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

Abhinav Natarajan

Doctoral student in mathematics

My research interests are in applied algebraic topology and geometry, statistics, and machine learning.