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Piecewise Constant Spectral Graph Neural Network

Vahan Martirosyan, Jhony H. Giraldo, Fragkiskos D. Malliaros

TL;DR

PieCoN addresses the rigidity of traditional spectral GNNs that rely on low-degree polynomial filters by introducing piecewise constant spectral filters combined with polynomial filters. It uses adaptive spectrum partitioning to isolate informative frequency bands and integrates interval-based constant filters with learned polynomial components, improving expressivity, especially on heterophilic graphs. Theoretical results provide error bounds for polynomial approximation and establish invariances to eigenvector sign flips and basis changes, while experiments across nine datasets show PieCoN achieving state-of-the-art or competitive results, with notable gains on heterophilic graphs. This approach enhances spectral GNNs by enabling sharper spectral control and robust learning from the graph spectrum, offering a practical pathway for leveraging spectral structure in diverse graph domains.

Abstract

Graph Neural Networks (GNNs) have achieved significant success across various domains by leveraging graph structures in data. Existing spectral GNNs, which use low-degree polynomial filters to capture graph spectral properties, may not fully identify the graph's spectral characteristics because of the polynomial's small degree. However, increasing the polynomial degree is computationally expensive and beyond certain thresholds leads to performance plateaus or degradation. In this paper, we introduce the Piecewise Constant Spectral Graph Neural Network(PieCoN) to address these challenges. PieCoN combines constant spectral filters with polynomial filters to provide a more flexible way to leverage the graph structure. By adaptively partitioning the spectrum into intervals, our approach increases the range of spectral properties that can be effectively learned. Experiments on nine benchmark datasets, including both homophilic and heterophilic graphs, demonstrate that PieCoN is particularly effective on heterophilic datasets, highlighting its potential for a wide range of applications.

Piecewise Constant Spectral Graph Neural Network

TL;DR

PieCoN addresses the rigidity of traditional spectral GNNs that rely on low-degree polynomial filters by introducing piecewise constant spectral filters combined with polynomial filters. It uses adaptive spectrum partitioning to isolate informative frequency bands and integrates interval-based constant filters with learned polynomial components, improving expressivity, especially on heterophilic graphs. Theoretical results provide error bounds for polynomial approximation and establish invariances to eigenvector sign flips and basis changes, while experiments across nine datasets show PieCoN achieving state-of-the-art or competitive results, with notable gains on heterophilic graphs. This approach enhances spectral GNNs by enabling sharper spectral control and robust learning from the graph spectrum, offering a practical pathway for leveraging spectral structure in diverse graph domains.

Abstract

Graph Neural Networks (GNNs) have achieved significant success across various domains by leveraging graph structures in data. Existing spectral GNNs, which use low-degree polynomial filters to capture graph spectral properties, may not fully identify the graph's spectral characteristics because of the polynomial's small degree. However, increasing the polynomial degree is computationally expensive and beyond certain thresholds leads to performance plateaus or degradation. In this paper, we introduce the Piecewise Constant Spectral Graph Neural Network(PieCoN) to address these challenges. PieCoN combines constant spectral filters with polynomial filters to provide a more flexible way to leverage the graph structure. By adaptively partitioning the spectrum into intervals, our approach increases the range of spectral properties that can be effectively learned. Experiments on nine benchmark datasets, including both homophilic and heterophilic graphs, demonstrate that PieCoN is particularly effective on heterophilic datasets, highlighting its potential for a wide range of applications.
Paper Structure (27 sections, 9 theorems, 28 equations, 13 figures, 4 tables, 1 algorithm)

This paper contains 27 sections, 9 theorems, 28 equations, 13 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background
  4. Graph Neural Networks
  5. Graph Fourier Transform and Spectral GNNs.
  6. Spatial GNNs.
  7. Piecewise Constant Spectral Graph Neural Network (PieCoN)
  8. Identifying Significant Points in the Spectrum
  9. Construction of Constant Filters
  10. Polynomial Filters
  11. Combining Constant and Polynomial Filters
  12. Computational Complexity
  13. Theoretical Analysis
  14. Error Analysis for Polynomial Approximation
  15. Sign and Basis Invariance
  16. ...and 12 more sections

Key Result

Theorem 1

Let $\boldsymbol{\hat{A}} \in \mathbb{R}^{n \times n}$ be a normalized adjacency matrix with spectrum $\{\lambda_i\}_{i=1}^n$ where $-1 \leq \lambda_1 \leq \cdots \leq \lambda_n \leq 1$. Assume that these eigenvalues are $\epsilon$-dense on $[-1,1]$ and that $d^2\epsilon<1$. Let $f:[-1,1] \to \mathb

Figures (13)

  • Figure 1: Comparison of JacobiConv and PieCoN trained filters on the Chameleon dataset.
  • Figure 2: Overview of the PieCoN model. Our method processes an input graph through eigenvalue segmentation (Alg. \ref{['a1']}) to create constant filters, while separately applying polynomial filters. These filters are trained and combined to create the final spectral filter.
  • Figure 3: Using the derivative of eigenvalues to identify significant points which show relatively high changes in the spectrum.
  • Figure 4: Constant filters.
  • Figure 5: Polynomial filters.
  • ...and 8 more figures

Theorems & Definitions (13)

  • Theorem 1: Approximation error for $\epsilon$-dense eigenvalues
  • Theorem 2: Approximation error for functions with jump discontinuities
  • Proposition 1: This follows directly from lim2023sign
  • Proposition 2
  • Theorem 1: Approximation error for $\epsilon$-dense eigenvalues
  • proof
  • Theorem 2: Approximation error for functions with jump discontinuities
  • proof
  • Proposition 1: This follows directly from lim2023sign
  • proof
  • ...and 3 more