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

TL;DR

L2G-Net introduces a novel spectral graph neural network that efficiently combines local subgraph spectral representations via Cauchy factorizations, enabling modeling of long-range dependencies with fewer parameters.

Contribution

The paper proposes a new spectral GNN architecture, L2G-Net, using Cauchy factorizations to efficiently integrate local spectral information into a global model without full eigendecomposition.

Findings

L2G-Net outperforms existing spectral methods on benchmarks with non-local dependencies.

L2G-Net achieves competitive accuracy with significantly fewer learnable parameters.

The method scales quadratically with the number of nodes, leveraging graph topology.

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.