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

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.