Tropical $k$-means clustering for phylogenetic trees

TL;DR

This paper introduces a novel clustering algorithm for equidistant phylogenetic trees using the asymmetric tropical distance, leveraging the mathematical structure of the Bergman fan of the complete graphical matroid.

Contribution

It defines and analyzes a new clustering method for phylogenetic trees based on tropical geometry and the Bergman fan, connecting tropical distances with phylogenetic tree space.

Findings

The algorithm effectively clusters phylogenetic trees based on tropical distances.

Mathematical analysis of the clustering method's properties.

Application to equidistant trees with N leaves.

Abstract

The asymmetric tropical distance is a distance measure on the tropical torus $\mathbb{R}^n/\mathbb{R}\mathbf{1}$ and in particular on the Bergman fan $B(K_N) \subseteq \mathbb{R}^{\binom{N}{2}}/\mathbb{R}\mathbf{1}$ of the complete graphical matroid. In this paper, we define and analyse a clustering algorithm for equidistant phylogenetic trees based on this distance, using the correspondence between $B(K_N)$ and the space of equidistant trees with $N$ leaves.