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

arXiv: https://arxiv.org/abs/2305.18467 · HTML: https://mlsanity.com/paper/2305.18467/

## 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 (most cited first, 30 of 30)

- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Point-GNN: Graph Neural Network for 3D Object Detection in a Point Cloud](https://mlsanity.com/paper/2003.01251/) (2020)
- [Stability Properties of Graph Neural Networks](https://mlsanity.com/paper/1905.04497/) (2019)
- [Graphon Neural Networks and the Transferability of Graph Neural Networks](https://mlsanity.com/paper/2006.03548/) (2020)
- [PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/2108.01938/) (2021)
- [Convolutional Neural Network Architectures for Signals Supported on Graphs](https://mlsanity.com/paper/1805.00165/) (2018)
- [Transfer Learning of Graph Neural Networks with Ego-graph Information Maximization](https://mlsanity.com/paper/2009.05204/) (2020)
- [Transferability of Spectral Graph Convolutional Neural Networks](https://mlsanity.com/paper/1907.12972/) (2019)
- [Graph Neural Networks for Decentralized Multi-Robot Path Planning](https://mlsanity.com/paper/1912.06095/) (2019)
- [Convergence and Stability of Graph Convolutional Networks on Large Random Graphs](https://mlsanity.com/paper/2006.01868/) (2020)
- [Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networks](https://mlsanity.com/paper/2003.03777/) (2020)
- [Transferability of Graph Neural Networks: an Extended Graphon Approach](https://mlsanity.com/paper/2109.10096/) (2021)
- [3D Point Cloud Denoising using Graph Laplacian Regularization of a Low Dimensional Manifold Model](https://mlsanity.com/paper/1803.07252/) (2018)
- [Graph Convolutional Neural Networks via Scattering](https://mlsanity.com/paper/1804.00099/) (2018)
- [Transferability Properties of Graph Neural Networks](https://mlsanity.com/paper/2112.04629/) (2021)
- [Graphon Signal Processing](https://mlsanity.com/paper/2003.05030/) (2020)
- [Multi-Robot Coverage and Exploration using Spatial Graph Neural Networks](https://mlsanity.com/paper/2011.01119/) (2020)
- [Intrinsic Geometric Information Transfer Learning on Multiple Graph-Structured Datasets](https://mlsanity.com/paper/1611.04687/) (2016)
- [Improved spectral convergence rates for graph Laplacians on epsilon-graphs and k-NN graphs](https://mlsanity.com/paper/1910.13476/) (2019)
- [Convolutional Neural Networks on Manifolds: From Graphs and Back](https://mlsanity.com/paper/2210.00376/) (2022)
- [Stability to Deformations of Manifold Filters and Manifold Neural Networks](https://mlsanity.com/paper/2106.03725/) (2021)
- [Stability of Neural Networks on Manifolds to Relative Perturbations](https://mlsanity.com/paper/2110.04702/) (2021)
- [A Convergence Rate for Manifold Neural Networks](https://mlsanity.com/paper/2212.12606/) (2022)
- [Stable and Transferable Wireless Resource Allocation Policies via Manifold Neural Networks](https://mlsanity.com/paper/2110.04706/) (2021)
- [Learning Globally Smooth Functions on Manifolds](https://mlsanity.com/paper/2210.00301/) (2022)
- [Convolutional Filtering on Sampled Manifolds](https://mlsanity.com/paper/2211.11058/) (2022)

## Built on by (most cited first, 9 of 9)

- [A Manifold Perspective on the Statistical Generalization of Graph Neural Networks](https://mlsanity.com/paper/2406.05225/) (2024)
- [Generalization of Geometric Graph Neural Networks with Lipschitz Loss Functions](https://mlsanity.com/paper/2409.05191/) (2024)
- [Generalization of Graph Neural Networks is Robust to Model Mismatch](https://mlsanity.com/paper/2408.13878/) (2024)
- [Manifold Filter-Combine Networks](https://mlsanity.com/paper/2307.04056/) (2023)
- [Improved Image Classification with Manifold Neural Networks](https://mlsanity.com/paper/2409.13063/) (2024)
- [Manifold limit for the training of shallow graph convolutional neural networks](https://mlsanity.com/paper/2601.06025/) (2026)
- [Size-Transferability of Graph Transformers with Convolutional Positional Encodings](https://mlsanity.com/paper/2602.15239/) (2026)
- [Graph Learning Should Move Beyond Restrictive Views of Spectral and Message-Passing GNNs](https://mlsanity.com/paper/2602.10031/) (2026)
- [Limit Analysis of Graph Neural Networks with Wireless Conflict Graphs](https://mlsanity.com/paper/2606.03794/) (2026)

## 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.