mlsanityData to 14 Oct 2026

Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-offs

arXiv 2305.18467 ↗First posted 29 May 2023cs.LG eess.SPcited by 9

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Abstract (arXiv)

This paper studies the relationship between a graph neural network (GNN) and a manifold neural network (MNN) when the graph is constructed from a set of points sampled from the manifold, thus encoding geometric information. We consider convolutional MNNs and GNNs where the manifold and the graph convolutions are respectively defined in terms of the Laplace-Beltrami operator and the graph Laplacian. Using the appropriate kernels, we analyze both dense and moderately sparse graphs. We prove non-asymptotic error bounds showing that convolutional filters and neural networks on these graphs converge to convolutional filters and neural networks on the continuous manifold. As a byproduct of this analysis, we observe an important trade-off between the discriminability of graph filters and their ability to approximate the desired behavior of manifold filters. We then discuss how this trade-off is ameliorated in neural networks due to the frequency mixing property of nonlinearities. We further derive a transferability corollary for geometric graphs sampled from the same manifold. We validate our results numerically on a navigation control problem and a point cloud classification task.

Key facts. Cited by 9 papers in the map (data through 14 Oct 2026). Builds on 30 of its 61 references that are in the map. First posted on arXiv 29 May 2023.

Builds on

30 of 61 references.

References · 30

  1. 2018Dynamic Graph CNN for Learning on Point Cloudscited by 2,223
  2. 2018Graph Neural Networks: A Review of Methods and Applicationscited by 1,576
  3. 2016Geometric deep learning: going beyond Euclidean datacited by 1,048
  4. 2016Geometric deep learning on graphs and manifolds using mixture model CNNscited by 468
  5. 2017Graph Signal Processing: Overview, Challenges and Applicationscited by 413
  6. 2020Point-GNN: Graph Neural Network for 3D Object Detection in a Point Cloudcited by 335
  7. 2019Stability Properties of Graph Neural Networkscited by 127
  8. 2020Graphon Neural Networks and the Transferability of Graph Neural Networkscited by 118
  9. 2021PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equationscited by 106
  10. 2018Convolutional Neural Network Architectures for Signals Supported on Graphscited by 95
  11. 2020Transfer Learning of Graph Neural Networks with Ego-graph Information Maximizationcited by 93
  12. 2019Transferability of Spectral Graph Convolutional Neural Networkscited by 92
18 more
  1. 2019Graph Neural Networks for Decentralized Multi-Robot Path Planningcited by 86
  2. 2020Convergence and Stability of Graph Convolutional Networks on Large Random Graphscited by 76
  3. 2020Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networkscited by 75
  4. 2021Transferability of Graph Neural Networks: an Extended Graphon Approachcited by 53
  5. 20183D Point Cloud Denoising using Graph Laplacian Regularization of a Low Dimensional Manifold Modelcited by 47
  6. 2018Graph Convolutional Neural Networks via Scatteringcited by 46
  7. 2021Transferability Properties of Graph Neural Networkscited by 43
  8. 2020Graphon Signal Processingcited by 37
  9. 2020Multi-Robot Coverage and Exploration using Spatial Graph Neural Networkscited by 19
  10. 2016Intrinsic Geometric Information Transfer Learning on Multiple Graph-Structured Datasetscited by 18
  11. 2019Improved spectral convergence rates for graph Laplacians on epsilon-graphs and k-NN graphscited by 18
  12. 2022Convolutional Neural Networks on Manifolds: From Graphs and Backcited by 14
  13. 2021Stability to Deformations of Manifold Filters and Manifold Neural Networkscited by 9
  14. 2021Stability of Neural Networks on Manifolds to Relative Perturbationscited by 9
  15. 2022A Convergence Rate for Manifold Neural Networkscited by 6
  16. 2021Stable and Transferable Wireless Resource Allocation Policies via Manifold Neural Networkscited by 4
  17. 2022Learning Globally Smooth Functions on Manifoldscited by 4
  18. 2022Convolutional Filtering on Sampled Manifoldscited by 4

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 1; 2024: 4; 2025: 0; 2026: 4.

9 papers.

  1. 2024A Manifold Perspective on the Statistical Generalization of Graph Neural Networkscited by 17
  2. 2024Generalization of Geometric Graph Neural Networks with Lipschitz Loss Functionscited by 6
  3. 2024Generalization of Graph Neural Networks is Robust to Model Mismatchcited by 4
  4. 2023Manifold Filter-Combine Networkscited by 2
  5. 2024Improved Image Classification with Manifold Neural Networkscited by 1
  6. 2026Manifold limit for the training of shallow graph convolutional neural networkscited by 1
  7. 2026Size-Transferability of Graph Transformers with Convolutional Positional Encodingscited by 1
  8. 2026Graph Learning Should Move Beyond Restrictive Views of Spectral and Message-Passing GNNscited by 0
  9. 2026Limit Analysis of Graph Neural Networks with Wireless Conflict Graphscited by 0