Alpha shapes in kernel density estimation

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.