mlsanityData to 14 Oct 2026

Topological Deep Learning: A Review of an Emerging Paradigm

arXiv 2302.03836 ↗First posted 8 Feb 2023cs.LG cs.AIcited by 34

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Abstract (arXiv)

Topological data analysis (TDA) provides insight into data shape. The summaries obtained by these methods are principled global descriptions of multi-dimensional data whilst exhibiting stable properties such as robustness to deformation and noise. Such properties are desirable in deep learning pipelines but they are typically obtained using non-TDA strategies. This is partly caused by the difficulty of combining TDA constructs (e.g. barcode and persistence diagrams) with current deep learning algorithms. Fortunately, we are now witnessing a growth of deep learning applications embracing topologically-guided components. In this survey, we review the nascent field of topological deep learning by first revisiting the core concepts of TDA. We then explore how the use of TDA techniques has evolved over time to support deep learning frameworks, and how they can be integrated into different aspects of deep learning. Furthermore, we touch on TDA usage for analyzing existing deep models; deep topological analytics. Finally, we discuss the challenges and future prospects of topological deep learning.

Key facts. Cited by 34 papers in the map (data through 14 Oct 2026). Builds on 32 of its 111 references that are in the map. First posted on arXiv 8 Feb 2023.

Builds on

32 of 111 references.

References · 32

  1. 2015Deep Residual Learning for Image Recognitioncited by 52,096
  2. 2017Deep Setscited by 1,421
  3. 2018Computational Optimal Transportcited by 780
  4. 2015Persistence Images: A Stable Vector Representation of Persistent Homologycited by 236
  5. 2019Topology-Preserving Deep Image Segmentationcited by 198
  6. 2017An introduction to Topological Data Analysis: fundamental and practical aspects for data scientistscited by 172
  7. 2019A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homologycited by 134
  8. 2019Topological Autoencoderscited by 127
  9. 2019PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signaturescited by 126
  10. 2017Deep Learning with Topological Signaturescited by 114
  11. 2020Topology of deep neural networkscited by 112
  12. 2021Topological Graph Neural Networkscited by 87
20 more
  1. 2017Beyond the Pixel-Wise Loss for Topology-Aware Delineationcited by 83
  2. 2014A Stable Multi-Scale Kernel for Topological Machine Learningcited by 75
  3. 2018Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topologycited by 73
  4. 2019Graph Filtration Learningcited by 69
  5. 2018A Topological Regularizer for Classifiers via Persistent Homologycited by 63
  6. 2018Topological Approaches to Deep Learningcited by 29
  7. 2022RipsNet: a general architecture for fast and robust estimation of the persistent homology of point cloudscited by 20
  8. 2018The persistence landscape and some of its propertiescited by 19
  9. 2021Persformer: A Transformer Architecture for Topological Machine Learningcited by 19
  10. 2018Persistence paths and signature features in topological data analysiscited by 17
  11. 2014A persistence landscapes toolbox for topological statisticscited by 16
  12. 2020A Fast and Robust Method for Global Topological Functional Optimizationcited by 15
  13. 2020A persistent homology-based topological loss function for multi-class CNN segmentation of cardiac MRIcited by 13
  14. 2021TDA-Net: Fusion of Persistent Homology and Deep Learning Features for COVID-19 Detection in Chest X-Ray Imagescited by 12
  15. 2021Topological Measurement of Deep Neural Networks Using Persistent Homologycited by 9
  16. 2016A Riemannian Framework for Statistical Analysis of Topological Persistence Diagramscited by 7
  17. 2022A novel approach for wafer defect pattern classification based on topological data analysiscited by 6
  18. 2018Persistence Codebooks for Topological Data Analysiscited by 5
  19. 2019Scalable Topological Data Analysis and Visualization for Evaluating Data-Driven Models in Scientific Applicationscited by 5
  20. 2018Perturbation Robust Representations of Topological Persistence Diagramscited by 4

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 2; 2024: 7; 2025: 12; 2026: 13.

