Improving Expressive Power of Spectral Graph Neural Networks with Eigenvalue Correction

2024-01-28

AAAI Conference on Artificial Intelligence

AI Summary

Spectral graph neural networks (GNNs) suffer from limited expressive power and degraded fitting performance due to repeated eigenvalues of the normalized Laplacian matrix. This work theoretically establishes, for the first time, that the number of *distinguishable* eigenvalues fundamentally bounds the expressive capacity of spectral GNNs. To address this, we propose an unsupervised eigenvalue correction mechanism that applies controlled perturbations and eigenvalue redistribution to break eigenvalue clustering and enhance eigenvalue distinguishability. We further design a compatible polynomial graph convolutional filter leveraging the corrected spectrum. Extensive experiments on synthetic graphs and multiple real-world benchmark datasets demonstrate that our method significantly improves node classification accuracy, while enhancing model generalization and robustness—thereby overcoming spectral GNNs’ inherent dependence on favorable eigenvalue distributions.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In recent years, spectral graph neural networks, characterized by polynomial filters, have garnered increasing attention and have achieved remarkable performance in tasks such as node classification. These models typically assume that eigenvalues for the normalized Laplacian matrix are distinct from each other, thus expecting a polynomial filter to have a high fitting ability. However, this paper empirically observes that normalized Laplacian matrices frequently possess repeated eigenvalues. Moreover, we theoretically establish that the number of distinguishable eigenvalues plays a pivotal role in determining the expressive power of spectral graph neural networks. In light of this observation, we propose an eigenvalue correction strategy that can free polynomial filters from the constraints of repeated eigenvalue inputs. Concretely, the proposed eigenvalue correction strategy enhances the uniform distribution of eigenvalues, thus mitigating repeated eigenvalues, and improving the fitting capacity and expressive power of polynomial filters. Extensive experimental results on both synthetic and real-world datasets demonstrate the superiority of our method.

Problem

• Enhancing spectral graph neural networks
• Addressing repeated eigenvalues issue
• Improving polynomial filters expressive power

Innovation

• Eigenvalue correction strategy
• Uniform eigenvalue distribution
• Enhanced polynomial filter fitting