Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Polynomial Selection in Spectral Graph Neural Networks: An Error-Sum of Function Slices Approach

Guoming Li, Jian Yang, Shangsong Liang, Dongsheng Luo

TL;DR

This work tackles the polynomial selection problem in spectral graph neural networks by introducing an error-sum of function slices framework that ties polynomial approximation errors to graph-convolution construction error. It proves theoretical bounds linking slice-wise polynomial errors to GNN performance and motivates using narrow-slice-friendly polynomials. Building on this theory, the authors propose a trigonometric filter with Taylor-based parameter decomposition, yielding a parameter-efficient, scalable filter design, and instantiate it in TFGNN, a decoupled spectral GNN. Extensive experiments on node classification and graph anomaly detection demonstrate that TFGNN achieves state-of-the-art or competitive results across diverse datasets while maintaining efficiency comparable to leading methods. The work provides a principled approach to polynomial design in spectral GNNs and highlights practical implications for large-scale graph learning.

Abstract

Spectral graph neural networks are proposed to harness spectral information inherent in graph-structured data through the application of polynomial-defined graph filters, recently achieving notable success in graph-based web applications. Existing studies reveal that various polynomial choices greatly impact spectral GNN performance, underscoring the importance of polynomial selection. However, this selection process remains a critical and unresolved challenge. Although prior work suggests a connection between the approximation capabilities of polynomials and the efficacy of spectral GNNs, there is a lack of theoretical insights into this relationship, rendering polynomial selection a largely heuristic process. To address the issue, this paper examines polynomial selection from an error-sum of function slices perspective. Inspired by the conventional signal decomposition, we represent graph filters as a sum of disjoint function slices. Building on this, we then bridge the polynomial capability and spectral GNN efficacy by proving that the construction error of graph convolution layer is bounded by the sum of polynomial approximation errors on function slices. This result leads us to develop an advanced filter based on trigonometric polynomials, a widely adopted option for approximating narrow signal slices. The proposed filter remains provable parameter efficiency, with a novel Taylor-based parameter decomposition that achieves streamlined, effective implementation. With this foundation, we propose TFGNN, a scalable spectral GNN operating in a decoupled paradigm. We validate the efficacy of TFGNN via benchmark node classification tasks, along with an example graph anomaly detection application to show its practical utility.

Polynomial Selection in Spectral Graph Neural Networks: An Error-Sum of Function Slices Approach

TL;DR

This work tackles the polynomial selection problem in spectral graph neural networks by introducing an error-sum of function slices framework that ties polynomial approximation errors to graph-convolution construction error. It proves theoretical bounds linking slice-wise polynomial errors to GNN performance and motivates using narrow-slice-friendly polynomials. Building on this theory, the authors propose a trigonometric filter with Taylor-based parameter decomposition, yielding a parameter-efficient, scalable filter design, and instantiate it in TFGNN, a decoupled spectral GNN. Extensive experiments on node classification and graph anomaly detection demonstrate that TFGNN achieves state-of-the-art or competitive results across diverse datasets while maintaining efficiency comparable to leading methods. The work provides a principled approach to polynomial design in spectral GNNs and highlights practical implications for large-scale graph learning.

Abstract

Spectral graph neural networks are proposed to harness spectral information inherent in graph-structured data through the application of polynomial-defined graph filters, recently achieving notable success in graph-based web applications. Existing studies reveal that various polynomial choices greatly impact spectral GNN performance, underscoring the importance of polynomial selection. However, this selection process remains a critical and unresolved challenge. Although prior work suggests a connection between the approximation capabilities of polynomials and the efficacy of spectral GNNs, there is a lack of theoretical insights into this relationship, rendering polynomial selection a largely heuristic process. To address the issue, this paper examines polynomial selection from an error-sum of function slices perspective. Inspired by the conventional signal decomposition, we represent graph filters as a sum of disjoint function slices. Building on this, we then bridge the polynomial capability and spectral GNN efficacy by proving that the construction error of graph convolution layer is bounded by the sum of polynomial approximation errors on function slices. This result leads us to develop an advanced filter based on trigonometric polynomials, a widely adopted option for approximating narrow signal slices. The proposed filter remains provable parameter efficiency, with a novel Taylor-based parameter decomposition that achieves streamlined, effective implementation. With this foundation, we propose TFGNN, a scalable spectral GNN operating in a decoupled paradigm. We validate the efficacy of TFGNN via benchmark node classification tasks, along with an example graph anomaly detection application to show its practical utility.
Paper Structure (35 sections, 3 theorems, 15 equations, 4 figures, 12 tables)

This paper contains 35 sections, 3 theorems, 15 equations, 4 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Backgrounds and Preliminaries
  3. Connecting Polynomial Capability with Spectral GNN Efficacy
  4. Theoretical insights
  5. Numerical validation
  6. Summary
  7. The proposed TFGNN
  8. Parameter-efficient trigonometric filter
  9. Taylor-based parameter decomposition
  10. Modeling TFGNN as decoupled paradigm
  11. Complexity analysis of TFGNN
  12. Empirical Studies
  13. Slice approximation and filter learning
  14. Benchmark node classification tasks
  15. Datasets and baselines
  16. ...and 20 more sections

Key Result

lemma 1

Let $f(x)$ be a function composed of function slices $f_{s}(x)$, $s=1,2,...,n$. Let $\mathbf{T}_{0:D}(x;f)$ be a $D$-degree polynomial that provides LSE approximation of $f(x)$ with error $\epsilon$. Specially, define $\epsilon_{s}$, $s=1,2,...,n$, as the polynomial approximation error of each slice

Figures (4)

  • Figure 1: Example of function slicing. $f(x)$ is dissected into three components, determined by its eigenvalues.
  • Figure 2: The functions served as target filters. Additional mathematical details are available in Appendix.
  • Figure 3: Ablation studies on $K$ and $\omega$. Darker shades indicate higher results. Additional results are in Appendix.
  • Figure 4: Additional ablation studies on $K$ and $\omega$. Darker shades indicate higher performance values.

Theorems & Definitions (6)

  • definition 1
  • definition 2
  • definition 3
  • lemma 1
  • theorem 1
  • theorem 2