A Finitely Stable Edit Distance for Merge Trees

TL;DR

This paper introduces a new edit distance for merge trees, demonstrating its stability and comparing it with existing metrics through theoretical analysis and practical simulations.

Contribution

It proposes a novel edit distance for merge trees and analyzes its stability properties, linking it to the 1-Wasserstein distance, with extensive comparisons to existing metrics.

Findings

The new metric is stable and comparable to the 1-Wasserstein distance.

The metric performs well in theoretical and practical evaluations.

Extensive comparison with existing metrics shows its advantages.

Abstract

In this paper we define a novel edit distance for merge trees, which we argue to be suitable for a good range of applications. Relying also on some technical results contained in other works, we investigate its stability properties, which end up being analogous to the ones of the 1-Wasserstein distance between persistence diagrams. In the appendix, we extensively compare our metric in relationship with other metrics appearing in the literature, with both theoretic and practical considerations and a simulation.