On the breakdown point of transport-based quantiles

2024-10-21

AI Summary

This paper investigates the robustness of optimal-transport-based center-outward quantiles in high-dimensional spaces (d ≥ 2), focusing on their breakdown point—the minimal contamination proportion that drives the estimator arbitrarily far from the true value. Method: Leveraging a novel theoretical connection between transport quantiles and Tukey depth with respect to a reference measure, the analysis integrates optimal transport theory, center-outward distribution functions, and depth function theory. Contribution/Results: The study establishes, for the first time, exact breakdown points: the transport median has breakdown point 1/2, and the τ-th depth contour point has breakdown point exactly τ for τ ∈ [0, 1/2]. These results demonstrate that multivariate transport depth fully inherits the optimal robustness of univariate quantiles and matches the breakdown behavior of classical Tukey depth. The work provides a new theoretical benchmark and geometrically interpretable characterization for high-dimensional robust statistics.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

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

Recent work has used optimal transport ideas to generalize the notion of (center-outward) quantiles to dimension $dgeq 2$. 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 $ auin [0,1/2]$ has breakdown point of $ au$. 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.

Problem

• Analyzing robustness of transport-based quantiles through breakdown points
• Establishing breakdown point properties for multivariate transport depth
• Connecting transport map breakdown to Tukey depth in reference measures

Innovation

• Optimal transport generalizes quantiles to multivariate data
• Breakdown point analysis measures robustness of transport quantiles
• Transport median has breakdown point of 1/2 like univariate quantiles