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.2022.31
URN: urn:nbn:de:0030-drops-160398
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16039/
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Contessoto, Marco ; Mémoli, Facundo ; Stefanou, Anastasios ; Zhou, Ling

Persistent Cup-Length

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LIPIcs-SoCG-2022-31.pdf (1 MB)

Abstract

Cohomological ideas have recently been injected into persistent homology and have for example been used for accelerating the calculation of persistence diagrams by the software Ripser.
The cup product operation which is available at cohomology level gives rise to a graded ring structure that extends the usual vector space structure and is therefore able to extract and encode additional rich information. The maximum number of cocycles having non-zero cup product yields an invariant, the cup-length, which is useful for discriminating spaces.
In this paper, we lift the cup-length into the persistent cup-length function for the purpose of capturing ring-theoretic information about the evolution of the cohomology (ring) structure across a filtration. We show that the persistent cup-length function can be computed from a family of representative cocycles and devise a polynomial time algorithm for its computation. We furthermore show that this invariant is stable under suitable interleaving-type distances.

BibTeX - Entry

@InProceedings{contessoto_et_al:LIPIcs.SoCG.2022.31,
  author =	{Contessoto, Marco and M\'{e}moli, Facundo and Stefanou, Anastasios and Zhou, Ling},
  title =	{{Persistent Cup-Length}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{31:1--31:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16039},
  URN =		{urn:nbn:de:0030-drops-160398},
  doi =		{10.4230/LIPIcs.SoCG.2022.31},
  annote =	{Keywords: cohomology, cup product, persistence, cup length, Gromov-Hausdorff distance}
}

Keywords: cohomology, cup product, persistence, cup length, Gromov-Hausdorff distance
Collection: 38th International Symposium on Computational Geometry (SoCG 2022)
Issue Date: 2022
Date of publication: 01.06.2022

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