Breakdown points of transport-based quantiles

Abstract

Optimal transport ideas have recently been used to extend the concept of (center-outward) quantiles to higher dimensions ($d\ge 2$). In this talk, we explore the robustness of these transport-based quantiles by analyzing their breakdown point—the smallest level of data contamination required for the quantiles to produce extreme values. In this talk we explain our recent work Avella-Medina and González-Sanz [2024]. We show that the transport median, as introduced by Hallin et al. [2021], achieves a breakdown point of $1/2$. Additionally, points on the transport depth contour of order $\tau \in [0,1/2]$ exhibit a breakdown point of $\tau$, aligning the robustness of multivariate transport depth with that of its univariate counterpart. Our approach leverages a connection between the breakdown point of transport maps and the Tukey depth of points in the reference measure. This provides new insights into the reliability of transport-based quantiles in high-dimensional settings.

Keywords: Depth – Optimal transport – Quantiles – Robustness