# Manifold limit for the training of shallow graph convolutional neural networks

arXiv: https://arxiv.org/abs/2601.06025 · HTML: https://mlsanity.com/paper/2601.06025/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 47 references that are in the map.
- First posted on arXiv 9 Jan 2026.

## Builds on (most cited first, 22 of 22)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters](https://mlsanity.com/paper/1705.07664/) (2017)
- [Transferability of Spectral Graph Convolutional Neural Networks](https://mlsanity.com/paper/1907.12972/) (2019)
- [Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networks](https://mlsanity.com/paper/2003.03777/) (2020)
- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Representation formulas and pointwise properties for Barron functions](https://mlsanity.com/paper/2006.05982/) (2020)
- [Transferability of Graph Neural Networks: an Extended Graphon Approach](https://mlsanity.com/paper/2109.10096/) (2021)
- [A variational approach to the consistency of spectral clustering](https://mlsanity.com/paper/1508.01928/) (2015)
- [Sparsity of solutions for variational inverse problems with finite-dimensional data](https://mlsanity.com/paper/1809.05045/) (2018)
- [Continuum limit of total variation on point clouds](https://mlsanity.com/paper/1403.6355/) (2014)
- [Large Data and Zero Noise Limits of Graph-Based Semi-Supervised Learning Algorithms](https://mlsanity.com/paper/1805.09450/) (2018)
- [Rates of Convergence for Laplacian Semi-Supervised Learning with Low Labeling Rates](https://mlsanity.com/paper/2006.02765/) (2020)
- [Limits, approximation and size transferability for GNNs on sparse graphs via graphops](https://mlsanity.com/paper/2306.04495/) (2023)
- [Generalization Bounds for Message Passing Networks on Mixture of Graphons](https://mlsanity.com/paper/2404.03473/) (2024)
- [Convolutional Neural Networks on Manifolds: From Graphs and Back](https://mlsanity.com/paper/2210.00376/) (2022)
- [Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-offs](https://mlsanity.com/paper/2305.18467/) (2023)
- [Generalization of Geometric Graph Neural Networks with Lipschitz Loss Functions](https://mlsanity.com/paper/2409.05191/) (2024)
- [On the Consistency of Graph-based Bayesian Learning and the Scalability of Sampling Algorithms](https://mlsanity.com/paper/1710.07702/) (2017)
- [Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators](https://mlsanity.com/paper/2509.26371/) (2025)

## Built on by (most cited first, 1 of 1)

- [Diffusion Processes on Implicit Manifolds](https://mlsanity.com/paper/2604.07213/) (2026)

## Abstract (arXiv)

We study the discrete-to-continuum consistency of the training of shallow graph convolutional neural networks (GCNNs) on proximity graphs of sampled point clouds under a manifold assumption. Graph convolution is defined spectrally via the graph Laplacian, whose low-frequency spectrum approximates that of the Laplace-Beltrami operator of the underlying smooth manifold, and shallow GCNNs of possibly infinite width are linear functionals on the space of measures on the parameter space. From this functional-analytic perspective, graph signals are seen as spatial discretizations of functions on the manifold, which leads to a natural notion of training data consistent across graph resolutions. To enable convergence results, the continuum parameter space is chosen as a weakly compact product of unit balls, with Sobolev regularity imposed on the output weight and bias, but not on the convolutional parameter. The corresponding discrete parameter spaces inherit the corresponding spectral decay, and are additionally restricted by a frequency cutoff adapted to the informative spectral window of the graph Laplacians. Under these assumptions, we prove Gamma-convergence of regularized empirical risk minimization functionals and corresponding convergence of their global minimizers, in the sense of weak convergence of the parameter measures and uniform convergence of the functions over compact sets. This provides a formalization of mesh and sample independence, and hence of transferability across discretizations, for the training of such networks.