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Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets

Authors Ángel Javier Alonso , Michael Kerber

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ACM Subject Classification
  • Mathematics of computing → Topology
  • Theory of computation → Computational geometry
Keywords
  • Multiparameter persistent homology
  • Clustering
  • Decomposition of persistence modules
  • Elder Rule

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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.

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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) https://doi.org/10.4230/LIPIcs.SoCG.2023.7

Author Details

Ángel Javier Alonso
  • Technische Universität Graz, Austria
Michael Kerber
  • Technische Universität Graz, Austria

Funding

This research has been supported by the Austrian Science Fund (FWF) grant P 33765-N.

Acknowledgements

The authors thank Jan Jendrysiak for helpful discussions. We are also grateful to the anonymous reviewers for their careful reading of our manuscript and their detailed comments and suggestions.

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