License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2023.51
URN: urn:nbn:de:0030-drops-179014
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17901/
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Mémoli, Facundo ; Zhou, Ling

Ephemeral Persistence Features and the Stability of Filtered Chain Complexes

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LIPIcs-SoCG-2023-51.pdf (0.9 MB)

Abstract

We strengthen the usual stability theorem for Vietoris-Rips (VR) persistent homology of finite metric spaces by building upon constructions due to Usher and Zhang in the context of filtered chain complexes. The information present at the level of filtered chain complexes includes ephemeral points, i.e. points with zero persistence, which provide additional information to that present at homology level. The resulting invariant, called verbose barcode, which has a stronger discriminating power than the usual barcode, is proved to be stable under certain metrics which are sensitive to these ephemeral points. In some situations, we provide ways to compute such metrics between verbose barcodes. We also exhibit several examples of finite metric spaces with identical (standard) VR barcodes yet with different verbose VR barcodes thus confirming that these ephemeral points strengthen the discriminating power of the standard VR barcode.

BibTeX - Entry

@InProceedings{memoli_et_al:LIPIcs.SoCG.2023.51,
  author =	{M\'{e}moli, Facundo and Zhou, Ling},
  title =	{{Ephemeral Persistence Features and the Stability of Filtered Chain Complexes}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{51:1--51:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/17901},
  URN =		{urn:nbn:de:0030-drops-179014},
  doi =		{10.4230/LIPIcs.SoCG.2023.51},
  annote =	{Keywords: filtered chain complexes, Vietoris-Rips complexes, barcode, bottleneck distance, matching distance, Gromov-Hausdorff distance}
}

Keywords: filtered chain complexes, Vietoris-Rips complexes, barcode, bottleneck distance, matching distance, Gromov-Hausdorff distance
Collection: 39th International Symposium on Computational Geometry (SoCG 2023)
Issue Date: 2023
Date of publication: 09.06.2023

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