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DualLaguerreNet: A Decoupled Spectral Filter GNN and the Uncovering of the Flexibility-Stability Trade-off

Huseyin Goksu

TL;DR

The paper addresses the LaguerreNet limitation of a single adaptive parameter by proposing DualLaguerreNet, which decouples the spectrum into low- and high-frequency operators and applies independent Laguerre filters with parameters $\alpha_1$ and $\alpha_2$. It demonstrates that this decoupled flexibility yields state-of-the-art results on heterophilic graphs (e.g., Texas and Cornell) but can underperform on homophilic graphs due to a bias-variance trade-off, acting as a regularizer in simpler settings. The work provides a data-driven analysis of how increased spectral flexibility affects generalization, supported by learned parameter separations and stability observations under LayerNorm. These findings offer practical guidance for designing adaptive spectral GNNs, balancing spectrum-specific expressiveness with model stability and capacity control.

Abstract

Graph Neural Networks (GNNs) based on spectral filters, such as the Adaptive Orthogonal Polynomial Filter (AOPF) class (e.g., LaguerreNet), have shown promise in unifying the solutions for heterophily and over-smoothing. However, these single-filter models suffer from a "compromise" problem, as their single adaptive parameter (e.g., alpha) must learn a suboptimal, averaged response across the entire graph spectrum. In this paper, we propose DualLaguerreNet, a novel GNN architecture that solves this by introducing "Decoupled Spectral Flexibility." DualLaguerreNet splits the graph Laplacian into two operators, L_low (low-frequency) and L_high (high-frequency), and learns two independent, adaptive Laguerre polynomial filters, parameterized by alpha_1 and alpha_2, respectively. This work, however, uncovers a deeper finding. While our experiments show DualLaguerreNet's flexibility allows it to achieve state-of-the-art results on complex heterophilic tasks (outperforming LaguerreNet), it simultaneously underperforms on simpler, homophilic tasks. We identify this as a fundamental "Flexibility-Stability Trade-off". The increased parameterization (2x filter parameters and 2x model parameters) leads to overfitting on simple tasks, demonstrating that the "compromise" of simpler models acts as a crucial regularizer. This paper presents a new SOTA architecture for heterophily while providing a critical analysis of the bias-variance trade-off inherent in adaptive GNN filter design.

DualLaguerreNet: A Decoupled Spectral Filter GNN and the Uncovering of the Flexibility-Stability Trade-off

TL;DR

The paper addresses the LaguerreNet limitation of a single adaptive parameter by proposing DualLaguerreNet, which decouples the spectrum into low- and high-frequency operators and applies independent Laguerre filters with parameters and . It demonstrates that this decoupled flexibility yields state-of-the-art results on heterophilic graphs (e.g., Texas and Cornell) but can underperform on homophilic graphs due to a bias-variance trade-off, acting as a regularizer in simpler settings. The work provides a data-driven analysis of how increased spectral flexibility affects generalization, supported by learned parameter separations and stability observations under LayerNorm. These findings offer practical guidance for designing adaptive spectral GNNs, balancing spectrum-specific expressiveness with model stability and capacity control.

Abstract

Graph Neural Networks (GNNs) based on spectral filters, such as the Adaptive Orthogonal Polynomial Filter (AOPF) class (e.g., LaguerreNet), have shown promise in unifying the solutions for heterophily and over-smoothing. However, these single-filter models suffer from a "compromise" problem, as their single adaptive parameter (e.g., alpha) must learn a suboptimal, averaged response across the entire graph spectrum. In this paper, we propose DualLaguerreNet, a novel GNN architecture that solves this by introducing "Decoupled Spectral Flexibility." DualLaguerreNet splits the graph Laplacian into two operators, L_low (low-frequency) and L_high (high-frequency), and learns two independent, adaptive Laguerre polynomial filters, parameterized by alpha_1 and alpha_2, respectively. This work, however, uncovers a deeper finding. While our experiments show DualLaguerreNet's flexibility allows it to achieve state-of-the-art results on complex heterophilic tasks (outperforming LaguerreNet), it simultaneously underperforms on simpler, homophilic tasks. We identify this as a fundamental "Flexibility-Stability Trade-off". The increased parameterization (2x filter parameters and 2x model parameters) leads to overfitting on simple tasks, demonstrating that the "compromise" of simpler models acts as a crucial regularizer. This paper presents a new SOTA architecture for heterophily while providing a critical analysis of the bias-variance trade-off inherent in adaptive GNN filter design.

Paper Structure

This paper contains 17 sections, 5 equations, 3 figures, 5 tables.

Table of Contents

  1. INTRODUCTION
  2. RELATED WORK
  3. Adaptive Orthogonal Polynomial Filters (AOPF)
  4. Split-Spectrum and Dual-Branch GNNs
  5. Bias-Variance and Overfitting in Adaptive GNNs
  6. DualLaguerreNet: Methodology
  7. Preliminaries: LaguerreNet and the "Compromise" Problem
  8. DualLaguerreNet: Decoupled Spectral Flexibility
  9. EXPERIMENTAL ANALYSIS
  10. Experimental Setup
  11. Main Results
  12. Discussion: The Flexibility-Stability Trade-off
  13. Hypothesis 1 (Heterophily): Decoupled Flexibility Wins
  14. The Proof: Two Brains Learn Differently
  15. Hypothesis 2 (Homophily): The Flexibility-Stability Trade-off
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Figure 1: Training Dynamics Comparison (K=3, H=16). On heterophilic datasets (Texas, Cornell), 'GAT' and 'APPNP' fail to converge (cyan, magenta). The AOPF family ('MeixnerNet', 'KrawtchoukNet', 'LaguerreNet', 'DualLaguerreNet') are all stable and converge to high accuracy, with 'DualLaguerreNet' (red) showing strong performance.
  • Figure 2: Figure 2: $K$ (Polynomial Degree) vs. Test Accuracy (PubMed). 'ChebyNet' (blue) collapses from over-smoothing. The AOPF filters, including 'LaguerreNet' (purple) and 'DualLaguerreNet' (red), are robust to high $K$ due to 'LayerNorm' stabilization, confirming our design tames $O(k^2)$ instability goksu2025meixnernet.
  • Figure 3: Figure 3: Hidden Dimension ($H$) vs. Test Accuracy (PubMed, K=3). 'LaguerreNet' (purple) shows signs of overfitting as capacity increases (0.7690 $\to$ 0.7390). 'DualLaguerreNet' (red), despite its higher parameter count, remains stable (0.7700 $\to$ 0.7680).