Morse theory for chromatic Delaunay triangulations

TL;DR

This paper introduces chromatic Delaunay-cech and Delaunay-Rips filtrations as efficient tools for topological data analysis of multi-species point clouds, establishing their theoretical relationships and stability.

Contribution

It generalizes discrete Morse theory to chromatic filtrations and demonstrates computational advantages and stability for multi-class data analysis.

Findings

Chromatic Delaunay-cech and Delaunay-Rips filtrations are related to chromatic alpha filtration via simplicial collapses.

The chromatic Delaunay-Rips filtration is locally stable to point cloud perturbations.

Numerical experiments show computational efficiency of the proposed filtrations.

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.