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Gromov Hyperbolicity, Geodesic Defect, and Apparent Pairs in Vietoris-Rips Filtrations

Authors Ulrich Bauer , Fabian Roll



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Ulrich Bauer
  • Department of Mathematics and Munich Data Science Institute, Technische Universität München, Germany
  • www.ulrich-bauer.org
Fabian Roll
  • Department of Mathematics, Technische Universität München, Germany

Acknowledgements

We thank Michael Bleher, Lukas Hahn, and Andreas Ott for stimulating discussions about applications of Vietoris-Rips complexes to the topological study of coronavirus evolution motivating our interest in the persistence of tree metrics, the organizers and participants of the AATRN Vietoris-Rips online seminar for sparking our interest in the Contractibility Lemma, and the anonymous reviewers for valuable feedback.

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Ulrich Bauer and Fabian Roll. Gromov Hyperbolicity, Geodesic Defect, and Apparent Pairs in Vietoris-Rips Filtrations. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 15:1-15:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.15

Abstract

Motivated by computational aspects of persistent homology for Vietoris–Rips filtrations, we generalize a result of Eliyahu Rips on the contractibility of Vietoris–Rips complexes of geodesic spaces for a suitable parameter depending on the hyperbolicity of the space. We consider the notion of geodesic defect to extend this result to general metric spaces in a way that is also compatible with the filtration. We further show that for finite tree metrics the Vietoris–Rips complexes collapse to their corresponding subforests. We relate our result to modern computational methods by showing that these collapses are induced by the apparent pairs gradient, which is used as an algorithmic optimization in Ripser, explaining its particularly strong performance on tree-like metric data.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Geometric topology
  • Mathematics of computing → Trees
  • Theory of computation → Computational geometry
Keywords
  • Vietoris–Rips complexes
  • persistent homology
  • discrete Morse theory
  • apparent pairs
  • hyperbolicity
  • geodesic defect
  • Ripser

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