Poisson approximation of large-lifetime cycles
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913623319773184 |
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| author | Hirsch, Christian Lundbye, Nikolaj Nyvold Otto, Moritz |
| author_facet | Hirsch, Christian Lundbye, Nikolaj Nyvold Otto, Moritz |
| contents | In topological data analysis, the notions of persistent homology, birthtime, lifetime, and deathtime are used to assign and capture relevant cycles (i.e., topological features) of a point cloud, such as loops and cavities. In particular, cycles with a large lifetime are of special interest. In this paper, we study such large-lifetime cycles when the point cloud is modeled as a Poisson point process. First, we consider the case with no bound on the deathtime, where we establish Poisson convergence of the centers of large-lifetime features on the 2-dimensional flat torus. Afterwards, by imposing a bound on the deathtime, we enter a sparse connectivity regime, and we prove joint Poisson convergence of the centers, lifetimes, and deathtimes in dimensions $d \geq 2$ under suitable model conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17482 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Poisson approximation of large-lifetime cycles Hirsch, Christian Lundbye, Nikolaj Nyvold Otto, Moritz Probability 55U10, 60D05, 60G55, 60F05, 60G70 In topological data analysis, the notions of persistent homology, birthtime, lifetime, and deathtime are used to assign and capture relevant cycles (i.e., topological features) of a point cloud, such as loops and cavities. In particular, cycles with a large lifetime are of special interest. In this paper, we study such large-lifetime cycles when the point cloud is modeled as a Poisson point process. First, we consider the case with no bound on the deathtime, where we establish Poisson convergence of the centers of large-lifetime features on the 2-dimensional flat torus. Afterwards, by imposing a bound on the deathtime, we enter a sparse connectivity regime, and we prove joint Poisson convergence of the centers, lifetimes, and deathtimes in dimensions $d \geq 2$ under suitable model conditions. |
| title | Poisson approximation of large-lifetime cycles |
| topic | Probability 55U10, 60D05, 60G55, 60F05, 60G70 |
| url | https://arxiv.org/abs/2412.17482 |