An Optimal Transport approach for the Schr\"odinger bridge problem and convergence of Sinkhorn algorithm

TL;DR

This paper introduces a novel optimal transport-based approach to the Schrödinger bridge problem, providing new proofs of existence and convergence results, including for multi-marginal cases, and offers insights into the Sinkhorn algorithm's behavior.

Contribution

It presents a new duality approach linking Schrödinger bridges with entropy-regularized optimal transport, extending convergence proofs of Sinkhorn algorithm to multiple marginals.

Findings

Established a priori estimates consistent as regularization vanishes

Provided a new proof of existence for entropic potentials and Schrödinger system solutions

Proved convergence of Sinkhorn algorithm in multi-marginal scenarios

Abstract

This paper exploit the equivalence between the Schr\"odinger Bridge problem and the entropy penalized optimal transport in order to find a different approach to the duality, in the spirit of optimal transport. This approach results in a priori estimates which are consistent in the limit when the regularization parameter goes to zero. In particular, we find a new proof of the existence of maximizing entropic-potentials and therefore, the existence of a solution of the Schr\"odinger system. Our method extends also when we have more than two marginals: we can provide an alternative proof of the convergence of the Sinkhorn algorithm with two marginals and we show that the Sinkhorn algorithm converges in the multi-marginal case.