Stability of Entropic Optimal Transport and Schr\"odinger Bridges

TL;DR

This paper proves the stability and well-posedness of solutions to entropic optimal transport and Schrödinger bridge problems, using geometric concepts like cyclical invariance, even under infinite cost conditions.

Contribution

It introduces a geometric framework for stability analysis of entropic optimal transport solutions, extending to Schrödinger bridges and infinite cost scenarios.

Findings

Solutions are stable with respect to marginals and cost functions.

Well-posedness holds even for infinite cost transports.

Results apply broadly to static Schrödinger bridge problems.

Abstract

We establish the stability of solutions to the entropically regularized optimal transport problem with respect to the marginals and the cost function. The result is based on the geometric notion of cyclical invariance and inspired by the use of $c$-cyclical monotonicity in classical optimal transport. As a consequence of stability, we obtain the wellposedness of the solution in this geometric sense, even when all transports have infinite cost. More generally, our results apply to a class of static Schr\"odinger bridge problems including entropic optimal transport.