Intrinsic Interleaving Distance for Merge Trees

TL;DR

This paper studies the interleaving distance for merge trees, establishing its intrinsic nature on labeled and unlabeled trees, and provides algorithms for metric centers, advancing statistical analysis of topological summaries.

Contribution

It proves the intrinsic property of the interleaving distance for merge trees and introduces algorithms for constructing metric 1-centers, enabling statistical methods on these structures.

Findings

Interleaving distance is intrinsic on labeled merge trees.

Algorithm for constructing metric 1-centers for collections of merge trees.

Intrinsic property also holds for unlabeled merge trees.

Abstract

Merge trees are a type of graph-based topological summary that tracks the evolution of connected components in the sublevel sets of scalar functions. They enjoy widespread applications in data analysis and scientific visualization. In this paper, we consider the problem of comparing two merge trees via the notion of interleaving distance in the metric space setting. We investigate various theoretical properties of such a metric. In particular, we show that the interleaving distance is intrinsic on the space of labeled merge trees and provide an algorithm to construct metric 1-centers for collections of labeled merge trees. We further prove that the intrinsic property of the interleaving distance also holds for the space of unlabeled merge trees. Our results are a first step toward performing statistics on graph-based topological summaries.