Stability to Deformations of Manifold Filters and Manifold Neural Networks

• URL: http://arxiv.org/abs/2106.03725v5
• Date: Thu, 14 Mar 2024 18:39:42 GMT
• Title: Stability to Deformations of Manifold Filters and Manifold Neural Networks
• Authors: Zhiyang Wang, Luana Ruiz, Alejandro Ribeiro,
• Abstract summary: The paper defines and studies manifold (M) convolutional filters and neural networks (NNs) The main technical contribution of the paper is to analyze the stability of manifold filters and MNNs to smooth deformations of the manifold.
• Score: 89.53585099149973
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: The paper defines and studies manifold (M) convolutional filters and neural networks (NNs). \emph{Manifold} filters and MNNs are defined in terms of the Laplace-Beltrami operator exponential and are such that \emph{graph} (G) filters and neural networks (NNs) are recovered as discrete approximations when the manifold is sampled. These filters admit a spectral representation which is a generalization of both the spectral representation of graph filters and the frequency response of standard convolutional filters in continuous time. The main technical contribution of the paper is to analyze the stability of manifold filters and MNNs to smooth deformations of the manifold. This analysis generalizes known stability properties of graph filters and GNNs and it is also a generalization of known stability properties of standard convolutional filters and neural networks in continuous time. The most important observation that follows from this analysis is that manifold filters, same as graph filters and standard continuous time filters, have difficulty discriminating high frequency components in the presence of deformations. This is a challenge that can be ameliorated with the use of manifold, graph, or continuous time neural networks. The most important practical consequence of this analysis is to shed light on the behavior of graph filters and GNNs in large-scale graphs.

Related papers

• Dual-Frequency Filtering Self-aware Graph Neural Networks for Homophilic and Heterophilic Graphs [60.82508765185161]
  We propose Dual-Frequency Filtering Self-aware Graph Neural Networks (DFGNN) DFGNN integrates low-pass and high-pass filters to extract smooth and detailed topological features. It dynamically adjusts filtering ratios to accommodate both homophilic and heterophilic graphs.
  arXiv  Detail & Related papers  (2024-11-18T04:57:05Z)
• GrassNet: State Space Model Meets Graph Neural Network [57.62885438406724]
  Graph State Space Network (GrassNet) is a novel graph neural network with theoretical support that provides a simple yet effective scheme for designing arbitrary graph spectral filters. To the best of our knowledge, our work is the first to employ SSMs for the design of graph GNN spectral filters. Extensive experiments on nine public benchmarks reveal that GrassNet achieves superior performance in real-world graph modeling tasks.
  arXiv  Detail & Related papers  (2024-08-16T07:33:58Z)
• Shape-aware Graph Spectral Learning [36.63516222161871]
  Spectral Graph Neural Networks (GNNs) are gaining attention for their ability to surpass the limitations of message-passing GNNs. Some works empirically show that the preferred graph frequency is related to the graph homophily level. This relationship between graph frequency and graphs with homophily/heterophily has not been systematically analyzed and considered in existing spectral GNNs. We propose shape-aware regularization on a Newton Interpolation-based spectral filter that can learn an arbitrary spectral filter and incorporate prior knowledge about the desired shape of the corresponding homophily level.
  arXiv  Detail & Related papers  (2023-10-16T04:57:30Z)
• Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-offs [122.06927400759021]
  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.
  arXiv  Detail & Related papers  (2023-05-29T08:27:17Z)
• Node-oriented Spectral Filtering for Graph Neural Networks [38.0315325181726]
  Graph neural networks (GNNs) have shown remarkable performance on homophilic graph data. In general, learning a universal spectral filter on the graph from the global perspective may still suffer from great difficulty in adapting to the variation of local patterns. We propose the node-oriented spectral filtering for graph neural network (namely NFGNN)
  arXiv  Detail & Related papers  (2022-12-07T14:15:28Z)
• Stability of Aggregation Graph Neural Networks [153.70485149740608]
  We study the stability properties of aggregation graph neural networks (Agg-GNNs) considering perturbations of the underlying graph. We prove that the stability bounds are defined by the properties of the filters in the first layer of the CNN that acts on each node. We also conclude that in Agg-GNNs the selectivity of the mapping operators is tied to the properties of the filters only in the first layer of the CNN stage.
  arXiv  Detail & Related papers  (2022-07-08T03:54:52Z)
• Graph Neural Networks with Adaptive Frequency Response Filter [55.626174910206046]
  We develop a graph neural network framework AdaGNN with a well-smooth adaptive frequency response filter. We empirically validate the effectiveness of the proposed framework on various benchmark datasets.
  arXiv  Detail & Related papers  (2021-04-26T19:31:21Z)
• Interpretable Stability Bounds for Spectral Graph Filters [12.590415345079991]
  We study filter stability and provide a novel and interpretable upper bound on the change of filter output. This upper bound allows us to reason, in terms of structural properties of the graph, when a spectral graph filter will be stable.
  arXiv  Detail & Related papers  (2021-02-18T19:25:52Z)
• Framework for Designing Filters of Spectral Graph Convolutional Neural Networks in the Context of Regularization Theory [1.0152838128195467]
  Graph convolutional neural networks (GCNNs) have been widely used in graph learning. It has been observed that the smoothness functional on graphs can be defined in terms of the graph Laplacian. In this work, we explore the regularization properties of graph Laplacian and proposed a generalized framework for regularized filter designs in spectral GCNNs.
  arXiv  Detail & Related papers  (2020-09-29T06:19:08Z)

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.