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.

翻译：最近的两项研究，Avella-Medina 和 González-Sanz (2026) 以及 Passeggeri 和 Paindaveine (2026)，通过崩溃点（即能够使映射取任意异常值的最小污染比例）研究了最优运输映射的鲁棒性。他们的主要发现如下：令 $P$ 和 $Q$ 分别表示目标和参考测度，并令 $T$ 为平方欧几里得成本下的最优运输映射。那么，当 $P$ 受到扰动而 $Q$ 固定时，$T(u)$ 的崩溃点与 $u$ 相对于 $Q$ 的 Tukey 深度一致。在本注记中，我们将此结果推广到一般凸成本函数，证明成本函数对最优运输映射的崩溃点没有任何影响。我们的贡献提供了最优运输映射崩溃点的明确刻画。特别地，它表明对于一大类正则成本函数，所有基于运输的分位数都具有相同的高崩溃点性质。