Alpha Shapes and Optimal Transport on the Sphere

Abstract In [3], the authors used the Legendre transform to give a tractable method for studying Topological Data Analysis (TDA) in terms of sums of Gaussian kernels. In this paper, we prove a variant for sums of cosine similarity-based kernel functions, which requires considering the more general “ c -transform” from optimal transport theory [18]. We then apply these methods to a point cloud arising from a recent breakthrough study, which exhibits a toroidal structure in the brain activity of rats [12]. A key part of this application is that the transport map and transformed density function arising from the theorem replace certain delicate preprocessing steps related to density-based denoising and subsampling.