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

• URL: http://arxiv.org/abs/2305.18467v2
• Date: Wed, 28 Jun 2023 03:07:55 GMT
• Title: Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-offs
• Authors: Zhiyang Wang and Luana Ruiz and Alejandro Ribeiro
• Abstract summary: We study 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. 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.
• Score: 122.06927400759021
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: 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.

Related papers

• Convolutional Neural Networks on Manifolds: From Graphs and Back [122.06927400759021]
  We propose a manifold neural network (MNN) composed of a bank of manifold convolutional filters and point-wise nonlinearities. To sum up, we focus on the manifold model as the limit of large graphs and construct MNNs, while we can still bring back graph neural networks by the discretization of MNNs.
  arXiv  Detail & Related papers  (2022-10-01T21:17:39Z)
• Learning Graph Structure from Convolutional Mixtures [119.45320143101381]
  We propose a graph convolutional relationship between the observed and latent graphs, and formulate the graph learning task as a network inverse (deconvolution) problem. In lieu of eigendecomposition-based spectral methods, we unroll and truncate proximal gradient iterations to arrive at a parameterized neural network architecture that we call a Graph Deconvolution Network (GDN) GDNs can learn a distribution of graphs in a supervised fashion, perform link prediction or edge-weight regression tasks by adapting the loss function, and they are inherently inductive.
  arXiv  Detail & Related papers  (2022-05-19T14:08:15Z)
• Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs [16.88394293874848]
  We use cellular sheaf theory to show that the underlying geometry of the graph is deeply linked with the performance of GNNs. By considering a hierarchy of increasingly general sheaves, we study how the ability of the sheaf diffusion process to achieve linear separation of the classes in the infinite time limit expands. We prove that when the sheaf is non-trivial, discretised parametric diffusion processes have greater control than GNNs over their behaviour.
  arXiv  Detail & Related papers  (2022-02-09T17:25:02Z)
• Overcoming Oversmoothness in Graph Convolutional Networks via Hybrid Scattering Networks [11.857894213975644]
  We propose a hybrid graph neural network (GNN) framework that combines traditional GCN filters with band-pass filters defined via the geometric scattering transform. Our theoretical results establish the complementary benefits of the scattering filters to leverage structural information from the graph, while our experiments show the benefits of our method on various learning tasks.
  arXiv  Detail & Related papers  (2022-01-22T00:47:41Z)
• Transferability Properties of Graph Neural Networks [125.71771240180654]
  Graph neural networks (GNNs) are provably successful at learning representations from data supported on moderate-scale graphs. We study the problem of training GNNs on graphs of moderate size and transferring them to large-scale graphs. Our results show that (i) the transference error decreases with the graph size, and (ii) that graph filters have a transferability-discriminability tradeoff that in GNNs is alleviated by the scattering behavior of the nonlinearity.
  arXiv  Detail & Related papers  (2021-12-09T00:08:09Z)
• Stability of Graph Convolutional Neural Networks to Stochastic Perturbations [122.12962842842349]
  Graph convolutional neural networks (GCNNs) are nonlinear processing tools to learn representations from network data. Current analysis considers deterministic perturbations but fails to provide relevant insights when topological changes are random. This paper investigates the stability of GCNNs to perturbed graph perturbations induced by link losses.
  arXiv  Detail & Related papers  (2021-06-19T16:25:28Z)
• Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networks [183.97265247061847]
  We leverage graph signal processing to characterize the representation space of graph neural networks (GNNs) We discuss the role of graph convolutional filters in GNNs and show that any architecture built with such filters has the fundamental properties of permutation equivariance and stability to changes in the topology. We also study the use of GNNs in recommender systems and learning decentralized controllers for robot swarms.
  arXiv  Detail & Related papers  (2020-03-08T13:02:15Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.