Universal distances for extended persistence

Ulrich Bauer*, Magnus Bakke Botnan, Benedikt Fluhr

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

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.

Original languageEnglish
Pages (from-to)475-530
Number of pages56
JournalJournal of Applied and Computational Topology
Volume8
Issue number3
Early online date1 Jul 2024
DOIs
Publication statusPublished - Sept 2024

Bibliographical note

Publisher Copyright:
© The Author(s) 2024.

Funding

FundersFunder number
Deutsche Forschungsgemeinschaft

    Keywords

    • 55N31
    • 62R40
    • Bottleneck distance
    • Persistence diagrams
    • Persistent homology
    • Reeb graphs
    • Topological data analysis

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