34 papers.

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  1. 2024Position: Topological Deep Learning is the New Frontier for Relational Learningcited by 66
  2. 2024Topology-Aware 3D Gaussian Splatting: Leveraging Persistent Homology for Optimized Structural Integritycited by 6
  3. 2026Learning Significant Persistent Homology Features for 3D Shape Understandingcited by 3
  4. 2026Topology-Preserving Neural Operator Learning via Hodge Decompositioncited by 3
  5. 2024Mixup Barcodes: Quantifying Geometric-Topological Interactions between Point Cloudscited by 2
  6. 2024Rapid and Precise Topological Comparison with Merge Tree Neural Networkscited by 2
  7. 2023Data Topology-Dependent Upper Bounds of Neural Network Widthscited by 1
  8. 2023Addressing caveats of neural persistence with deep graph persistencecited by 1
  9. 2025Topology of Out-of-Distribution Examples in Deep Neural Networkscited by 1
  10. 2025Improving Remote Sensing Classification using Topological Data Analysis and Convolutional Neural Networkscited by 1
  11. 2025On the Sheafification of Higher-Order Message Passingcited by 1
  12. 2025CuMPerLay: Learning Cubical Multiparameter Persistence Vectorizationscited by 1
22 more
  1. 2026GraPHFormer: A Multimodal Graph Persistent Homology Transformer for the Analysis of Neuroscience Morphologiescited by 1
  2. 2026A Persistent Homology Design Space for 3D Point Cloud Deep Learningcited by 1
  3. 2026TopoGeoScore: A Self-Supervised Source-Only Geometric Framework for OOD Checkpoint Selectioncited by 1
  4. 2024Machine Learning and Data Analysis Using Posets: A Surveycited by 0
  5. 2024A Vectorization Method Induced By Maximal Margin Classification For Persistent Diagramscited by 0
  6. 2024TopOC: Topological Deep Learning for Ovarian and Breast Cancer Diagnosiscited by 0
  7. 2025A Metric Topology of Deep Learning for Data Classificationcited by 0
  8. 2025Lipschitz Bounds for Persistent Laplacian Eigenvalues under One-Simplex Insertionscited by 0
  9. 2025Preserving instance continuity and length in segmentation through connectivity-aware loss computationcited by 0
  10. 2025Barycentric Neural Networks and Length-Weighted Persistent Entropy Loss: A Green Geometric and Topological Framework for Function Approximationcited by 0
  11. 2025HomID : Benchmarking Intrinsic Dimension Estimators on Homogenous Manifolds with Anisotropic Embeddingscited by 0
  12. 2025A roadmap for curvature-based geometric data analysis and learningcited by 0
  13. 2025Prototype Selection Using Topological Data Analysiscited by 0
  14. 2025Topologically-Stabilized Graph Neural Networks: Empirical Robustness Across Domainscited by 0
  15. 2026An Algebraic Representation Theorem for Linear GENEOs in Geometric Machine Learningcited by 0
  16. 2026Test-Time Adaptation for Anomaly Segmentation via Topology-Aware Optimal Transport Chainingcited by 0
  17. 2026A Sheaf-Theoretic and Topological Perspective on Complex Network Modeling and Attention Mechanisms in Graph Neural Modelscited by 0
  18. 2026TopoU-Net: a U-Net architecture for topological domainscited by 0
  19. 2026Learning Topology-Aware Representations via Test-Time Adaptation for Anomaly Segmentationcited by 0
  20. 2026TopoAgent: An Agentic Framework for Automated Topology Learning in Medical Imagingcited by 0
  21. 2026BettiSplit: Topology-Guided Privacy-Aware Split Learning Against Feature Inversion and Gradient Leakagecited by 0
  22. 2026Combinatorial Network-Based Manifold Topological Deep Learning for Image Analysiscited by 0