eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-06-09
7:1
7:16
10.4230/LIPIcs.SoCG.2023.7
article
Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets
Alonso, Ángel Javier
1
https://orcid.org/0000-0002-5822-546X
Kerber, Michael
1
https://orcid.org/0000-0002-8030-9299
Technische Universität Graz, Austria
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol258-socg2023/LIPIcs.SoCG.2023.7/LIPIcs.SoCG.2023.7.pdf
Multiparameter persistent homology
Clustering
Decomposition of persistence modules
Elder Rule