Decomposition of zero-dimensional persistence modules via rooted subsets
| Autor: | Alonso, Ángel Javier, Kerber, Michael |
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| Jazyk: | angličtina |
| Rok vydání: | 2023 |
| Předmět: |
Multiparameter persistent homology
Computational Geometry (cs.CG) FOS: Computer and information sciences Decomposition of persistence modules Mathematics of computing → Topology Theory of computation → Computational geometry FOS: Mathematics Elder Rule Algebraic Topology (math.AT) Computer Science - Computational Geometry Mathematics - Algebraic Topology Clustering |
| Popis: | 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, M\'emoli 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. Comment: 16 pages, 5 figures, 1 table |
| Databáze: | OpenAIRE |
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