Which Graph Shift Operator? A Spectral Answer to an Empirical Question

• URL: http://arxiv.org/abs/2602.06557v1
• Date: Fri, 06 Feb 2026 09:59:54 GMT
• Title: Which Graph Shift Operator? A Spectral Answer to an Empirical Question
• Authors: Yassine Abbahaddou,
• Abstract summary: We introduce a novel alignment gain metric that quantifies the geometric distortion between the input signal and label subspaces.<n>Our theoretical analysis connects this alignment directly to generalization bounds via a spectral proxy for the Lipschitz constant.<n>This yields a principled, efficient criterion to rank computation and select the optimal GSO for any prediction task prior to training.
• Score: 2.0917449835910404
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Graph Neural Networks (GNNs) have established themselves as the leading models for learning on graph-structured data, generally categorized into spatial and spectral approaches. Central to these architectures is the Graph Shift Operator (GSO), a matrix representation of the graph structure used to filter node signals. However, selecting the optimal GSO, whether fixed or learnable, remains largely empirical. In this paper, we introduce a novel alignment gain metric that quantifies the geometric distortion between the input signal and label subspaces. Crucially, our theoretical analysis connects this alignment directly to generalization bounds via a spectral proxy for the Lipschitz constant. This yields a principled, computation-efficient criterion to rank and select the optimal GSO for any prediction task prior to training, eliminating the need for extensive search.

Related papers

• Advection-Diffusion on Graphs: A Bakry-Emery Laplacian for Spectral Graph Neural Networks [2.6124895681424323]
  Graph Neural Networks (GNNs) often struggle to propagate information across long distances due to oversmoothing and oversquashing.<n>We introduce a Bakry-Emery graph Laplacian that integrates diffusion and advection through a learnable node-wise potential.<n>We develop mu-ChebNet, a spectral architecture that jointly learns the potential and Chebyshev filters.
  arXiv  Detail & Related papers  (2026-02-20T11:01:12Z)
• A Spectral Interpretation of Redundancy in a Graph Reservoir [51.40366905583043]
  This work revisits the definition of the reservoir in the Multiresolution Reservoir Graph Neural Network (MRGNN)<n>It proposes a variant based on a Fairing algorithm originally introduced in the field of surface design in computer graphics.<n>The core contribution of the paper lies in the theoretical analysis of the algorithm from a random walks perspective.
  arXiv  Detail & Related papers  (2025-07-17T10:02:57Z)
• Centrality Graph Shift Operators for Graph Neural Networks [21.136895833789442]
  We study Centrality GSOs (CGSOs) which normalize adjacency matrices by global centrality metrics. We show how our CGSO can act as the message passing operator in any Graph Neural Network.
  arXiv  Detail & Related papers  (2024-11-07T12:32:24Z)
• What Improves the Generalization of Graph Transformers? A Theoretical Dive into the Self-attention and Positional Encoding [67.59552859593985]
  Graph Transformers, which incorporate self-attention and positional encoding, have emerged as a powerful architecture for various graph learning tasks. This paper introduces first theoretical investigation of a shallow Graph Transformer for semi-supervised classification.
  arXiv  Detail & Related papers  (2024-06-04T05:30:16Z)
• Spectral Greedy Coresets for Graph Neural Networks [61.24300262316091]
  The ubiquity of large-scale graphs in node-classification tasks hinders the real-world applications of Graph Neural Networks (GNNs) This paper studies graph coresets for GNNs and avoids the interdependence issue by selecting ego-graphs based on their spectral embeddings. Our spectral greedy graph coreset (SGGC) scales to graphs with millions of nodes, obviates the need for model pre-training, and applies to low-homophily graphs.
  arXiv  Detail & Related papers  (2024-05-27T17:52:12Z)
• Graph Spectral Embedding using the Geodesic Betweeness Centrality [76.27138343125985]
  We introduce the Graph Sylvester Embedding (GSE), an unsupervised graph representation of local similarity, connectivity, and global structure. GSE uses the solution of the Sylvester equation to capture both network structure and neighborhood proximity in a single representation.
  arXiv  Detail & Related papers  (2022-05-07T04:11:23Z)
• Optimal Propagation for Graph Neural Networks [51.08426265813481]
  We propose a bi-level optimization approach for learning the optimal graph structure. We also explore a low-rank approximation model for further reducing the time complexity.
  arXiv  Detail & Related papers  (2022-05-06T03:37:00Z)
• Pointspectrum: Equivariance Meets Laplacian Filtering for Graph Representation Learning [3.7875603451557063]
  Graph Representation Learning (GRL) has become essential for modern graph data mining and learning tasks. While Graph Neural Networks (GNNs) have been used in state-of-the-art GRL architectures, they have been shown to suffer from over smoothing. We propose PointSpectrum, a spectral method that incorporates a set equivariant network to account for a graph's structure.
  arXiv  Detail & Related papers  (2021-09-06T10:59:11Z)
• Graph Networks with Spectral Message Passing [1.0742675209112622]
  We introduce the Spectral Graph Network, which applies message passing to both the spatial and spectral domains. Our results show that the Spectral GN promotes efficient training, reaching high performance with fewer training iterations despite having more parameters.
  arXiv  Detail & Related papers  (2020-12-31T21:33:17Z)
• Bridging the Gap Between Spectral and Spatial Domains in Graph Neural Networks [8.563354084119062]
  We show some equivalence of the graph convolution process regardless it is designed in the spatial or the spectral domain. The proposed framework is used to design new convolutions in spectral domain with a custom frequency profile while applying them in the spatial domain.
  arXiv  Detail & Related papers  (2020-03-26T01:49:24Z)
• Graphon Pooling in Graph Neural Networks [169.09536309161314]
  Graph neural networks (GNNs) have been used effectively in different applications involving the processing of signals on irregular structures modeled by graphs. We propose a new strategy for pooling and sampling on GNNs using graphons which preserves the spectral properties of the graph.
  arXiv  Detail & Related papers  (2020-03-03T21:04:20Z)

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.