Empirical Regularized Optimal Transport: Statistical Theory and Applications

Understand

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.