Breakdown properties of optimal transport maps: general transportation costs

TL;DR

This paper extends the understanding of the robustness of optimal transport maps, showing that the breakdown point is unaffected by the choice of convex cost functions, thus generalizing previous results beyond squared Euclidean costs.

Contribution

It provides a generalization of the breakdown point characterization for optimal transport maps to a broad class of convex cost functions, confirming robustness properties.

Findings

Breakdown point of transport maps is unaffected by the cost function choice.

Transport-based quantiles share high breakdown point properties.

Theoretical extension from squared Euclidean cost to general convex costs.

Abstract

Two recent works, Avella-Medina and Gonz\'alez-Sanz (2026) and Passeggeri and Paindaveine (2026), studied the robustness of the optimal transport map through its breakdown point, i.e., the smallest fraction of contamination that can make the map take arbitrarily aberrant values. Their main finding is the following: let $P$ and $Q$ denote the target and reference measures, respectively, and let $T$ be the optimal transport map for the squared Euclidean cost. Then, the breakdown point of $T(u)$, when $P$ is perturbed and $Q$ is fixed, coincides with the Tukey depth of $u$ relative to $Q$. In this note, we extend this result to general convex cost functions, demonstrating that the cost function does not have any impact on the breakdown point of the optimal transport map. Our contribution provides a definitive characterization of the breakdown point of the optimal transport map. In particular, it shows that for a broad class of regular cost functions, all transport-based quantiles enjoy the same high breakdown point properties.