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A roadmap for the computation of persistent homology

arXiv 1506.08903 ↗First posted 30 Jun 2015math.AT cs.CG physics.data-ancited by 127

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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.

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

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 18; 2022: 25; 2023: 20; 2024: 16; 2025: 30; 2026: 18.

127 papers.

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  1. 2022Topologically faithful image segmentation via induced matching of persistence barcodescited by 49
  2. 2021A persistent homology-based topological loss for CNN-based multi-class segmentation of CMRcited by 21
  3. 2022On the effectiveness of persistent homologycited by 20
  4. 2023On the Expressivity of Persistent Homology in Graph Learningcited by 19
  5. 2022ToDD: Topological Compound Fingerprinting in Computer-Aided Drug Discoverycited by 13
  6. 2024Topology-Aware Latent Diffusion for 3D Shape Generationcited by 13
  7. 2023Topology-Aware Focal Loss for 3D Image Segmentationcited by 12
  8. 2021Improving Metric Dimensionality Reduction with Distributed Topologycited by 11
  9. 2025Topological Signatures vs. Gradient Histograms: A Comparative Study for Medical Image Classificationcited by 10
  10. 2021Topological Measurement of Deep Neural Networks Using Persistent Homologycited by 9
  11. 2023Dirac signal processing of higher-order topological signalscited by 9
  12. 2023Topological Pooling on Graphscited by 9
38 more
  1. 2021The decomposition of the higher-order homology embedding constructed from the k-Laplaciancited by 8
  2. 2022Approximating Persistent Homology for Large Datasetscited by 8
  3. 2023Adaptive Topological Feature via Persistent Homology: Filtration Learning for Point Cloudscited by 8
  4. 2024Graphcode: Learning from multiparameter persistent homology using graph neural networkscited by 8
  5. 2021Helmholtzian Eigenmap: Topological feature discovery & edge flow learning from point cloud datacited by 7
  6. 2025Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Modelscited by 7
  7. 2021Quantitative Performance Assessment of CNN Units via Topological Entropy Calculationcited by 6
  8. 2021Minimal Cycle Representatives in Persistent Homology using Linear Programming: an Empirical Study with User's Guidecited by 6
  9. 2023Stability and Machine Learning Applications of Persistent Homology Using the Delaunay-Rips Complexcited by 6
  10. 2023Improving Self-supervised Molecular Representation Learning using Persistent Homologycited by 6
  11. 2021Topologically Regularized Data Embeddingscited by 5
  12. 2021Fast Topological Clustering with Wasserstein Distancecited by 5
  13. 2025The Shape of Adversarial Influence: Characterizing LLM Latent Spaces with Persistent Homologycited by 5
  14. 2026Four-Stage Alzheimer's Disease Classification from MRI Using Topological Feature Extraction, Feature Selection, and Ensemble Learningcited by 5
  15. 2022Persistent Homology for Breast Tumor Classification using Mammogram Scanscited by 4
  16. 2022Position-prior Clustering-based Self-attention Module for Knee Cartilage Segmentationcited by 4
  17. 2023Synthetic Data Generation and Deep Learning for the Topological Analysis of 3D Datacited by 4
  18. 2024EMP: Effective Multidimensional Persistence for Graph Representation Learningcited by 4
  19. 2024MOUNTAINEER: Topology-Driven Visual Analytics for Comparing Local Explanationscited by 4
  20. 2022Classification of Stochastic Processes with Topological Data Analysiscited by 3
  21. 2023Singular value decomposition based matrix surgerycited by 3
  22. 2024Cracking Factual Knowledge: A Comprehensive Analysis of Degenerate Knowledge Neurons in Large Language Modelscited by 3
  23. 2024Conditional Uncertainty Quantification for Tensorized Topological Neural Networkscited by 3
  24. 2025Time-Series Forecasting via Topological Information Supervised Framework with Efficient Topological Feature Learningcited by 3
  25. 2021Time Series Classification via Topological Data Analysiscited by 2
  26. 2021Classification based on Topological Data Analysiscited by 2
  27. 2021Classification of COVID-19 via Homology of CT-SCANcited by 2
  28. 2021A Multi-parameter Persistence Framework for Mathematical Morphologycited by 2
  29. 2021Topological Information Retrieval with Dilation-Invariant Bottleneck Comparative Measurescited by 2
  30. 2022Topological EEG Nonlinear Dynamics Analysis for Emotion Recognitioncited by 2
  31. 2022Topological Data Analysis in Time Series: Temporal Filtration and Application to Single-Cell Genomicscited by 2
  32. 2022Persistent Homology of Coarse Grained State Space Networkscited by 2
  33. 2022Reduction Algorithms for Persistence Diagrams of Networks: CoralTDA and PrunITcited by 2
  34. 2023Analysing Multiscale Clusterings with Persistent Homologycited by 2
  35. 2023Topology Estimation of Simulated 4D Image Data by Combining Downscaling and Convolutional Neural Networkscited by 2
  36. 2023Multiparameter Persistent Homology for Molecular Property Predictioncited by 2
  37. 2024Local Topology Measures of Contextual Language Model Latent Spaces With Applications to Dialogue Term Extractioncited by 2
  38. 2025Topological Alignment of Shared Vision-Language Embedding Spacecited by 2

50 most cited of 127 All 127 (JSON)