Entropic Optimal Transport: Convergence of Potentials

TL;DR

This paper investigates the convergence of Schr"odinger potentials, used in entropic regularization of optimal transport, to Kantorovich potentials as the regularization parameter approaches zero, establishing strong convergence results.

Contribution

It proves the $L^{1}$ convergence of Schr"odinger potentials to Kantorovich potentials in the small-noise limit for all continuous, integrable cost functions on Polish spaces.

Findings

Schr"odinger potentials converge in $L^{1}$ to Kantorovich potentials as regularization vanishes.

Strong compactness in $L^{1}$ is established for potentials in the limit.

Results hold for all continuous, integrable cost functions on Polish spaces.

Abstract

We study the potential functions that determine the optimal density for $\varepsilon$-entropically regularized optimal transport, the so-called Schr\"odinger potentials, and their convergence to the counterparts in classical optimal transport, the Kantorovich potentials. In the limit $\varepsilon\to0$ of vanishing regularization, strong compactness holds in $L^{1}$ and cluster points are Kantorovich potentials. In particular, the Schr\"odinger potentials converge in $L^{1}$ to the Kantorovich potentials as soon as the latter are unique. These results are proved for all continuous, integrable cost functions on Polish spaces. In the language of Schr\"odinger bridges, the limit corresponds to the small-noise regime.