Empirical Regularized Optimal Transport: Statistical Theory and Applications

TL;DR

This paper develops the statistical theory for empirical regularized optimal transport distances, including limit distributions and bootstrap consistency, with applications to confidence bands in biological network analysis.

Contribution

It provides the first comprehensive asymptotic distributional results for empirical regularized optimal transport, including the Sinkhorn divergence, using the implicit function theorem.

Findings

Empirical regularized transport plans follow a Gaussian distribution asymptotically.

The bootstrap method is consistent for these transport distances.

Monte Carlo simulations validate the theoretical results.

Abstract

We derive limit distributions for certain 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.