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A Topology Layer for Machine Learning

arXiv 1905.12200 ↗First posted 29 May 2019cs.LG math.AT stat.MLcited by 79

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Abstract (arXiv)

Topology applied to real world data using persistent homology has started to find applications within machine learning, including deep learning. We present a differentiable topology layer that computes persistent homology based on level set filtrations and edge-based filtrations. We present three novel applications: the topological layer can (i) regularize data reconstruction or the weights of machine learning models, (ii) construct a loss on the output of a deep generative network to incorporate topological priors, and (iii) perform topological adversarial attacks on deep networks trained with persistence features. The code (www.github.com/bruel-gabrielsson/TopologyLayer) is publicly available and we hope its availability will facilitate the use of persistent homology in deep learning and other gradient based applications.

Key facts. Cited by 79 papers in the map (data through 7 Oct 2026). First posted on arXiv 29 May 2019.

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Citing papers per year: 2021: 21; 2022: 8; 2023: 11; 2024: 13; 2025: 10; 2026: 16.

79 papers.

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  1. 2021Priors in Bayesian Deep Learning: A Reviewcited by 101
  2. 2021Intrinsic Dimension, Persistent Homology and Generalization in Neural Networkscited by 70
  3. 2024Position: Topological Deep Learning is the New Frontier for Relational Learningcited by 65
  4. 2021Artificial Text Detection via Examining the Topology of Attention Mapscited by 49
  5. 2021A persistent homology-based topological loss for CNN-based multi-class segmentation of CMRcited by 21
  6. 2022On the effectiveness of persistent homologycited by 20
  7. 2023Differentiable Euler Characteristic Transforms for Shape Classificationcited by 18
  8. 2021Activation Landscapes as a Topological Summary of Neural Network Performancecited by 17
  9. 2024TopoCellGen: Generating Histopathology Cell Topology with a Diffusion Modelcited by 16
  10. 2023Metric Space Magnitude and Generalisation in Neural Networkscited by 15
  11. 2023GRIL: A 2-parameter Persistence Based Vectorization for Machine Learningcited by 13
  12. 2024Geometry-Informed Neural Networkscited by 12
38 more
  1. 2021Persistent Homology with Improved Locality Information for more Effective Delineationcited by 10
  2. 2024TopoDiffusionNet: A Topology-aware Diffusion Modelcited by 10
  3. 2021Cycle Registration in Persistent Homology with Applications in Topological Bootstrapcited by 8
  4. 2021Augmenting Molecular Deep Generative Models with Topological Data Analysis Representationscited by 6
  5. 2024Topology-Aware 3D Gaussian Splatting: Leveraging Persistent Homology for Optimized Structural Integritycited by 6
  6. 2025TopoLiDM: Topology-Aware LiDAR Diffusion Models for Interpretable and Realistic LiDAR Point Cloud Generationcited by 6
  7. 2021Topological Deep Learning: Classification Neural Networkscited by 5
  8. 2021A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysiscited by 5
  9. 2021Topologically Regularized Data Embeddingscited by 5
  10. 2021Topologically penalized regression on manifoldscited by 5
  11. 2022BERTops: Studying BERT Representations under a Topological Lenscited by 5
  12. 2023Topological Node2vec: Enhanced Graph Embedding via Persistent Homologycited by 5
  13. 2022Rethinking Persistent Homology for Visual Recognitioncited by 4
  14. 2024Learning Persistent Community Structures in Dynamic Networks via Topological Data Analysiscited by 4
  15. 2024Revisiting Point Cloud Completion: Are We Ready For The Real-World?cited by 4
  16. 2024STITCH: Surface reconstrucTion using Implicit neural representations with Topology Constraints and persistent Homologycited by 4
  17. 2025The Shape of Adversarial Influence: Characterizing LLM Latent Spaces with Persistent Homologycited by 4
  18. 2022Topology-Preserving Dimensionality Reduction via Interleaving Optimizationcited by 3
  19. 2026A Closed-Form Persistence-Landmark Pipeline for Certified Point-Cloud and Graph Classificationcited by 3
  20. 2021Optimisation of Spectral Wavelets for Persistence-based Graph Classificationcited by 2
  21. 2021Domain Adaptation by Topology Regularizationcited by 2
  22. 2021Classification based on Topological Data Analysiscited by 2
  23. 2021A Multi-parameter Persistence Framework for Mathematical Morphologycited by 2
  24. 2021Topological Information Retrieval with Dilation-Invariant Bottleneck Comparative Measurescited by 2
  25. 2022On the Convergence of Optimizing Persistent-Homology-Based Lossescited by 2
  26. 2023Topology-Aware Loss for Aorta and Great Vessel Segmentation in Computed Tomography Imagescited by 2
  27. 2023GLiDR: Topologically Regularized Graph Generative Network for Sparse LiDAR Point Cloudscited by 2
  28. 2024Mixup Barcodes: Quantifying Geometric-Topological Interactions between Point Cloudscited by 2
  29. 2024When Witnesses Defend: A Witness Graph Topological Layer for Adversarial Graph Learningcited by 2
  30. 2025Topological Signatures of ReLU Neural Network Activation Patternscited by 2
  31. 2026Topology-Guaranteed Image Segmentation: Enforcing Connectivity, Genus, and Width Constraintscited by 2
  32. 2026A Closed-Form Adaptive-Landmark Kernel for Certified Point-Cloud and Graph Classificationcited by 2
  33. 2021Component Tree Loss Function: Definition and Optimizationcited by 1
  34. 2023Importance attribution in neural networks by means of persistence landscapes of time seriescited by 1
  35. 2023Algebraic Topological Networks via the Persistent Local Homology Sheafcited by 1
  36. 2024Mix-GENEO: A Flexible Filtration for Multiparameter Persistent Homology Detects Digital Imagescited by 1
  37. 2025Quasi Zigzag Persistence: A Topological Framework for Analyzing Time-Varying Datacited by 1
  38. 2025Disconnect to Connect: A Data Augmentation Method for Improving Topology Accuracy in Image Segmentationcited by 1

50 most cited of 79 All 79 (JSON)