Tropical Fermat-Weber points

TL;DR

This paper studies the computation of Fermat-Weber points under the tropical metric in a quotient space relevant to phylogenetics, revealing their convex polytope structure and conditions for uniqueness, with applications and numerical analysis.

Contribution

It characterizes the set of tropical Fermat-Weber points as a convex polytope and provides a combinatorial formula, advancing understanding in tropical geometry and phylogenetics.

Findings

The set of tropical Fermat-Weber points forms a convex polytope.

Conditions for the set to be a singleton are identified.

Numerical experiments illustrate the properties of these points in phylogenetic spaces.

Abstract

In a metric space, the Fermat-Weber points of a sample are statistics to measure the central tendency of the sample and it is well-known that the Fermat-Weber point of a sample is not necessarily unique in the metric space. We investigate the computation of Fermat-Weber points under the tropical metric on the quotient space $\mathbb{R}^{n} \!/ \mathbb{R} {\bf 1}$ with a fixed $n \in \mathbb{N}$, motivated by its application to the space of equidistant phylogenetic trees with $N$ leaves (in this case $n=\binom{N}{2}$) realized as the tropical linear space of all ultrametrics. We show that the set of all tropical Fermat-Weber points of a finite sample is always a classical convex polytope, and we present a combinatorial formula for a key value associated to this set. We identify conditions under which this set is a singleton. We apply numerical experiments to analyze the set of the tropical Fermat-Weber points within a space of phylogenetic trees. We discuss the issues in the computation of the tropical Fermat-Weber points.