Convexity in Tree Spaces

TL;DR

This paper explores the geometric and convexity properties of the space of phylogenetic trees, comparing tropical and CAT(0) metrics, and highlights the advantages of tropical convexity for statistical applications.

Contribution

It provides a detailed analysis of convexity structures in tree spaces, emphasizing the benefits of tropical metrics over CAT(0) metrics for geometric statistics.

Findings

Tropical convexity exhibits favorable properties for statistical analysis.

Geodesic triangles in CAT(0) space can have arbitrarily high dimension.

Tropical metrics have geodesics of small depth, aiding statistical methods.

Abstract

We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all \ ultrametrics. The ${\rm CAT}(0)$-metric of Billera-Holmes-Vogtman arises from the theory of orthant spaces. While its geodesics can be computed by the Owen-Provan algorithm, geodesic triangles are complicated. We show that the dimension of such a triangle can be arbitrarily high. Tropical convexity and the tropical metric behave better. They exhibit properties desirable for geometric statistics, such as geodesics of small depth.