Breakdown points of Fermat-Weber problems under gauge distances

TL;DR

This work analyzes the robustness of Fermat--Weber points under general gauge distances, establishing a precise breakdown point of $1/(1+σ)$ where $σ$ measures gauge asymmetry. The authors prove the upper bound via a skewness-driven contamination argument and the lower bound by bounding the shifted FW solutions within a fixed region, and they introduce the elementary hull to study uniform robustness. They show that polyhedral gauges enjoy uniform robustness, while norms that are locally strictly convex do not, and in the plane a uniform robust gauge must be polyhedral. The contamination locus and majority-rule connections reveal structural facets of robustness and offer insights for consensus methods in settings like phylogenetics. Overall, the paper provides a unified, geometry-based framework for robust location under skewed distances and highlights conditions under which robust behavior is predictable and controllable.

Abstract

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

Figures (7)

• Figure 1: Three unit balls of gauges of skewness 2 and their skewness directions (bold arrows or bold segments; see text for details)
• Figure 2: Case of an uncorrupted $a\in A$ in the proof of theorem \ref{['th:lowerBound']}
• Figure 3: On the left the normal cones and on the right the elementary hull of a 5-point set for the triangular unit ball (b) of Figure \ref{['fig:ballsSD']}
• Figure 4: Non-polyhedral convex set described in Example \ref{['ex:strictlyLocalConvSetFromOneSide']} which is not strictly locally convex at any point. The point $u$ is an exposed point that is the limit of the exposed points $v_n$, but is also contained in a larger exposed face of the polyhedral set
• Figure 5: How the elementary convex set $C_\pi$, from the proof of Proposition \ref{['prop:nonPolyhGauge2Dim']}, with respect to $a$ and $b$ can be constructed. It depends on a face $F$ of $B_\gamma$, which may be either the singleton $\{u\}$ or a segment with $u$ as one endpoint (dotted case), extending downward. The set $C_\pi$ is the single point $\{y'\}$ when $u$ is exposed, and the line segment $[x',y']$ otherwise

Theorems & Definitions (71)

• Definition 1
• Remark 1
• Remark 1
• Definition 2
• Remark 2
• Example 1
• Lemma 2.1
• proof
• Remark 3
• Lemma 3.1