mlsanityData to 7 Oct 2026

Stability of Neural Networks on Manifolds to Relative Perturbations

arXiv 2110.04702 ↗First posted 10 Oct 2021eess.SP cs.LGcited by 9

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Abstract (arXiv)

Graph Neural Networks (GNNs) show impressive performance in many practical scenarios, which can be largely attributed to their stability properties. Empirically, GNNs can scale well on large size graphs, but this is contradicted by the fact that existing stability bounds grow with the number of nodes. Graphs with well-defined limits can be seen as samples from manifolds. Hence, in this paper, we analyze the stability properties of convolutional neural networks on manifolds to understand the stability of GNNs on large graphs. Specifically, we focus on stability to relative perturbations of the Laplace-Beltrami operator. To start, we construct frequency ratio threshold filters which separate the infinite-dimensional spectrum of the Laplace-Beltrami operator. We then prove that manifold neural networks composed of these filters are stable to relative operator perturbations. As a product of this analysis, we observe that manifold neural networks exhibit a trade-off between stability and discriminability. Finally, we illustrate our results empirically in a wireless resource allocation scenario where the transmitter-receiver pairs are assumed to be sampled from a manifold.

Key facts. Cited by 9 papers in the map (data through 7 Oct 2026). Builds on 20 of its 23 references that are in the map. First posted on arXiv 10 Oct 2021.

Builds on

20 of 23 references.

References · 20

  1. 2018How Powerful are Graph Neural Networks?cited by 4,355
  2. 2018Graph Neural Networks: A Review of Methods and Applicationscited by 1,571
  3. 2016Geometric deep learning: going beyond Euclidean datacited by 1,042
  4. 2019Graph Neural Networks for Social Recommendationcited by 711
  5. 2019Stability Properties of Graph Neural Networkscited by 125
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  8. 2019Transferability of Spectral Graph Convolutional Neural Networkscited by 90
  9. 2019Optimal Wireless Resource Allocation with Random Edge Graph Neural Networkscited by 90
  10. 2020Convergence and Stability of Graph Convolutional Networks on Large Random Graphscited by 75
  11. 2019Learning Decentralized Controllers for Robot Swarms with Graph Neural Networkscited by 62
  12. 2020Unfolding WMMSE using Graph Neural Networks for Efficient Power Allocationcited by 61
8 more
  1. 2018Graph Convolutional Neural Networks via Scatteringcited by 46
  2. 2020Graphon Signal Processingcited by 37
  3. 2019Stability of Graph Scattering Transformscited by 34
  4. 2019Improved spectral convergence rates for graph Laplacians on epsilon-graphs and k-NN graphscited by 18
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  6. 2017Surface Networkscited by 14
  7. 2020Unsupervised Learning for Asynchronous Resource Allocation in Ad-hoc Wireless Networkscited by 9
  8. 2021Stability of Neural Networks on Riemannian Manifoldscited by 7

Built on by

Cited by year

Citing papers per year: 2022: 4; 2023: 3; 2024: 1; 2025: 1.

9 papers.

  1. 2022Geometric Scattering on Measure Spacescited by 17
  2. 2022Convolutional Neural Networks on Manifolds: From Graphs and Backcited by 13
  3. 2022The Manifold Scattering Transform for High-Dimensional Point Cloud Datacited by 9
  4. 2023Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-offscited by 9
  5. 2022A Convergence Rate for Manifold Neural Networkscited by 6
  6. 2023Manifold Filter-Combine Networkscited by 2
  7. 2025VDW-GNNs: Vector diffusion wavelets for geometric graph neural networkscited by 1
  8. 2023Spectral Convergence of Complexon Shift Operatorscited by 0
  9. 2024Convergence of Manifold Filter-Combine Networkscited by 0