A regularized transportation cost stemming from entropic approximation

TL;DR

This paper investigates entropic regularizations of optimal transport problems, establishing minimal conditions for convergence and characterizing the variational limits, including multi-marginal cases.

Contribution

It identifies the weakest compactness conditions needed for convergence of entropic regularized optimal transport functionals and characterizes their variational limits, even without convergence to the original problem.

Findings

Weakest compactness conditions suffice for convergence

Characterization of variational limits of regularized functionals

Applicability to multi-marginal optimal transport problems

Abstract

We study the entropic regularizations of optimal transport problems under suitable summability assumptions on the point-wise transport cost. These summability assumptions already appear in the literature. However, we show that the weakest compactness conditions that can be derived are already enough to obtain the convergence of the regularized functionals. This approach allows us to characterize the variational limit of the regularization even when it does not converge to the original problem. The results apply also to problems with more than two marginals.