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

arXiv: https://arxiv.org/abs/2303.01501 · HTML: https://mlsanity.com/paper/2303.01501/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 50 references that are in the map.
- First posted on arXiv 2 Mar 2023.

## Builds on (most cited first, 10 of 10)

- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodes](https://mlsanity.com/paper/2003.07989/) (2020)
- [A persistent homology approach to heart rate variability analysis with an application to sleep-wake classification](https://mlsanity.com/paper/1908.06856/) (2019)
- [Persistent Homology of Attractors For Action Recognition](https://mlsanity.com/paper/1603.05310/) (2016)
- [The Morse theory of vCech and Delaunay complexes](https://mlsanity.com/paper/1312.1231/) (2013)
- [Sleep-wake classification via quantifying heart rate variability by convolutional neural network](https://mlsanity.com/paper/1808.00142/) (2018)
- [The Stability of Delaunay Triangulations](https://mlsanity.com/paper/1304.2947/) (2013)

## Built on by (most cited first, 6 of 6)

- [The Flood Complex: Large-Scale Persistent Homology on Millions of Points](https://mlsanity.com/paper/2509.22432/) (2025)
- [A Pipeline for Data-Driven Learning of Topological Features with Applications to Protein Stability Prediction](https://mlsanity.com/paper/2408.04847/) (2024)
- [Topology of Currencies: Persistent Homology for FX Co-movements: A Comparative Clustering Study](https://mlsanity.com/paper/2510.19306/) (2025)
- [TopoPrune: Robust Data Pruning via Unified Latent Space Topology](https://mlsanity.com/paper/2602.02739/) (2026)
- [A Persistent Homology Design Space for 3D Point Cloud Deep Learning](https://mlsanity.com/paper/2604.04299/) (2026)
- [The sublevel Flood bifiltration: towards scalable 2-parameter persistent homology](https://mlsanity.com/paper/2610.05441/) (2026)

## Abstract (arXiv)

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.