Title: Alpha shapes in kernel density estimation

Abstract: A foundational problem in topology and data is to determine the topological type (e.g. the persistent homology groups) of the superlevel sets $\mathcal{L}(a)=f^{-1}[e^{-a},\infty)$ of a sum of Gaussian kernels $f(x)=\sum_i a_i \exp(-\lVert x-x_i\rVert^2/2h^2)$ for $\{x_i\}\subset \mathbb{R}^d$. In this paper, we show that each $\mathcal{L}(a)$ coincides with the union of a certain power-shifted covering by balls, whose centers range over a closed subspace of the convex hull $\mathcal{L}(a)\subset conv(\{x_i\})$. We then present an explicit homotopy equivalence $p:\mathcal{L}(a)\rightarrow \mathcal{S}(a)$, realizing $\mathcal{S}(a)$ as a continuous version of the alpha shape. This leads to a prescription for modeling noisy point clouds by density-weighted alpha complexes which, in addition to computing persistent homology, give rise to refined geometric models. In order to compute alpha complexes in higher dimension, we used a recent algorithm due to the present authors based on the duality principle.