Breakdown points of Fermat-Weber problems under gauge distances

• 成果类型: Article; Early Access
• 署名作者: Comaneci, Andrei; Plastria, Frank
• 署名单位: Technical University of Berlin; Vrije Universiteit Brussel
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-025-02302-4
• 发表日期: 2025-11-24
• 关键词: fermat-weber problem; Gauge distance; breakdown point; Robust location estimator; Elementary hull; Contamination locus; MAJORITY RULES; M-ESTIMATORS; location

摘要

We compute the robustness of Fermat-Weber points with respect to any finite gauge. We show a breakdown point of 1/(1+sigma)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/(1+\sigma )$$\end{document} where sigma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} is the asymmetry measure of the gauge. We obtain quantitative results indicating how far a corrupted Fermat-Weber point can lie from the true value in terms of the original sample and the size of the corrupted part. If the distance from the true value depends only on the original sample, then we call the gauge 'uniformly robust.' We show that polyhedral gauges are uniformly robust, but locally strictly convex norms are not, while in dimension 2 any uniform robust gauge is polyhedral.