L2G-Net: Local to Global Spectral Graph Neural Networks via Cauchy Factorizations

2026-02-21

AI Summary

Traditional spectral graph neural networks struggle to effectively model long-range dependencies due to the high computational cost of full-graph eigendecomposition and the lack of vertex-domain locality. This work proposes a local–global spectral graph neural network that circumvents full-graph decomposition by performing localized spectral analysis on subgraphs and efficiently integrating global information through Cauchy-structured matrices. Driven by graph topology, the method operates with only quadratic computational complexity while preserving both local structural awareness and the capacity to capture global dependencies. On benchmark tasks emphasizing non-local relationships, the approach achieves state-of-the-art performance with orders of magnitude fewer parameters than existing spectral methods.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Search and Optimization: Local Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Social Networks and Social Media: Computational social science

Abstract

Despite their theoretical advantages, spectral methods based on the graph Fourier transform (GFT) are seldom used in graph neural networks (GNNs) due to the cost of computing the eigenbasis and the lack of vertex-domain locality in spectral representations. As a result, most GNNs rely on local approximations such as polynomial Laplacian filters or message passing, which limit their ability to model long-range dependencies. In this paper, we introduce a novel factorization of the GFT into operators acting on subgraphs, which are then combined via a sequence of Cauchy matrices. We use this factorization to propose a new class of spectral GNNs, which we term L2G-Net (Local-to-Global Net). Unlike existing spectral methods, which are either fully global (when they use the GFT) or local (when they use polynomial filters), L2G-Net operates by processing the spectral representations of subgraphs and then combining them via structured matrices. Our algorithm avoids full eigendecompositions, exploiting graph topology to construct the factorization with quadratic complexity in the number of nodes, scaled by the subgraph interface size. Experiments on benchmarks stressing non-local dependencies show that L2G-Net outperforms existing spectral techniques and is competitive with the state-of-the-art with orders of magnitude fewer learnable parameters.

Problem

• spectral graph neural networks
• graph Fourier transform
• long-range dependencies
• vertex-domain locality
• eigenbasis computation

Innovation

• spectral graph neural networks
• Cauchy factorization
• graph Fourier transform
• local-to-global learning
• subgraph decomposition