mlsanityData to 14 Oct 2026

Geometric Scattering on Measure Spaces

arXiv 2208.08561 ↗First posted 17 Aug 2022stat.ML cs.LG math.SPcited by 17

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Abstract (arXiv)

The scattering transform is a multilayered, wavelet-based transform initially introduced as a model of convolutional neural networks (CNNs) that has played a foundational role in our understanding of these networks' stability and invariance properties. Subsequently, there has been widespread interest in extending the success of CNNs to data sets with non-Euclidean structure, such as graphs and manifolds, leading to the emerging field of geometric deep learning. In order to improve our understanding of the architectures used in this new field, several papers have proposed generalizations of the scattering transform for non-Euclidean data structures such as undirected graphs and compact Riemannian manifolds without boundary. In this paper, we introduce a general, unified model for geometric scattering on measure spaces. Our proposed framework includes previous work on geometric scattering as special cases but also applies to more general settings such as directed graphs, signed graphs, and manifolds with boundary. We propose a new criterion that identifies to which groups a useful representation should be invariant and show that this criterion is sufficient to guarantee that the scattering transform has desirable stability and invariance properties. Additionally, we consider finite measure spaces that are obtained from randomly sampling an unknown manifold. We propose two methods for constructing a data-driven graph on which the associated graph scattering transform approximates the scattering transform on the underlying manifold. Moreover, we use a diffusion-maps based approach to prove quantitative estimates on the rate of convergence of one of these approximations as the number of sample points tends to infinity. Lastly, we showcase the utility of our method on spherical images, directed graphs, and on high-dimensional single-cell data.

Key facts. Cited by 17 papers in the map (data through 14 Oct 2026). Builds on 44 of its 81 references that are in the map. First posted on arXiv 17 Aug 2022.

Builds on

44 of 81 references.

References · 44

  1. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,361
  2. 2017Graph Attention Networkscited by 8,923
  3. 2017Inductive Representation Learning on Large Graphscited by 6,320
  4. 2018How Powerful are Graph Neural Networks?cited by 4,393
  5. 2016Convolutional Neural Networks on Graphs with Fast Localized Spectral Filteringcited by 2,089
  6. 2016node2vec: Scalable Feature Learning for Networkscited by 1,918
  7. 2013Spectral Networks and Locally Connected Networks on Graphscited by 1,102
  8. 2016Geometric deep learning: going beyond Euclidean datacited by 1,048
  9. 2018Predict then Propagate: Graph Neural Networks meet Personalized PageRankcited by 993
  10. 2018Hypergraph Neural Networkscited by 488
  11. 2017graph2vec: Learning Distributed Representations of Graphscited by 230
  12. 2017CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filterscited by 203
32 more
  1. 2021MagNet: A Neural Network for Directed Graphscited by 107
  2. 2016Learning shape correspondence with anisotropic convolutional neural networkscited by 95
  3. 2021Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyondcited by 82
  4. 2020Directed Graph Convolutional Networkcited by 73
  5. 2022Generalization Analysis of Message Passing Neural Networks on Large Random Graphscited by 59
  6. 2018Geometric Scattering for Graph Data Analysiscited by 49
  7. 2015A Mathematical Theory of Deep Convolutional Neural Networks for Feature Extractioncited by 46
  8. 2018Graph Convolutional Neural Networks via Scatteringcited by 46
  9. 2018Diffusion Scattering Transforms on Graphscited by 44
  10. 2019Spectral-based Graph Convolutional Network for Directed Graphscited by 38
  11. 2016Magnetic Eigenmaps for the Visualization of Directed Networkscited by 25
  12. 2022Can Hybrid Geometric Scattering Networks Help Solve the Maximum Clique Problem?cited by 25
  13. 2021Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolationcited by 24
  14. 2022SigMaNet: One Laplacian to Rule Them Allcited by 24
  15. 2016Magnetic eigenmaps for community detection in directed networkscited by 22
  16. 2021Scattering Networks on the Sphere for Scalable and Rotationally Equivariant Spherical CNNscited by 18
  17. 2022Overcoming Oversmoothness in Graph Convolutional Networks via Hybrid Scattering Networkscited by 18
  18. 2017Path-Based Spectral Clustering: Guarantees, Robustness to Outliers, and Fast Algorithmscited by 17
  19. 2020Balancing Geometry and Density: Path Distances on High-Dimensional Datacited by 16
  20. 2018Multiscale Sparse Microcanonical Modelscited by 14
  21. 2019Spectral Analysis Of Weighted Laplacians Arising In Data Clusteringcited by 11
  22. 2020Regularized spectral methods for clustering signed networkscited by 9
  23. 2021Stability of Neural Networks on Manifolds to Relative Perturbationscited by 9
  24. 2020Characterization and comparison of large directed graphs through the spectra of the magnetic Laplaciancited by 7
  25. 2021Molecular Graph Generation via Geometric Scatteringcited by 7
  26. 2015Deep Convolutional Neural Networks Based on Semi-Discrete Framescited by 6
  27. 2016A Note on Markov Normalized Magnetic Eigenmapscited by 6
  28. 2016Deep Convolutional Neural Networks on Cartoon Functionscited by 4
  29. 2017Underwater object classification using scattering transform of sonar signalscited by 4
  30. 2022Nonbacktracking spectral clustering of nonuniform hypergraphscited by 4
  31. 2020Data-Driven Learning of Geometric Scattering Networkscited by 3
  32. 2014Audio Source Separation with Discriminative Scattering Networkscited by 1

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 1; 2023: 5; 2024: 3; 2025: 6; 2026: 2.

17 papers.

Sort:

  1. 2024A Manifold Perspective on the Statistical Generalization of Graph Neural Networkscited by 17
  2. 2022A Convergence Rate for Manifold Neural Networkscited by 6
  3. 2023DYMAG: Rethinking Message Passing Using Dynamical-systems-based Waveformscited by 3
  4. 2023BLIS-Net: Classifying and Analyzing Signals on Graphscited by 3
  5. 2023Manifold Filter-Combine Networkscited by 2
  6. 2023Directed Scattering for Knowledge Graph-based Cellular Signaling Analysiscited by 2
  7. 2025Wavelet Scattering Transform and Fourier Representation for Offline Detection of Malicious Clients in Federated Learningcited by 2
  8. 2023Multiscale Hodge Scattering Networks for Data Analysiscited by 1
  9. 2024Energy Propagation in Scattering Convolution Networks Can Be Arbitrarily Slowcited by 1
  10. 2025VDW-GNNs: Vector diffusion wavelets for geometric graph neural networkscited by 1
  11. 2024Convergence of Manifold Filter-Combine Networkscited by 0
  12. 2025Digital implementations of deep feature extractors are intrinsically informativecited by 0
5 more
  1. 2025InfoGain Wavelets: Furthering the Design of Graph Diffusion Waveletscited by 0
  2. 2025Scattering Networks on Noncommutative Finite Groupscited by 0
  3. 2025SlepNet: Spectral Subgraph Representation Learning for Neural Dynamicscited by 0
  4. 2026Can Computational Reducibility Lead to Transferable Models for Graph Combinatorial Optimization?cited by 0
  5. 2026Transversal Pooling Neural Networkscited by 0