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The Localized Union-Of-Balls Bifiltration

Authors Michael Kerber , Matthias Söls

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Algebraic topology
Keywords
  • Topological Data Analysis
  • Multi-Parameter Persistence
  • Persistent Local Homology

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Abstract

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.

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Michael Kerber and Matthias Söls. The Localized Union-Of-Balls Bifiltration. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 45:1-45:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.SoCG.2023.45

Author Details

Michael Kerber
  • Institute of Geometry, Technische Universität Graz, Austria
Matthias Söls
  • Institute of Geometry, Technische Universität Graz, Austria

Funding

The authors acknowledge the support of the Austrian Science Fund (FWF).
  • Kerber, Michael: Austrian Science Fund (FWF) grant P 33765-N.
  • Söls, Matthias: Austrian Science Fund (FWF): grant W1230.

Acknowledgements

The authors thank Anton Gfrerrer and Thomas Pock for helpful discussions.

Supplementary Materials

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