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A Stable Multi-Scale Kernel for Topological Machine Learning

arXiv 1412.6821 ↗First posted 21 Dec 2014stat.ML cs.CV cs.LGcited by 74

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Abstract (arXiv)

Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes.

Key facts. Cited by 74 papers in the map (data through 7 Oct 2026). First posted on arXiv 21 Dec 2014.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 16; 2022: 8; 2023: 13; 2024: 9; 2025: 11; 2026: 17.

74 papers.

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  1. 2021Topology-Aware Segmentation Using Discrete Morse Theorycited by 75
  2. 2024Cross-Scale MAE: A Tale of Multi-Scale Exploitation in Remote Sensingcited by 47
  3. 2021Link Prediction with Persistent Homology: An Interactive Viewcited by 37
  4. 2021Topological Detection of Trojaned Neural Networkscited by 34
  5. 2023Topological Deep Learning: A Review of an Emerging Paradigmcited by 34
  6. 2022RipsNet: a general architecture for fast and robust estimation of the persistent homology of point cloudscited by 20
  7. 2021Persformer: A Transformer Architecture for Topological Machine Learningcited by 19
  8. 2023On the Expressivity of Persistent Homology in Graph Learningcited by 18
  9. 2023Differentiable Euler Characteristic Transforms for Shape Classificationcited by 18
  10. 2022Neural Approximation of Graph Topological Featurescited by 16
  11. 2023Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measurescited by 15
  12. 2021Topological Attention for Time Series Forecastingcited by 14
38 more
  1. 2021TopoTxR: A Topological Biomarker for Predicting Treatment Response in Breast Cancercited by 13
  2. 2024Tensor-view Topological Graph Neural Networkcited by 12
  3. 2025Topological Signatures vs. Gradient Histograms: A Comparative Study for Medical Image Classificationcited by 10
  4. 2021Estimation and Quantization of Expected Persistence Diagramscited by 8
  5. 2021Graph Filtration Kernelscited by 8
  6. 2022Approximating Persistent Homology for Large Datasetscited by 8
  7. 2021Learning Hyperbolic Representations of Topological Featurescited by 7
  8. 2022A Primer on Topological Data Analysis to Support Image Analysis Tasks in Environmental Sciencecited by 7
  9. 2021Augmenting Molecular Deep Generative Models with Topological Data Analysis Representationscited by 6
  10. 2024Graphcode: Learning from multiparameter persistent homology using graph neural networkscited by 6
  11. 2025Enhancing Graph Representation Learning with Localized Topological Featurescited by 6
  12. 2024D-GRIL: End-to-End Topological Learning with 2-parameter Persistencecited by 5
  13. 2021Topology Applied to Machine Learning: From Global to Localcited by 4
  14. 2021A computationally efficient framework for vector representation of persistence diagramscited by 3
  15. 2023Wasserstein Auto-Encoders of Merge Trees (and Persistence Diagrams)cited by 3
  16. 2023Landslide Topology Uncovers Failure Movementscited by 3
  17. 2024SIFM: A Foundation Model for Multi-granularity Arctic Sea Ice Forecastingcited by 3
  18. 2021Topological Information Retrieval with Dilation-Invariant Bottleneck Comparative Measurescited by 2
  19. 2021A Domain-Oblivious Approach for Learning Concise Representations of Filtered Topological Spaces for Clusteringcited by 2
  20. 2022Do Neural Networks Trained with Topological Features Learn Different Internal Representations?cited by 2
  21. 2023Statistical learning on measures: an application to persistence diagramscited by 2
  22. 2023Image Classification using Combination of Topological Features and Neural Networkscited by 2
  23. 2025TopoImages: Incorporating Local Topology Encoding into Deep Learning Models for Medical Image Classificationcited by 2
  24. 2025TACO-Net: Topological Signatures Triumph in 3D Object Classificationcited by 2
  25. 2026A Closed-Form Adaptive-Landmark Kernel for Certified Point-Cloud and Graph Classificationcited by 2
  26. 2022k-Means Clustering for Persistent Homologycited by 1
  27. 2022Reliable Malware Analysis and Detection using Topology Data Analysiscited by 1
  28. 2023Vector Summaries of Persistence Diagrams for Permutation-based Hypothesis Testingcited by 1
  29. 2023Stability for Inference with Persistent Homology Rank Functionscited by 1
  30. 2023Higher-order topological kernels via quantum computationcited by 1
  31. 2023Addressing caveats of neural persistence with deep graph persistencecited by 1
  32. 2023GeoTop: Advancing Image Classification with Geometric-Topological Analysiscited by 1
  33. 2024A Pipeline for Data-Driven Learning of Topological Features with Applications to Protein Stability Predictioncited by 1
  34. 2024Functional connectomes of neural networkscited by 1
  35. 2025Topological Signatures of Adversaries in Multimodal Alignmentscited by 1
  36. 2025From Betti Numbers to Persistence Diagrams: A Hybrid Quantum Algorithm for Topological Data Analysiscited by 1
  37. 2021Statistical embedding: Beyond principal componentscited by 0
  38. 2021Nonparametric Topological Layers in Neural Networkscited by 0

50 most cited of 74 All 74 (JSON)