From ChebNet to ChebGibbsNet

TL;DR

This paper identifies the Gibbs phenomenon as a key factor limiting ChebNet's performance and proposes ChebGibbsNet, which incorporates Gibbs damping to improve spectral graph convolutional network accuracy on various graph types.

Contribution

It introduces Gibbs damping to mitigate the Gibbs phenomenon in ChebNet, and reorganizes the model into ChebGibbsNet, achieving superior performance over existing methods.

Findings

ChebGibbsNet outperforms GPR-GNN and BernNet on multiple graph types.

Gibbs damping significantly reduces approximation errors caused by the Gibbs phenomenon.

Decoupling feature propagation and transformation enhances model effectiveness.

Abstract

Recent advancements in Spectral Graph Convolutional Networks (SpecGCNs) have led to state-of-the-art performance in various graph representation learning tasks. To exploit the potential of SpecGCNs, we analyze corresponding graph filters via polynomial interpolation, the cornerstone of graph signal processing. Different polynomial bases, such as Bernstein, Chebyshev, and monomial basis, have various convergence rates that will affect the error in polynomial interpolation. Although adopting Chebyshev basis for interpolation can minimize maximum error, the performance of ChebNet is still weaker than GPR-GNN and BernNet. \textbf{We point out it is caused by the Gibbs phenomenon, which occurs when the graph frequency response function approximates the target function.} It reduces the approximation ability of a truncated polynomial interpolation. In order to mitigate the Gibbs phenomenon, we propose to add the Gibbs damping factor with each term of Chebyshev polynomials on ChebNet. As a result, our lightweight approach leads to a significant performance boost. Afterwards, we reorganize ChebNet via decoupling feature propagation and transformation. We name this variant as \textbf{ChebGibbsNet}. Our experiments indicate that ChebGibbsNet is superior to other advanced SpecGCNs, such as GPR-GNN and BernNet, in both homogeneous graphs and heterogeneous graphs.