mlsanityData to 7 Oct 2026

A Convergence Rate for Manifold Neural Networks

arXiv 2212.12606 ↗First posted 23 Dec 2022cs.LG cs.NA eess.SPcited by 6

Not on the map. Why?

Abstract (arXiv)

High-dimensional data arises in numerous applications, and the rapidly developing field of geometric deep learning seeks to develop neural network architectures to analyze such data in non-Euclidean domains, such as graphs and manifolds. Recent work by Z. Wang, L. Ruiz, and A. Ribeiro has introduced a method for constructing manifold neural networks using the spectral decomposition of the Laplace Beltrami operator. Moreover, in this work, the authors provide a numerical scheme for implementing such neural networks when the manifold is unknown and one only has access to finitely many sample points. The authors show that this scheme, which relies upon building a data-driven graph, converges to the continuum limit as the number of sample points tends to infinity. Here, we build upon this result by establishing a rate of convergence that depends on the intrinsic dimension of the manifold but is independent of the ambient dimension. We also discuss how the rate of convergence depends on the depth of the network and the number of filters used in each layer.

Key facts. Cited by 6 papers in the map (data through 7 Oct 2026). Builds on 12 of its 25 references that are in the map. First posted on arXiv 23 Dec 2022.

Builds on

12 of 25 references.

References · 12

  1. 2019A Comprehensive Survey on Graph Neural Networkscited by 2,906
  2. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,131
  3. 2016Geometric deep learning: going beyond Euclidean datacited by 1,042
  4. 2021ETA Prediction with Graph Neural Networks in Google Mapscited by 130
  5. 2021Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolationcited by 24
  6. 2022Geometric Scattering on Measure Spacescited by 17
  7. 2020Balancing Geometry and Density: Path Distances on High-Dimensional Datacited by 16
  8. 2022Convolutional Neural Networks on Manifolds: From Graphs and Backcited by 13
  9. 2021Stability of Neural Networks on Manifolds to Relative Perturbationscited by 9
  10. 2022The Manifold Scattering Transform for High-Dimensional Point Cloud Datacited by 9
  11. 2022Robust Inference of Manifold Density and Geometry by Doubly Stochastic Scalingcited by 7
  12. 2022Convolutional Filtering on Sampled Manifoldscited by 4

Built on by

Cited by year

Citing papers per year: 2023: 2; 2024: 3; 2025: 1.

6 papers.

  1. 2024A Manifold Perspective on the Statistical Generalization of Graph Neural Networkscited by 16
  2. 2023Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-offscited by 9
  3. 2024Generalization of Graph Neural Networks is Robust to Model Mismatchcited by 4
  4. 2023Manifold Filter-Combine Networkscited by 2
  5. 2025VDW-GNNs: Vector diffusion wavelets for geometric graph neural networkscited by 1
  6. 2024Convergence of Manifold Filter-Combine Networkscited by 0