The Universal $\ell^p$-Metric on Merge Trees

TL;DR

This paper introduces a new $ ext{ell}^p$-type metric on merge trees, extending interleaving distances, and proves its stability and universality properties, providing a unified framework for comparing merge trees.

Contribution

It defines a universal $ ext{ell}^p$-metric on merge trees, generalizing existing distances and establishing stability and universality results across all $p$ values.

Findings

The $ ext{ell}^p$-metric is a true metric on merge trees.

The distance upper-bounds the $p$-Wasserstein distance between barcodes.

The metric is stable under cellular sublevel filtrations.

Abstract

Adapting a definition given by Bjerkevik and Lesnick for multiparameter persistence modules, we introduce an $\ell^p$-type extension of the interleaving distance on merge trees. We show that our distance is a metric, and that it upper-bounds the $p$-Wasserstein distance between the associated barcodes. For each $p\in[1,\infty]$, we prove that this distance is stable with respect to cellular sublevel filtrations and that it is the universal (i.e., largest) distance satisfying this stability property. In the $p=\infty$ case, this gives a novel proof of universality for the interleaving distance on merge trees.