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Documents authored by Alonso, Ángel Javier

Document
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.7
Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets

Authors: Ángel Javier Alonso and Michael Kerber

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
We study the decomposition of zero-dimensional persistence modules, viewed as functors valued in the category of vector spaces factorizing through sets. Instead of working directly at the level of vector spaces, we take a step back and first study the decomposition problem at the level of sets. This approach allows us to define the combinatorial notion of rooted subsets. In the case of a filtered metric space M, rooted subsets relate the clustering behavior of the points of M with the decomposition of the associated persistence module. In particular, we can identify intervals in such a decomposition quickly. In addition, rooted subsets can be understood as a generalization of the elder rule, and are also related to the notion of constant conqueror of Cai, Kim, Mémoli and Wang. As an application, we give a lower bound on the number of intervals that we can expect in the decomposition of zero-dimensional persistence modules of a density-Rips filtration in Euclidean space: in the limit, and under very general circumstances, we can expect that at least 25% of the indecomposable summands are interval modules.
Cite as

Ángel Javier Alonso and Michael Kerber. Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{alonso_et_al:LIPIcs.SoCG.2023.7,
  author =	{Alonso, \'{A}ngel Javier and Kerber, Michael},
  title =	{{Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{7:1--7:16},
  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/entities/document/10.4230/LIPIcs.SoCG.2023.7},
  URN =		{urn:nbn:de:0030-drops-178570},
  doi =		{10.4230/LIPIcs.SoCG.2023.7},
  annote =	{Keywords: Multiparameter persistent homology, Clustering, Decomposition of persistence modules, Elder Rule}
}
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