Fast and accurate computation of ellipse tangency time for anisotropic persistent homology

Abstract

Persistent homology (PH) is a cornerstone of topological data analysis; however, standard filtrations often overlook the anisotropic structure in empirical data. We introduce Ellipse Cloud, a preprocessing pipeline that accentuates anisotropy within PH. The method builds a Vietoris–Rips filtration from ellipse tangency times rather than pairwise Euclidean distances, thereby accommodating anisotropy while preserving compatibility with existing computational tools. The central computational challenge—identifying the first contact times of expanding ellipses—is resolved with an efficient and provably stable numerical algorithm.