用可学习的连续拉盖尔多项式解决图神经网络异质性与过平滑问题
LaguerreNet: Advancing a Unified Solution for Heterophily and Over-smoothing with Adaptive Continuous Polynomials
- 基于可训练参数的连续拉盖尔多项式构建自适应图滤波器
- 在K=10时性能最优，远超传统ChebyNet的稳定上限
- 适合处理异质图且对过平滑鲁棒，适合高阶聚合场景
谱图神经网络在异质图上表现差，且在高阶多项式（K）时易发生过平滑。标准滤波器（如ChebyNet）因静态低通特性导致此问题。虽有离散自适应滤波器（如MeixnerNet）出现，但其在连续域的扩展及无界系数稳定性仍未知。本文提出基于连续拉盖尔多项式的LaguerreNet，通过可训练的alpha参数动态调整滤波器频谱形状。采用基于LayerNorm的稳定技术解决O(k²)数值不稳定性问题。实验表明：1）LaguerreNet在挑战性异质图基准上达到顶尖性能；2）对过平滑极强鲁棒，性能峰值出现在K=10，较ChebyNet的崩溃点高出一个数量级。
原文摘要 · Abstract (English)
Spectral Graph Neural Networks (GNNs) suffer from two critical limitations: poor performance on "heterophilic" graphs and performance collapse at high polynomial degrees (K), known as over-smoothing. Both issues stem from the static, low-pass nature of standard filters (e.g., ChebyNet). While adaptive polynomial filters, such as the discrete MeixnerNet, have emerged as a potential unified solution, their extension to the continuous domain and stability with unbounded coefficients remain open questions. In this work, we propose `LaguerreNet`, a novel GNN filter based on continuous Laguerre polynomials. `LaguerreNet` learns the filter's spectral shape by making its core alpha parameter trainable, thereby advancing the adaptive polynomial approach. We solve the severe O(k^2) numerical instability of these unbounded polynomials using a `LayerNorm`-based stabilization technique. We demonstrate experimentally that this approach is highly effective: 1) `LaguerreNet` achieves state-of-the-art results on challenging heterophilic benchmarks. 2) It is exceptionally robust to over-smoothing, with performance peaking at K=10, an order of magnitude beyond where ChebyNet collapses.
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