A General Graph Spectral Wavelet Convolution via Chebyshev Order Decomposition

Nian Liu, Xiaoxin He, Thomas Laurent, Francesco Di Giovanni, Michael M. Bronstein, Xavier Bresson

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.

Lay Summary:

Graphs are a powerful way to represent complex systems — from molecules and transportation networks to social interactions. A key challenge in machine learning is teaching models to recognize patterns on these graphs, especially across both local and global structures.

In this work, we introduce WaveGC, a new method for learning on graphs using wavelets — mathematical tools that can capture information at multiple scales. Traditional graph learning methods often rely on fixed mathematical bases and limited filters. WaveGC instead learns flexible, multi-resolution wavelet bases tailored to the structure of each graph, enabling it to more effectively process both short-range and long-range interactions.

We achieve this by decomposing the mathematical building blocks (Chebyshev polynomials) into two components that separately handle broad background patterns and detailed variations. This design satisfies key mathematical criteria for wavelets and allows for efficient learning.

WaveGC consistently outperforms existing techniques across diverse graph learning tasks, from classifying scientific datasets to analyzing large-scale networks. Our work improves the ability of machine learning systems to make sense of structured, interconnected data.

Link To Code: https://github.com/liun-online/WaveGC

Primary Area: Deep Learning->Graph Neural Networks

Keywords: graph wavelet neural network, graph signal processing

Submission Number: 9412