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

TL;DR

DualLaguerreNet introduces a decoupled spectral filter GNN that enhances flexibility for heterophilic tasks but reveals a trade-off where increased model complexity can impair performance on simpler, homophilic tasks.

Contribution

It proposes a novel decoupled spectral filter GNN architecture, DualLaguerreNet, that improves heterophilic task performance and analyzes the inherent bias-variance trade-off in adaptive spectral filters.

Findings

Achieves state-of-the-art results on heterophilic tasks.

Underperforms on homophilic tasks due to overfitting.

Identifies the flexibility-stability trade-off in spectral GNNs.

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.