Mathematics > Algebraic Topology
[Submitted on 21 Mar 2023 (v1), last revised 30 Apr 2024 (this version, v3)]
Title:Alpha shapes in kernel density estimation
View PDF HTML (experimental)Abstract: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.
Submission history
From: Erik Carlsson [view email][v1] Tue, 21 Mar 2023 22:22:26 UTC (5,171 KB)
[v2] Wed, 27 Mar 2024 17:12:52 UTC (16,507 KB)
[v3] Tue, 30 Apr 2024 23:20:01 UTC (17,538 KB)
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