A General Graph Spectral Wavelet Convolution via Chebyshev Order Decomposition

2024-05-22

AI Summary

Existing graph spectral convolution methods suffer from limited flexibility in spectral basis selection and kernel parameterization, hindering effective modeling of large-scale spatial signal distributions and restricting the expressivity of spectral filters. To address these limitations, we propose WaveGC—a novel graph convolutional model that integrates multi-resolution graph wavelet bases with matrix-valued spectral filter kernels for efficient graph convolution. Our key innovation lies in constructing general-purpose graph wavelets satisfying the admissibility condition via odd-even Chebyshev polynomial decomposition, enabling theoretical decoupling of short-range and long-range information and supporting their adaptive fusion. Extensive experiments demonstrate that WaveGC consistently outperforms state-of-the-art graph wavelet neural networks on both short-range and long-range graph learning tasks, validating its enhanced representational capacity and superior generalization performance.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Spectral graph convolution, an important tool of data filtering on graphs, relies on two essential decisions: selecting spectral bases for signal transformation and parameterizing the kernel for frequency analysis. While recent techniques mainly focus on standard Fourier transform and vector-valued spectral functions, they fall short in flexibility to model signal distributions over large spatial ranges, and capacity of spectral function. In this paper, we present a novel wavelet-based graph convolution network, namely WaveGC, which integrates multi-resolution spectral bases and a matrix-valued filter kernel. Theoretically, we establish that WaveGC can effectively capture and decouple short-range and long-range information, providing superior filtering flexibility, surpassing existing graph wavelet neural networks. To instantiate WaveGC, we introduce a novel technique for learning general graph wavelets by separately combining odd and even terms of Chebyshev polynomials. This approach strictly satisfies wavelet admissibility criteria. Our numerical experiments showcase the consistent improvements in both short-range and long-range tasks. This underscores the effectiveness of the proposed model in handling different scenarios. Our code is available at https://github.com/liun-online/WaveGC.

Problem

• Enhancing flexibility in spectral graph convolution for large spatial ranges
• Improving spectral function capacity in graph signal filtering
• Capturing and decoupling short-range and long-range information effectively

Innovation

• Wavelet-based graph convolution network
• Multi-resolution spectral bases integration
• Chebyshev polynomial decomposition technique