Breakdown properties of optimal transport maps: general transportation costs

2026-03-16

AI Summary

This study investigates the robustness of optimal transport maps when the target measure is contaminated while the reference measure remains fixed, with a focus on how the breakdown point depends on the transport cost function. By integrating optimal transport theory, Tukey depth analysis, and a general framework for convex cost functions, the authors establish that the breakdown point is independent of the specific choice of cost function and is instead determined solely by the Tukey depth of the reference measure. This result unifies and extends existing findings, providing the first complete characterization of the high breakdown point shared by all transport-based quantiles under a broad class of regular cost functions.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• optimal transport
• breakdown point
• robustness
• transport map
• convex cost

Innovation

• optimal transport
• breakdown point
• Tukey depth
• robustness
• convex cost functions