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-Cech 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 Cech, chromatic Delaunay-Cech, 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-Cech and chromatic Delaunay-Rips filtrations in applications, and we demonstrate their computational advantage with numerical experiments.
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