A General Graph Spectral Wavelet Convolution via Chebyshev Order Decomposition

TL;DR

This paper introduces WaveGC, a novel graph convolution network that uses multi-resolution spectral bases and Chebyshev polynomial decomposition to improve flexibility and effectiveness in modeling both short-range and long-range graph signals.

Contribution

The paper proposes a new wavelet-based graph convolution method, WaveGC, with a Chebyshev polynomial technique for learning wavelets that satisfies admissibility, enhancing spectral filtering capabilities.

Findings

WaveGC outperforms existing methods in various tasks.

It effectively captures short-range and long-range information.

Numerical experiments show consistent improvements.

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.