bigot et al (2017) and Klatt et al (2018) have shown CLTs for entropic OT when the measures lie in a finite metric space (or, equivalently, when P and Q are finitely supported). Apart from being restrictive in practice, these results do not shed much light on the general situation because OT on finite metric spaces behaves quite differently from OT on R d .…”

Asymptotic distribution and convergence rates of stochastic algorithms for entropic optimal transportation between probability measures

“…The asymptotic behavior of empirical Wasserstein distances when both µ and ν are absolutely continuous measures has been extensively studied over the last years [15,16,17,20,37]. For probability measures supported on finite spaces, limiting distributions for empirical Wasserstein distance have been obtained in [41], while the asymptotic distribution of empirical Sinkhorn divergence has been recently considered in [8,29]. 1.3.…”