A General Graph Spectral Wavelet Convolution via Chebyshev Order Decomposition
This work addresses a domain-specific problem in graph neural networks by offering a more flexible and effective method for graph signal filtering, though it appears incremental as it builds on existing wavelet-based approaches.
The authors tackled the problem of limited flexibility and capacity in spectral graph convolution by proposing WaveGC, a wavelet-based graph convolution network that integrates multi-resolution spectral bases and a matrix-valued filter kernel, achieving consistent improvements in both short-range and long-range tasks.
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.