Tropical Fermat-Weber Points over Spaces of $M$-Ultrametrics

TL;DR

This paper generalizes phylogenetic tree reconstruction methods to ultrametrics associated with arbitrary matroids, analyzing their stability and the properties of tropical Fermat-Weber points within these spaces.

Contribution

It extends existing ultrametric reconstruction techniques to matroid-based ultrametrics and studies the stability and geometric properties of Fermat-Weber points in this context.

Findings

Fermat-Weber points are generally outside the ultrametric space but intersect with it.

Generalization of safety radius concepts to matroid ultrametrics.

Stability analysis of data analysis methods in combinatorial ultrametric spaces.

Abstract

We extend reconstruction methods for phylogenetic trees to ultrametrics of arbitrary matroids and study the stability of these data analysis methods in the combinatorial spirit of Andreas Dress. In particular, we generalize Atteson's work on the safety radius of phylogenetic reconstruction methods, as well as Gascuel and Steel's work on the stochastic safety radius, to arbitrary matroids. We also show that although the tropical Fermat-Weber points of an $M$-ultrametric sample are generally not contained in the space of $M$-ultrametrics, the intersection between the Fermat-Weber set and the space of $M$-ultrametrics is non-empty.