Delaunay-Rips filtration: a study and an algorithm

2025-12-19

AI Summary

For low-dimensional Euclidean point clouds, this paper introduces the Delaunay–Rips filtration (DRF), a novel topological summarization tool that balances theoretical guarantees with computational efficiency, serving as a scalable alternative to the computationally expensive Vietoris–Rips filtration. Theoretically, we establish the first provable approximation error bound between DRF and the Vietoris–Rips filtration and analyze its stability under input perturbations. Algorithmically, we propose the first general-purpose persistent homology computation framework supporting arbitrary dimensions, built upon Delaunay complex pruning and incremental updates. Our C++ implementation—with a Python interface—achieves several-fold speedups over standard Rips-based methods on low-dimensional data, while substantially reducing memory consumption. The open-source code is publicly available on GitHub.

Technology Category

• Computer Vision: Other Foundations of Computer Vision
• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Data Visualization & Summarization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Data and user privacy-enhancing technologies for the Web
• Web Mining and Content Analysis: Web data integration and cleaning

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.

Problem

• Analyzing Delaunay-Rips filtration for low-dimensional Euclidean point clouds
• Comparing its persistence diagrams with those from Rips filtration
• Developing an efficient algorithm to compute its persistence diagrams

Innovation

• Delaunay-Rips filtration is faster and lighter than Rips filtration
• Algorithm computes persistence diagrams in any dimension efficiently
• Implementation includes C++ code with Python bindings for accessibility