Universal Distances for Extended Persistence

TL;DR

This paper demonstrates that a more discriminative, universal stable distance exists for extended persistence diagrams, surpassing the bottleneck distance, with implications for Reeb graphs and sheaf theory.

Contribution

It introduces a universal stable distance for extended persistence diagrams and establishes its properties, contrasting it with the non-intrinsic interleaving distance.

Findings

A more discriminative, universal stable distance is identified.

The interleaving distance of sheaves and Reeb graphs is shown not to be intrinsic.

The paper develops a functorial construction based on relative interlevel set homology.

Abstract

The extended persistence diagram is an invariant of piecewise linear functions, which is known to be stable under perturbations of functions with respect to the bottleneck distance as introduced by Cohen-Steiner, Edelsbrunner, and Harer. We address the question of universality, which asks for the largest possible stable distance on extended persistence diagrams, showing that a more discriminative variant of the bottleneck distance is universal. Our result applies more generally to settings where persistence diagrams are considered only up to a certain degree. We achieve our results by establishing a functorial construction and several characteristic properties of relative interlevel set homology, which mirror the classical Eilenberg--Steenrod axioms. Finally, we contrast the bottleneck distance with the interleaving distance of sheaves on the real line by showing that the latter is not intrinsic, let alone universal. This particular result has the further implication that the interleaving distance of Reeb graphs is not intrinsic either.