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Convergence of Manifold Filter-Combine Networks

arXiv 2410.14639 ↗First posted 18 Oct 2024cs.LG eess.SP stat.MLcited by 0

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Abstract (arXiv)

In order to better understand manifold neural networks (MNNs), we introduce Manifold Filter-Combine Networks (MFCNs). The filter-combine framework parallels the popular aggregate-combine paradigm for graph neural networks (GNNs) and naturally suggests many interesting families of MNNs which can be interpreted as the manifold analog of various popular GNNs. We then propose a method for implementing MFCNs on high-dimensional point clouds that relies on approximating the manifold by a sparse graph. We prove that our method is consistent in the sense that it converges to a continuum limit as the number of data points tends to infinity.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 17 of its 33 references that are in the map. First posted on arXiv 18 Oct 2024.

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17 of 33 references.

References · 17

  1. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,282
  2. 2018How Powerful are Graph Neural Networks?cited by 4,355
  3. 2016Convolutional Neural Networks on Graphs with Fast Localized Spectral Filteringcited by 2,085
  4. 2018Graph Neural Networks: A Review of Methods and Applicationscited by 1,571
  5. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,131
  6. 2016Geometric deep learning: going beyond Euclidean datacited by 1,042
  7. 2017CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filterscited by 203
  8. 2018Geometric Scattering for Graph Data Analysiscited by 49
  9. 2018Graph Convolutional Neural Networks via Scatteringcited by 46
  10. 2018Diffusion Scattering Transforms on Graphscited by 43
  11. 2019Stability of Graph Scattering Transformscited by 34
  12. 2021Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolationcited by 24
5 more
  1. 2022Overcoming Oversmoothness in Graph Convolutional Networks via Hybrid Scattering Networkscited by 18
  2. 2022Geometric Scattering on Measure Spacescited by 17
  3. 2021Stability of Neural Networks on Manifolds to Relative Perturbationscited by 9
  4. 2022The Manifold Scattering Transform for High-Dimensional Point Cloud Datacited by 9
  5. 2022A Convergence Rate for Manifold Neural Networkscited by 6

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