Alpha shapes in kernel density estimation
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911860885815296 |
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| author | Carlsson, Erik Carlsson, John |
| author_facet | Carlsson, Erik Carlsson, John |
| contents | For every Gaussian kernel density estimator $f(x)=\sum_i a_i \exp(-\lVert x-x_i\rVert^2/2h^2)$ associated to a point cloud $\mathcal{D}=\{x_1,...,x_N\}\subset \mathbb{R}^d$, we define a nested family of closed subspaces $\mathcal{S}(a)\subset\mathbb{R}^d$, which we interpret as a continuous version of an alpha shape. Using arguments based on Fenchel duality, we prove that $\mathcal{S}(a)$ is homotopy equivalent to the superlevel set $\mathcal{L}(a)=f^{-1}[e^{-a},\infty)$, and that $\mathcal{L}(a)$ can be realized as the union of a certain power-shifted covering by balls with centers in $\mathcal{S}(a)$. By extracting finite alpha complexes with vertices in $\mathcal{S}(a)$, we obtain refined geometric models of noisy point clouds, as well as density-filtered persistent homology calculations. In order to compute alpha complexes in higher dimension, we used a recent algorithm due to the present authors based on the duality principle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_12213 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Alpha shapes in kernel density estimation Carlsson, Erik Carlsson, John Algebraic Topology 55N31, 62R40, 62G07 For every Gaussian kernel density estimator $f(x)=\sum_i a_i \exp(-\lVert x-x_i\rVert^2/2h^2)$ associated to a point cloud $\mathcal{D}=\{x_1,...,x_N\}\subset \mathbb{R}^d$, we define a nested family of closed subspaces $\mathcal{S}(a)\subset\mathbb{R}^d$, which we interpret as a continuous version of an alpha shape. Using arguments based on Fenchel duality, we prove that $\mathcal{S}(a)$ is homotopy equivalent to the superlevel set $\mathcal{L}(a)=f^{-1}[e^{-a},\infty)$, and that $\mathcal{L}(a)$ can be realized as the union of a certain power-shifted covering by balls with centers in $\mathcal{S}(a)$. By extracting finite alpha complexes with vertices in $\mathcal{S}(a)$, we obtain refined geometric models of noisy point clouds, as well as density-filtered persistent homology calculations. In order to compute alpha complexes in higher dimension, we used a recent algorithm due to the present authors based on the duality principle. |
| title | Alpha shapes in kernel density estimation |
| topic | Algebraic Topology 55N31, 62R40, 62G07 |
| url | https://arxiv.org/abs/2303.12213 |