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From ChebNet to ChebGibbsNet

Jie Zhang, Min-Te Sun

TL;DR

This work analyzes Spectral Graph Convolutional Networks through polynomial interpolation of graph frequency responses and identifies Gibbs phenomenon as a limiting factor for ChebNet. By introducing Gibbs damping factors to Chebyshev terms and decoupling propagation from transformation, the authors formulate ChebGibbsNet, which adapts to graph homophily and supports both homogeneous and heterogeneous graphs. The approach yields substantial performance gains over state-of-the-art SpecGCNs on multiple datasets, supported by spectral-domain analysis and ablation studies. The results highlight the practical value of damping-based polynomial filters and point to future exploration of other orthogonal polynomial families for graph representation learning.

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.

From ChebNet to ChebGibbsNet

TL;DR

This work analyzes Spectral Graph Convolutional Networks through polynomial interpolation of graph frequency responses and identifies Gibbs phenomenon as a limiting factor for ChebNet. By introducing Gibbs damping factors to Chebyshev terms and decoupling propagation from transformation, the authors formulate ChebGibbsNet, which adapts to graph homophily and supports both homogeneous and heterogeneous graphs. The approach yields substantial performance gains over state-of-the-art SpecGCNs on multiple datasets, supported by spectral-domain analysis and ablation studies. The results highlight the practical value of damping-based polynomial filters and point to future exploration of other orthogonal polynomial families for graph representation learning.

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.

Paper Structure

This paper contains 15 sections, 1 theorem, 15 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminary and Background
  4. Homogeneous Graphs and Heterogeneous Graphs
  5. Graph Signal Processing
  6. From ChebNet to GCN
  7. Proposed Method
  8. Polynomial Interpolation for Spectral Graph Convolutional Networks
  9. Chebyshev Polynomial Interpolation
  10. Gibbs Phenomenon and Gibbs Damping Factors
  11. ChebGibbsNet and Analysis from Spectral Domain
  12. Experiments
  13. Datasets and Experimental Setup
  14. Experiment results and analysis
  15. Conclusion and Future Work

Key Result

Theorem 4.1

Suppose $f$ be a continuous function on $[a, b]$. For every $\epsilon > 0$, there exists a polynomial $p$ such that $\lVert{f(x)-p(x)}\rVert_{\infty}<\epsilon$.

Figures (2)

  • Figure 1: The target function is discontinuous
  • Figure 2: The target function is singular

Theorems & Definitions (9)

  • Definition 3.1: Node Homophily
  • Definition 3.2: Chebyshev Polynomials of the first kind
  • Theorem 4.1: Weierstrass Approximation Theorem weierstrass_2013
  • Definition 4.2: Graph Polynomial Interpolation
  • Definition 4.3: Graph Polynomial Interpolation Error Function
  • Definition 4.4: Chebyshev Polynomial Interpolation
  • Definition 4.5: ChebGibbsNet
  • Definition 4.6: Spectral gap
  • Definition 4.7: Graph diffusion distance