TY - JOUR
T1 - Universal distances for extended persistence
AU - Bauer, Ulrich
AU - Botnan, Magnus Bakke
AU - Fluhr, Benedikt
N1 - Publisher Copyright:
© The Author(s) 2024.
PY - 2024/9
Y1 - 2024/9
N2 - 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.
AB - 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.
KW - 55N31
KW - 62R40
KW - Bottleneck distance
KW - Persistence diagrams
KW - Persistent homology
KW - Reeb graphs
KW - Topological data analysis
UR - http://www.scopus.com/inward/record.url?scp=85197281438&partnerID=8YFLogxK
U2 - 10.1007/s41468-024-00184-7
DO - 10.1007/s41468-024-00184-7
M3 - Article
AN - SCOPUS:85197281438
SN - 2367-1726
VL - 8
SP - 475
EP - 530
JO - Journal of Applied and Computational Topology
JF - Journal of Applied and Computational Topology
IS - 3
ER -