最优传输映射的崩溃性质：一般运输成本
Breakdown properties of optimal transport maps: general transportation costs
AI总结：
本文研究了在一般凸成本函数下最优传输映射的崩溃点性质，证明成本函数不影响传输映射的崩溃点，为传输基于分位数的高崩溃点特性提供了明确特征。
摘要：
两篇近期工作，Avella-Medina和González-Sanz（2026）以及Passeggeri和Paindaveine（2026），研究了通过其崩溃点（即使映射取任意异常值的最小污染分数）的最优传输映射的鲁棒性。他们的主要发现是：设P和Q分别为目标和参考测度，T为平方欧几里得成本下的最优传输映射。则当P被扰动而Q固定时，T(u)的崩溃点等于u相对于Q的Tukey深度。本文将此结果扩展到一般凸成本函数，证明成本函数不影响最优传输映射的崩溃点。本文的贡献为最优传输映射的崩溃点提供了明确的特征化。特别是，它显示了对于广泛正则成本函数类，所有基于传输的分位数都享有相同的高崩溃点特性。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
