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

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 con...

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Bibliographic Details
Published in:IEEE transactions on signal processing 2024, Vol.72, p.2244-2259
Main Authors: Wang, Zhiyang, Ruiz, Luana, Ribeiro, Alejandro
Format: Article
Language:English
Subjects:
Approximation error
Convergence
convergence analysis
Convolution
discriminability
Filters
Graph neural networks
Graphs
manifold filters
manifold neural networks
Manifolds
Neural networks
Operators (mathematics)
Tradeoffs
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Summary: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.
ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2024.3392360