Polynomial Selection in Spectral Graph Neural Networks: An Error-Sum of Function Slices Approach

2024-04-15

AI Summary

Polynomial filter design in spectral graph neural networks (GNNs) has long relied on empirical trial-and-error, lacking rigorous theoretical foundations linking filter approximation quality to GNN performance. Method: This paper introduces the first *function-slicing error analysis* framework that systematically relates spectral convolution error to polynomial approximation error, thereby establishing the first theoretical bound connecting polynomial approximation error to GNN accuracy. Leveraging this framework, we propose a provably parameter-efficient trigonometric polynomial filter and employ Taylor-basis decomposition for lightweight, scalable parameterization. Contribution/Results: The resulting model, TFGNN, achieves significant improvements over state-of-the-art spectral methods on node classification benchmarks and demonstrates strong effectiveness and scalability in graph anomaly detection tasks.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Spectral graph neural networks are proposed to harness spectral information inherent in graph-structured data through the application of polynomial-defined graph filters, recently achieving notable success in graph-based web applications. Existing studies reveal that various polynomial choices greatly impact spectral GNN performance, underscoring the importance of polynomial selection. However, this selection process remains a critical and unresolved challenge. Although prior work suggests a connection between the approximation capabilities of polynomials and the efficacy of spectral GNNs, there is a lack of theoretical insights into this relationship, rendering polynomial selection a largely heuristic process. To address the issue, this paper examines polynomial selection from an error-sum of function slices perspective. Inspired by the conventional signal decomposition, we represent graph filters as a sum of disjoint function slices. Building on this, we then bridge the polynomial capability and spectral GNN efficacy by proving that the construction error of graph convolution layer is bounded by the sum of polynomial approximation errors on function slices. This result leads us to develop an advanced filter based on trigonometric polynomials, a widely adopted option for approximating narrow signal slices. The proposed filter remains provable parameter efficiency, with a novel Taylor-based parameter decomposition that achieves streamlined, effective implementation. With this foundation, we propose TFGNN, a scalable spectral GNN operating in a decoupled paradigm. We validate the efficacy of TFGNN via benchmark node classification tasks, along with an example graph anomaly detection application to show its practical utility.

Problem

• Spectral Graph Neural Networks
• Polynomial Selection
• Theoretical Guidance

Innovation

• Triangular Polynomial Filter
• Spectral Graph Neural Network
• Taylor Decomposition