Delaunay-Rips filtration: a study and an algorithm

TL;DR

This paper analyzes the Delaunay-Rips filtration, a faster and more memory-efficient alternative to Rips filtration for low-dimensional Euclidean point clouds, providing theoretical insights and an efficient algorithm.

Contribution

It offers a thorough theoretical and empirical analysis, including approximation quality, stability, and an efficient algorithm with implementation for computing persistence diagrams.

Findings

Delaunay-Rips diagrams approximate Rips diagrams effectively.

The proposed algorithm is faster and uses less memory in low dimensions.

The implementation is available with Python bindings.

Abstract

The Delaunay-Rips filtration is a lighter and faster alternative to the well-known Rips filtration for low-dimensional Euclidean point clouds. Despite these advantages, it has seldom been studied. In this paper, we aim to bridge this gap by providing a thorough theoretical and empirical analysis of this construction. From a theoretical perspective, we show how the persistence diagrams associated with the Delaunay-Rips filtration approximate those obtained with the Rips filtration. Additionally, we describe the instabilities of the Delaunay-Rips persistence diagrams when the input point cloud is perturbed. Finally, we introduce an algorithm that computes persistence diagrams of Delaunay-Rips filtrations in any dimension. We show that our method is faster and has a lower memory footprint than traditional approaches in low dimensions. Our C++ implementation, which comes with Python bindings, is available at https://github.com/MClemot/GeoPH.