# A roadmap for the computation of persistent homology

arXiv: https://arxiv.org/abs/1506.08903 · HTML: https://mlsanity.com/paper/1506.08903/

## Key facts

- Cited by 127 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jun 2015.

## Built on by (most cited first, 30 of 127)

- [Topologically faithful image segmentation via induced matching of persistence barcodes](https://mlsanity.com/paper/2211.15272/) (2022)
- [A persistent homology-based topological loss for CNN-based multi-class segmentation of CMR](https://mlsanity.com/paper/2107.12689/) (2021)
- [On the effectiveness of persistent homology](https://mlsanity.com/paper/2206.10551/) (2022)
- [On the Expressivity of Persistent Homology in Graph Learning](https://mlsanity.com/paper/2302.09826/) (2023)
- [ToDD: Topological Compound Fingerprinting in Computer-Aided Drug Discovery](https://mlsanity.com/paper/2211.03808/) (2022)
- [Topology-Aware Latent Diffusion for 3D Shape Generation](https://mlsanity.com/paper/2401.17603/) (2024)
- [Topology-Aware Focal Loss for 3D Image Segmentation](https://mlsanity.com/paper/2304.12223/) (2023)
- [Improving Metric Dimensionality Reduction with Distributed Topology](https://mlsanity.com/paper/2106.07613/) (2021)
- [Topological Signatures vs. Gradient Histograms: A Comparative Study for Medical Image Classification](https://mlsanity.com/paper/2507.03006/) (2025)
- [Topological Measurement of Deep Neural Networks Using Persistent Homology](https://mlsanity.com/paper/2106.03016/) (2021)
- [Dirac signal processing of higher-order topological signals](https://mlsanity.com/paper/2301.10137/) (2023)
- [Topological Pooling on Graphs](https://mlsanity.com/paper/2303.14543/) (2023)
- [The decomposition of the higher-order homology embedding constructed from the k-Laplacian](https://mlsanity.com/paper/2107.10970/) (2021)
- [Approximating Persistent Homology for Large Datasets](https://mlsanity.com/paper/2204.09155/) (2022)
- [Adaptive Topological Feature via Persistent Homology: Filtration Learning for Point Clouds](https://mlsanity.com/paper/2307.09259/) (2023)
- [Graphcode: Learning from multiparameter persistent homology using graph neural networks](https://mlsanity.com/paper/2405.14302/) (2024)
- [Helmholtzian Eigenmap: Topological feature discovery & edge flow learning from point cloud data](https://mlsanity.com/paper/2103.07626/) (2021)
- [Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models](https://mlsanity.com/paper/2505.18230/) (2025)
- [Quantitative Performance Assessment of CNN Units via Topological Entropy Calculation](https://mlsanity.com/paper/2103.09716/) (2021)
- [Minimal Cycle Representatives in Persistent Homology using Linear Programming: an Empirical Study with User's Guide](https://mlsanity.com/paper/2105.07025/) (2021)
- [Stability and Machine Learning Applications of Persistent Homology Using the Delaunay-Rips Complex](https://mlsanity.com/paper/2303.01501/) (2023)
- [Improving Self-supervised Molecular Representation Learning using Persistent Homology](https://mlsanity.com/paper/2311.17327/) (2023)
- [Topologically Regularized Data Embeddings](https://mlsanity.com/paper/2110.09193/) (2021)
- [Fast Topological Clustering with Wasserstein Distance](https://mlsanity.com/paper/2112.00101/) (2021)
- [The Shape of Adversarial Influence: Characterizing LLM Latent Spaces with Persistent Homology](https://mlsanity.com/paper/2505.20435/) (2025)
- [Four-Stage Alzheimer's Disease Classification from MRI Using Topological Feature Extraction, Feature Selection, and Ensemble Learning](https://mlsanity.com/paper/2601.00918/) (2026)
- [Persistent Homology for Breast Tumor Classification using Mammogram Scans](https://mlsanity.com/paper/2201.02295/) (2022)
- [Position-prior Clustering-based Self-attention Module for Knee Cartilage Segmentation](https://mlsanity.com/paper/2206.10286/) (2022)
- [Synthetic Data Generation and Deep Learning for the Topological Analysis of 3D Data](https://mlsanity.com/paper/2309.16968/) (2023)
- [EMP: Effective Multidimensional Persistence for Graph Representation Learning](https://mlsanity.com/paper/2401.13713/) (2024)

## Abstract (arXiv)

Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. The computation of PH is an open area with numerous important and fascinating challenges. The field of PH computation is evolving rapidly, and new algorithms and software implementations are being updated and released at a rapid pace. The purposes of our article are to (1) introduce theory and computational methods for PH to a broad range of computational scientists and (2) provide benchmarks of state-of-the-art implementations for the computation of PH. We give a friendly introduction to PH, navigate the pipeline for the computation of PH with an eye towards applications, and use a range of synthetic and real-world data sets to evaluate currently available open-source implementations for the computation of PH. Based on our benchmarking, we indicate which algorithms and implementations are best suited to different types of data sets. In an accompanying tutorial, we provide guidelines for the computation of PH. We make publicly available all scripts that we wrote for the tutorial, and we make available the processed version of the data sets used in the benchmarking.