Recent work has used optimal transport ideas to generalize the notion of (center-outward) quantiles to dimension $d\geq 2$. We study the robustness properties of these transport-based quantiles by deriving their breakdown point, roughly, the smallest amount of contamination required to make these quantiles take arbitrarily aberrant values. We prove that the transport median defined in Chernozhukov et al.~(2017) and Hallin et al.~(2021) has breakdown point of $1/2$. Moreover, a point in the transport depth contour of order $\tau\in [0,1/2]$ has breakdown point of $\tau$. This shows that the multivariate transport depth shares the same breakdown properties as its univariate counterpart. Our proof relies on a general argument connecting the breakdown point of transport maps evaluated at a point to the Tukey depth of that point in the reference measure.

翻译：近期研究利用最优传输思想将（中心向外）分位点的概念推广至维度$d\\geq 2$的情形。本文通过推导其崩溃点——即使这些分位点产生任意异常值所需的最小污染量，研究此类基于传输的分位点的稳健性。我们证明Chernozhukov等人（2017）与Hallin等人（2021）定义的传输中位数具有$1/2$的崩溃点。此外，阶数为$\\tau\\in [0,1/2]$的传输深度轮廓上的点具有$\\tau$的崩溃点。这表明多元传输深度具有与一元情形相同的崩溃特性。我们的证明依赖于一个通用论证，将传输映射在某点处的崩溃点与该点在参考测度中的Tukey深度联系起来。