Stability and Machine Learning Applications of Persistent Homology Using the Delaunay-Rips Complex

TL;DR

This paper introduces the Delaunay-Rips complex for efficient persistent homology computation, analyzes its stability, and demonstrates its practical robustness and effectiveness in machine learning applications involving point cloud data.

Contribution

The paper defines and implements the Delaunay-Rips complex, providing theoretical stability analysis and empirical validation for its use in persistent homology and machine learning.

Findings

DR speeds up persistence calculations compared to other complexes.

DR persistence diagrams are stable under certain conditions.

In practice, DR performs comparably in ML tasks with other complexes.

Abstract

In this paper we define, implement, and investigate a simplicial complex construction for computing persistent homology of Euclidean point cloud data, which we call the Delaunay-Rips complex (DR). Assigning the Vietoris-Rips weights to simplices, DR experiences speed-up in the persistence calculations by only considering simplices that appear in the Delaunay triangulation of the point cloud. We document and compare a Python implementation of DR with other simplicial complex constructions for generating persistence diagrams. By imposing sufficient conditions on point cloud data, we are able to theoretically justify the stability of the persistence diagrams produced using DR. When the Delaunay triangulation of the point cloud changes under perturbations of the points, we prove that DR-produced persistence diagrams exhibit instability. Since we cannot guarantee that real-world data will satisfy our stability conditions, we demonstrate the practical robustness of DR for persistent homology in comparison with other simplicial complexes in machine learning applications. We find in our experiments that using DR for an ML-TDA pipeline performs comparatively well as using other simplicial complex constructions.