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DOI: 10.4230/LIPIcs.SoCG.2024.6

Probabilistic Analysis of Multiparameter Persistence Decompositions into Intervals

Authors: Ángel Javier Alonso, Michael Kerber, and Primoz Skraba

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract

Multiparameter persistence modules can be uniquely decomposed into indecomposable summands. Among these indecomposables, intervals stand out for their simplicity, making them preferable for their ease of interpretation in practical applications and their computational efficiency. Empirical observations indicate that modules that decompose into only intervals are rare. To support this observation, we show that for numerous common multiparameter constructions, such as density- or degree-Rips bifiltrations, and across a general category of point samples, the probability of the homology-induced persistence module decomposing into intervals goes to zero as the sample size goes to infinity.

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Ángel Javier Alonso, Michael Kerber, and Primoz Skraba. Probabilistic Analysis of Multiparameter Persistence Decompositions into Intervals. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{alonso_et_al:LIPIcs.SoCG.2024.6,
  author =	{Alonso, \'{A}ngel Javier and Kerber, Michael and Skraba, Primoz},
  title =	{{Probabilistic Analysis of Multiparameter Persistence Decompositions into Intervals}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{6:1--6:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.6},
  URN =		{urn:nbn:de:0030-drops-199510},
  doi =		{10.4230/LIPIcs.SoCG.2024.6},
  annote =	{Keywords: Topological Data Analysis, Multi-Parameter Persistence, Decomposition of persistence modules, Poisson point processes}
}

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Probabilistic Analysis of Multiparameter Persistence Decompositions into Intervals

Abstract

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