Maximum Persistent Betti Numbers of Čech Complexes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918348494733312 |
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| author | Edelsbrunner, Herbert Kahle, Matthew Kanazawa, Shu |
| author_facet | Edelsbrunner, Herbert Kahle, Matthew Kanazawa, Shu |
| contents | This note proves that only a linear number of holes in a Čech complex of $n$ points in $\mathbb{R}^d$ can persist over an interval of constant length. Specifically, for any fixed dimension $p < d$ and fixed $\varepsilon > 0$, the number of $p$-dimensional holes in the Čech complex at radius $1$ that persist to radius $1 + \varepsilon$ is bounded above by a constant times $n$, where $n$ is the number of points. The proof uses a packing argument supported by relating the Čech complexes with corresponding snap complexes over the cells in a partition of space. The argument is self-contained and elementary, relying on geometric and combinatorial constructions rather than on the existing theory of sparse approximations or interleavings. The bound also applies to Alpha complexes and Vietoris-Rips complexes. While our result can be inferred from prior work on sparse filtrations, to our knowledge, no explicit statement or direct proof of this bound appears in the literature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_05241 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximum Persistent Betti Numbers of Čech Complexes Edelsbrunner, Herbert Kahle, Matthew Kanazawa, Shu Combinatorics 55U10, 55N31, 52C45 This note proves that only a linear number of holes in a Čech complex of $n$ points in $\mathbb{R}^d$ can persist over an interval of constant length. Specifically, for any fixed dimension $p < d$ and fixed $\varepsilon > 0$, the number of $p$-dimensional holes in the Čech complex at radius $1$ that persist to radius $1 + \varepsilon$ is bounded above by a constant times $n$, where $n$ is the number of points. The proof uses a packing argument supported by relating the Čech complexes with corresponding snap complexes over the cells in a partition of space. The argument is self-contained and elementary, relying on geometric and combinatorial constructions rather than on the existing theory of sparse approximations or interleavings. The bound also applies to Alpha complexes and Vietoris-Rips complexes. While our result can be inferred from prior work on sparse filtrations, to our knowledge, no explicit statement or direct proof of this bound appears in the literature. |
| title | Maximum Persistent Betti Numbers of Čech Complexes |
| topic | Combinatorics 55U10, 55N31, 52C45 |
| url | https://arxiv.org/abs/2409.05241 |