A Sparse Multicover Bifiltration of Linear Size
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866918060887113728 |
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| author | Alonso, Ángel Javier |
| author_facet | Alonso, Ángel Javier |
| contents | The $k$-cover of a point cloud $X$ in $\mathbb{R}^{d}$ at radius $r$ is the set of all points within distance $r$ of at least $k$ points of $X$. By varying $r$ and $k$ we obtain a two-parameter filtration known as the multicover bifiltration. This bifiltration has received attention recently due to being choice-free and robust to outliers. However, it is hard to compute: the smallest known equivalent simplicial bifiltration has $O(|X|^{d+1})$ simplices. We introduce a $(1+ε)$-approximation of the multicover bifiltration of linear size $O(|X|)$, for fixed $d$ and $ε$. The methods also apply to the subdivision Rips bifiltration on metric spaces of bounded doubling dimension, yielding analogous results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06986 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Sparse Multicover Bifiltration of Linear Size Alonso, Ángel Javier Computational Geometry Algebraic Topology The $k$-cover of a point cloud $X$ in $\mathbb{R}^{d}$ at radius $r$ is the set of all points within distance $r$ of at least $k$ points of $X$. By varying $r$ and $k$ we obtain a two-parameter filtration known as the multicover bifiltration. This bifiltration has received attention recently due to being choice-free and robust to outliers. However, it is hard to compute: the smallest known equivalent simplicial bifiltration has $O(|X|^{d+1})$ simplices. We introduce a $(1+ε)$-approximation of the multicover bifiltration of linear size $O(|X|)$, for fixed $d$ and $ε$. The methods also apply to the subdivision Rips bifiltration on metric spaces of bounded doubling dimension, yielding analogous results. |
| title | A Sparse Multicover Bifiltration of Linear Size |
| topic | Computational Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2411.06986 |