Breakdown points of Fermat-Weber problems under gauge distances

TL;DR

This paper analyzes the robustness of Fermat-Weber points under gauge distances, quantifying their breakdown points and robustness properties, especially focusing on polyhedral gauges and convex norms.

Contribution

It introduces a formula for the breakdown point of Fermat-Weber points under any finite gauge and characterizes robustness for polyhedral gauges and convex norms.

Findings

Breakdown point is 1/(1+σ), with σ as the gauge's asymmetry measure.

Polyhedral gauges are uniformly robust, while strictly convex norms are not.

In dimension 2, all uniformly robust gauges are polyhedral.

Abstract

We compute the robustness of Fermat-Weber points with respect to any finite gauge. We show a breakdown point of $1/(1+\sigma)$ where $\sigma$ 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.