Quantitative Stability in Discrete Optimal Transport

TL;DR

This paper explores the stability and uniqueness properties of optimal transport solutions, extending theoretical inequalities and analyzing discrete problem behavior under perturbations, with implications for dual problem solutions.

Contribution

It extends stability and convexity results in optimal transport, analyzes discrete problem behavior under perturbations, and broadens dual solution uniqueness criteria.

Findings

Stability of optimal transport plans with respect to Wasserstein distance.

Extension of convexity inequalities for the Kantorovich functional.

Uniqueness of dual optimizers under specific support conditions.

Abstract

This work investigates several aspects related to quantitative stability in optimal transport, as well as uniqueness of the dual transport problem. Our main contributions are as follows. Chapter 1: Observations regarding the quantitative stability of optimal transport plans with respect to Wasserstein distance on the product space. Chapter 2: Extention of strong convexity inequalities for the Kantorovich functional to a larger class of source measures, using glueing arguments recently used for the quantitative stability of optimal transport maps. Chapters 3/4: A qualitative description of the behaviour of the fully discrete transport problem under perturbation of the support positions, as well as quantitative stability under uniqueness assumptions. Chapter 5: Extention of known uniqueness criteria for the dual transport problem. We show that when one marginal measure has Lipschitz-path connected support and the other has bounded support, the values of dual optimisers are unique up to a constant for a large family of costs, including $p$-costs for all $p>1$.