Probabilistic Analysis of Multiparameter Persistence Decompositions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913269979021312 |
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| author | Alonso, Ángel Javier Kerber, Michael Skraba, Primoz |
| author_facet | Alonso, Ángel Javier Kerber, Michael Skraba, Primoz |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_11939 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Probabilistic Analysis of Multiparameter Persistence Decompositions Alonso, Ángel Javier Kerber, Michael Skraba, Primoz Algebraic Topology Computational Geometry 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. |
| title | Probabilistic Analysis of Multiparameter Persistence Decompositions |
| topic | Algebraic Topology Computational Geometry |
| url | https://arxiv.org/abs/2403.11939 |