Quantitative stability in optimal transport for general power costs

TL;DR

This paper presents new quantitative stability results for optimal transport problems with power costs, providing explicit bounds on how solutions change with perturbations in the target measure.

Contribution

It extends stability analysis to a broad class of costs beyond quadratic, using assumptions on the source measure like log-concavity and bounded support.

Findings

Explicit bounds on stability of transport potentials and maps.

Results apply to Wasserstein distances with power cost exponent > 1.

Advances understanding of stability beyond quadratic costs.

Abstract

We establish novel quantitative stability results for optimal transport problems with respect to perturbations in the target measure. We provide explicit bounds on the stability of optimal transport potentials and maps, which are relevant for both theoretical and practical applications. Our results apply to a wide range of costs, including all Wasserstein distances with power cost exponent strictly larger than $1$ and leverage mostly assumptions on the source measure, such as log-concavity and bounded support. Our work provides a significant step forward in the understanding of stability of optimal transport problems, as previous results where mostly limited to the case of the quadratic cost.