@inproceedings{8a3491b5ec8f49f297a259c74b0e4793,
title = "Gromov Hyperbolicity, Geodesic Defect, and Apparent Pairs in Vietoris-Rips Filtrations",
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.",
keywords = "Ripser, Vietoris-Rips complexes, apparent pairs, discrete Morse theory, geodesic defect, hyperbolicity, persistent homology",
author = "Ulrich Bauer and Fabian Roll",
note = "Publisher Copyright: {\textcopyright} Ulrich Bauer and Fabian Roll; licensed under Creative Commons License CC-BY 4.0; 38th International Symposium on Computational Geometry, SoCG 2022 ; Conference date: 07-06-2022 Through 10-06-2022",
year = "2022",
month = jun,
day = "1",
doi = "10.4230/LIPIcs.SoCG.2022.15",
language = "English",
series = "Leibniz International Proceedings in Informatics, LIPIcs",
publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",
editor = "Xavier Goaoc and Michael Kerber",
booktitle = "38th International Symposium on Computational Geometry, SoCG 2022",
}