Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-offs
This work addresses theoretical foundations for geometric deep learning, providing insights into the behavior of GNNs on sampled data, which is incremental but important for applications like navigation control and point cloud classification.
The paper analyzes the convergence of graph neural networks (GNNs) to manifold neural networks (MNNs) when graphs are sampled from manifolds, proving non-asymptotic error bounds and identifying a trade-off between discriminability and approximation in graph filters, which is mitigated by nonlinearities in neural networks.
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.