Empirical Regularized Optimal Transport: Statistical Theory and Applications

Abstract

We derive limit distributions for empirical regularized optimal transport distances between probability distributions supported on a finite metric space and show consistency of the (naive) bootstrap. In particular, we prove that the empirical regularized transport plan itself asymptotically follows a Gaussian law. The theory includes the Boltzmann-Shannon entropy regularization and hence a limit law for the widely applied Sinkhorn divergence. Our approach is based on an application of the implicit function theorem to necessary and sufficient optimality conditions for the regularized transport problem. The asymptotic results are investigated in Monte Carlo simulations. We further discuss computational and statistical applications, e.g. confidence bands for colocalization analysis of protein interaction networks based on regularized optimal transport.

Auditing ML Models for Individual Bias and Unfairness

“…Inference for the optimal value of an optimization problem (2.2) is generally a hard task, and we focus on finite sample spaces. This simplification is common in the literature on inferential tools for optimal transport problems (Sommerfeld and Munk, 2018;Klatt et al, 2018). As we shall see, the restriction of finite spaces is sufficient for many practical problems, including evaluating the algorithmic fairness of the COMPAS recidivism prediction instrument.…”

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 .…”

Minimax Confidence Intervals for the Sliced Wasserstein Distance

“…In contrast, the confidence intervals derived in the present paper are valid under either no assumptions or mild moment assumptions on P and Q, and are applied more generally to the Sliced Wasserstein distance in arbitrary dimension. Other inferential results for Wasserstein distances include those of Sommerfeld and Munk (2018); Tameling et al (2017); Klatt et al (2018) when the support of P and Q are finite or countable sets, and the work of Rippl et al (2016) when P and Q only differ by a location-scale transformation.…”