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ERGNN: Spectral Graph Neural Network With Explicitly-Optimized Rational Graph Filters

Guoming Li, Jian Yang, Shangsong Liang

TL;DR

ERGNN addresses the practical deployment of rational graph filters in spectral GNNs by separating the rational filter into a polynomial numerator and an MLP-based denominator in a two-step framework. This design enables explicit optimization of both components while avoiding costly matrix inversions, underpinned by Appel's method and Chebyshev best-fit theory to justify improved approximation over polynomials. The approach is validated through extensive experiments on node classification, scalability to large graphs, and learning filters from signals, where ERGNN achieves state-of-the-art performance and favorable efficiency. Together, these results demonstrate that rational-based GNNs can be made practical and effective for real-world graph learning tasks.

Abstract

Approximation-based spectral graph neural networks, which construct graph filters with function approximation, have shown substantial performance in graph learning tasks. Despite their great success, existing works primarily employ polynomial approximation to construct the filters, whereas another superior option, namely ration approximation, remains underexplored. Although a handful of prior works have attempted to deploy the rational approximation, their implementations often involve intensive computational demands or still resort to polynomial approximations, hindering full potential of the rational graph filters. To address the issues, this paper introduces ERGNN, a novel spectral GNN with explicitly-optimized rational filter. ERGNN adopts a unique two-step framework that sequentially applies the numerator filter and the denominator filter to the input signals, thus streamlining the model paradigm while enabling explicit optimization of both numerator and denominator of the rational filter. Extensive experiments validate the superiority of ERGNN over state-of-the-art methods, establishing it as a practical solution for deploying rational-based GNNs.

ERGNN: Spectral Graph Neural Network With Explicitly-Optimized Rational Graph Filters

TL;DR

ERGNN addresses the practical deployment of rational graph filters in spectral GNNs by separating the rational filter into a polynomial numerator and an MLP-based denominator in a two-step framework. This design enables explicit optimization of both components while avoiding costly matrix inversions, underpinned by Appel's method and Chebyshev best-fit theory to justify improved approximation over polynomials. The approach is validated through extensive experiments on node classification, scalability to large graphs, and learning filters from signals, where ERGNN achieves state-of-the-art performance and favorable efficiency. Together, these results demonstrate that rational-based GNNs can be made practical and effective for real-world graph learning tasks.

Abstract

Approximation-based spectral graph neural networks, which construct graph filters with function approximation, have shown substantial performance in graph learning tasks. Despite their great success, existing works primarily employ polynomial approximation to construct the filters, whereas another superior option, namely ration approximation, remains underexplored. Although a handful of prior works have attempted to deploy the rational approximation, their implementations often involve intensive computational demands or still resort to polynomial approximations, hindering full potential of the rational graph filters. To address the issues, this paper introduces ERGNN, a novel spectral GNN with explicitly-optimized rational filter. ERGNN adopts a unique two-step framework that sequentially applies the numerator filter and the denominator filter to the input signals, thus streamlining the model paradigm while enabling explicit optimization of both numerator and denominator of the rational filter. Extensive experiments validate the superiority of ERGNN over state-of-the-art methods, establishing it as a practical solution for deploying rational-based GNNs.
Paper Structure (13 sections, 2 theorems, 6 equations, 1 figure, 3 tables)

This paper contains 13 sections, 2 theorems, 6 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Background and Preliminaries
  3. Spectral GNNs
  4. Augmented MLPs as Substitutions of GNNs
  5. Methodology
  6. Model Architecture of ERGNN
  7. Optimization Target
  8. Theoretical Insights
  9. Empirical Studies
  10. Node Classification on Real-world Graphs
  11. Evaluating Scalability on Large Graphs
  12. Learning Graph Filters from Signals
  13. Conclusions

Key Result

Lemma 1

(Restatement of Appel's methodfixed_numerator_learned_denominator-1) Let decay-type function $F(x)$ be the approximation target. Given a fixed function $P(x)$, we can derive a polynomial function $Q(x)=\sum_{i=1}^{n}w_{i}x^{i}$ with coefficients $w_{i}$ computed via Appel's algorithm, resulting in $

Figures (1)

  • Figure 1: Overall framework of ERGNN.

Theorems & Definitions (2)

  • Lemma 1
  • Theorem 1