The $\ell^\infty$-Cophenetic Metric for Phylogenetic Trees as an Interleaving Distance

TL;DR

This paper demonstrates that the $\, ext{ extlbrackdbl} ext{ extperthousand} ext{ extbrackdbl} ext{-cophenetic}$$ metric for phylogenetic trees can be viewed as an interleaving distance, unifying biological tree comparison with topological data analysis concepts.

Contribution

It establishes that the $\, ext{ extlbrackdbl} ext{ extperthousand} ext{ extbrackdbl} ext{-cophenetic}$$ metric is an interleaving distance by framing phylogenetic trees categorically and showing an isometric embedding.

Findings

The $\, ext{ extlbrackdbl} ext{ extperthousand} ext{ extbrackdbl} ext{-cophenetic}$$ metric is an interleaving distance.

The cophenetic vector map is an isometric embedding.

Categorical framework for phylogenetic trees aligns with topological data analysis.

Abstract

There are many metrics available to compare phylogenetic trees since this is a fundamental task in computational biology. In this paper, we focus on one such metric, the $\ell^\infty$-cophenetic metric introduced by Cardona et al. This metric works by representing a phylogenetic tree with $n$ labeled leaves as a point in $\mathbb{R}^{n(n+1)/2}$ known as the cophenetic vector, then comparing the two resulting Euclidean points using the $\ell^\infty$ distance. Meanwhile, the interleaving distance is a formal categorical construction generalized from the definition of Chazal et al., originally introduced to compare persistence modules arising from the field of topological data analysis. We show that the $\ell^\infty$-cophenetic metric is an example of an interleaving distance. To do this, we define phylogenetic trees as a category of merge trees with some additional structure; namely labelings on the leaves plus a requirement that morphisms respect these labels. Then we can use the definition of a flow on this category to give an interleaving distance. Finally, we show that, because of the additional structure given by the categories defined, the map sending a labeled merge tree to the cophenetic vector is, in fact, an isometric embedding, thus proving that the $\ell^\infty$-cophenetic metric is, in fact, an interleaving distance.