Quantitative Uniform Stability of the Iterative Proportional Fitting Procedure

TL;DR

This paper proves that the Sinkhorn algorithm, used for entropy-regularized optimal transport, remains stable over time with respect to marginals, providing quantitative bounds in Wasserstein distance, and extends results to Schrödinger bridges.

Contribution

It offers the first quantitative uniform stability analysis of the Sinkhorn algorithm in Wasserstein distance, with implications for Schrödinger bridge problems.

Findings

Proves uniform stability of Sinkhorn algorithm over time.

Provides explicit bounds in 1-Wasserstein metric.

Extends stability results to Schrödinger bridges.

Abstract

We establish the uniform in time stability, w.r.t. the marginals, of the Iterative Proportional Fitting Procedure, also known as Sinkhorn algorithm, used to solve entropy-regularised Optimal Transport problems. Our result is quantitative and stated in terms of the 1-Wasserstein metric. As a corollary we establish a quantitative stability result for Schr\"odinger bridges.