TY - GEN
T1 - The Localized Union-Of-Balls Bifiltration
AU - Kerber, Michael
AU - Söls, Matthias
N1 - Publisher Copyright:
© Michael Kerber and Matthias Söls; licensed under Creative Commons License CC-BY 4.0.
PY - 2023/6/1
Y1 - 2023/6/1
N2 - We propose an extension of the classical union-of-balls filtration of persistent homology: fixing a point q, we focus our attention to a ball centered at q whose radius is controlled by a second scale parameter. We discuss an absolute variant, where the union is just restricted to the q-ball, and a relative variant where the homology of the q-ball relative to its boundary is considered. Interestingly, these natural constructions lead to bifiltered simplicial complexes which are not k-critical for any finite k. Nevertheless, we demonstrate that these bifiltrations can be computed exactly and efficiently, and we provide a prototypical implementation using the CGAL library. We also argue that some of the recent algorithmic advances for 2-parameter persistence (which usually assume k-criticality for some finite k) carry over to the ∞-critical case.
AB - We propose an extension of the classical union-of-balls filtration of persistent homology: fixing a point q, we focus our attention to a ball centered at q whose radius is controlled by a second scale parameter. We discuss an absolute variant, where the union is just restricted to the q-ball, and a relative variant where the homology of the q-ball relative to its boundary is considered. Interestingly, these natural constructions lead to bifiltered simplicial complexes which are not k-critical for any finite k. Nevertheless, we demonstrate that these bifiltrations can be computed exactly and efficiently, and we provide a prototypical implementation using the CGAL library. We also argue that some of the recent algorithmic advances for 2-parameter persistence (which usually assume k-criticality for some finite k) carry over to the ∞-critical case.
KW - Multi-Parameter Persistence
KW - Persistent Local Homology
KW - Topological Data Analysis
UR - http://www.scopus.com/inward/record.url?scp=85163524915&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.SoCG.2023.45
DO - 10.4230/LIPIcs.SoCG.2023.45
M3 - Conference paper
AN - SCOPUS:85163524915
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 39th International Symposium on Computational Geometry, SoCG 2023
A2 - Chambers, Erin W.
A2 - Gudmundsson, Joachim
PB - Schloss Dagstuhl - Leibniz-Zentrum für Informatik
T2 - 39th International Symposium on Computational Geometry
Y2 - 12 June 2023 through 15 June 2023
ER -