Morse Theory for Chromatic Delaunay Triangulations

Abstract: The chromatic alpha filtration is a generalization of the alpha filtration that can encode spatial relationships among classes of labelled point cloud data, and has applications in topological data analysis of multi-species data. In this paper we introduce the chromatic Delaunay--\v{C}ech and chromatic Delaunay--Rips filtrations, which are computationally favourable alternatives to the chromatic alpha filtration. We use generalized discrete Morse theory to show that the \v{C}ech, chromatic Delaunay--\v{C}ech, and chromatic alpha filtrations are related by simplicial collapses. Our result generalizes a result of Bauer and Edelsbrunner from the non-chromatic to the chromatic setting. We also show that the chromatic Delaunay--Rips filtration is locally stable to perturbations of the underlying point cloud. Our results provide theoretical justification for the use of chromatic Delaunay--\v{C}ech and chromatic Delaunay--Rips filtrations in applications, and we demonstrate their computational advantage with numerical experiments.