Coefficient Decomposition for Spectral Graph Convolution

TL;DR

The paper addresses the unification and enhancement of spectral graph convolutions by treating the polynomial-filter coefficients as a learnable third-order tensor and applying tensor decompositions. It shows that many existing SGCLs can be viewed as special cases of coefficient decomposition and introduces two new architectures, CoDeSGC-CP and CoDeSGC-Tucker, based on CP and Tucker decompositions, respectively. Empirical results on node classification across 10 real-world graphs demonstrate that the proposed methods outperform a strong baseline (JacobiConv) on 8 of 10 datasets, with notable gains on certain datasets, illustrating the practical value of the coefficient-tensor perspective. The work provides a general, extensible framework for designing spectral GNNs with richer, multilinear coefficient relationships that improve expressivity without abandoning the spectral-convolution foundation.

Abstract

Spectral graph convolutional network (SGCN) is a kind of graph neural networks (GNN) based on graph signal filters, and has shown compelling expressivity for modeling graph-structured data. Most SGCNs adopt polynomial filters and learn the coefficients from the training data. Many of them focus on which polynomial basis leads to optimal expressive power and models' architecture is little discussed. In this paper, we propose a general form in terms of spectral graph convolution, where the coefficients of polynomial basis are stored in a third-order tensor. Then, we show that the convolution block in existing SGCNs can be derived by performing a certain coefficient decomposition operation on the coefficient tensor. Based on the generalized view, we develop novel spectral graph convolutions CoDeSGC-CP and -Tucker by tensor decomposition CP and Tucker on the coefficient tensor. Extensive experimental results demonstrate that the proposed convolutions achieve favorable performance improvements.