Morse Theory for the k-NN Distance Function
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866917617996922880 |
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| author | Reani, Yohai Bobrowski, Omer |
| author_facet | Reani, Yohai Bobrowski, Omer |
| contents | We study the $k$-th nearest neighbor distance function from a finite point-set in $\mathbb{R}^d$. We provide a Morse theoretic framework to analyze the sub-level set topology. In particular, we present a simple combinatorial-geometric characterization for critical points and their indices, along with detailed information about the possible changes in homology at the critical levels. We conclude by computing the expected number of critical points for a homogeneous Poisson process. Our results deliver significant insights and tools for the analysis of persistent homology in order-$k$ Delaunay mosaics, and random $k$-fold coverage. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_12792 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Morse Theory for the k-NN Distance Function Reani, Yohai Bobrowski, Omer Computational Geometry Algebraic Topology Combinatorics We study the $k$-th nearest neighbor distance function from a finite point-set in $\mathbb{R}^d$. We provide a Morse theoretic framework to analyze the sub-level set topology. In particular, we present a simple combinatorial-geometric characterization for critical points and their indices, along with detailed information about the possible changes in homology at the critical levels. We conclude by computing the expected number of critical points for a homogeneous Poisson process. Our results deliver significant insights and tools for the analysis of persistent homology in order-$k$ Delaunay mosaics, and random $k$-fold coverage. |
| title | Morse Theory for the k-NN Distance Function |
| topic | Computational Geometry Algebraic Topology Combinatorics |
| url | https://arxiv.org/abs/2403.12792 |