Universality in Random Persistent Homology and Scale-Invariant Functionals
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914908359098368 |
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| author | Bobrowski, Omer Skraba, Primoz |
| author_facet | Bobrowski, Omer Skraba, Primoz |
| contents | In this paper, we prove a universality result for the limiting distribution of persistence diagrams arising from geometric filtrations over random point processes. Specifically, we consider the distribution of the ratio of persistence values (death/birth), and show that for fixed dimension, homological degree and filtration type (Cech or Vietoris-Rips), the limiting distribution is independent of the underlying point process distribution, i.e., universal. In proving this result, we present a novel general framework for universality in scale-invariant functionals on point processes. Finally, we also provide a number of new results related to Morse theory in random geometric complexes, which may be of an independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05553 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Universality in Random Persistent Homology and Scale-Invariant Functionals Bobrowski, Omer Skraba, Primoz Probability Algebraic Topology In this paper, we prove a universality result for the limiting distribution of persistence diagrams arising from geometric filtrations over random point processes. Specifically, we consider the distribution of the ratio of persistence values (death/birth), and show that for fixed dimension, homological degree and filtration type (Cech or Vietoris-Rips), the limiting distribution is independent of the underlying point process distribution, i.e., universal. In proving this result, we present a novel general framework for universality in scale-invariant functionals on point processes. Finally, we also provide a number of new results related to Morse theory in random geometric complexes, which may be of an independent interest. |
| title | Universality in Random Persistent Homology and Scale-Invariant Functionals |
| topic | Probability Algebraic Topology |
| url | https://arxiv.org/abs/2406.05553 |