Quantitative Uniqueness of Kantorovich Potentials

TL;DR

This paper investigates the uniqueness of solutions in the dual optimal transport problem, providing both qualitative and quantitative bounds, especially when measures are supported on lower-dimensional or nearly connected sets.

Contribution

It establishes the first uniqueness results for cases with lower-dimensional supports and introduces a novel way to quantify the near-uniqueness of optimizers.

Findings

Optimal dual potentials are unique up to a constant when one measure's support is rectifiably connected.

The diameter of the set of dual potentials can be bounded by the Hausdorff distance to a regular connected set.

A new characterization of optimal dual potentials as intersection of explicit half-spaces is provided.

Abstract

This paper studies the uniqueness of solutions to the dual optimal transport problem, both qualitatively and quantitatively (bounds on the diameter of the set of optimisers). On the qualitative side, we prove that when one marginal measure's support is rectifiably connected (path-connected by rectifiable paths), the optimal dual potentials are unique up to a constant. This represents the first uniqueness result applicable even when both marginal measures are concentrated on lower-dimensional subsets of the ambient space, and also applies in cases where optimal potentials are nowhere differentiable on the supports of the marginals. On the quantitative side, we control the diameter of the set of optimal dual potentials by the Hausdorff distance between the support of one of the marginal measures and a regular connected set. In this way, we quantify the extent to which optimisers are almost unique when the support of one marginal measure is almost connected. This is a consequence of a novel characterisation of the set of optimal dual potentials as the intersection of an explicit family of half-spaces.