Breakdown properties of optimal transport maps: general transportation costs

Abstract

Two recent works, Avella-Medina and González-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.

Figures (2)

• Figure 1: In this picture we provide a geometric description of \ref{['lemma:technical']}. If $\partial^c f_n (u)$ escapes to the horizon, then there exists a sequence $\{(w_n,q_n)\}_n$ with $q_n\in \partial^c f_n(w_n)$ and such that $\|u-w_n\|\to 0$ and $\|q_n\|~\to~\infty$. Furthermore, part (ii) states that $\partial^c f_n(K)$ (purple part on the right in the picture) escapes to the horizon, for any compact set $K$ (purple part on the left in the picture) contained in the green region $R_{n_0}=\bigcap_{n\geq n_0}\{ x: h(x-q_n) \leq h(w_n-q_n) \}$ for some $n_0$ large enough. Part (iii) states that the green region converges as $n_0\to \infty$ to a halfspace containing $u$.
• Figure 2: Illustration of the geometric argument used in the proof of \ref{['lem:finite_sample_UB']}. If $w$ defines a half-space with border passing through more than one point, then we can find a slight modification $w_\beta$ defining a half-space with the same number of interior points, but with border passing only through $u_{(\ell_1)}$.

Theorems & Definitions (22)

• Theorem 2.1
• Theorem 2.2
• Definition 2.3
• Theorem 2.4
• Remark 2.5: The direction of the perturbation
• Definition 2.6
• Theorem 2.7
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3: Lower bound