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.

翻译：谱图神经网络（GNNs）存在两个关键局限：在“异质性”图上的性能不佳，以及在高多项式阶数（K）时出现性能崩溃，即过平滑现象。这两个问题均源于标准滤波器（如ChebyNet）静态、低通的本质。虽然自适应多项式滤波器（如离散MeixnerNet）已成为潜在的统一解决方案，但其向连续域的扩展以及无界系数下的稳定性仍是待解难题。本研究提出`LaguerreNet`——一种基于连续拉盖尔多项式的新型GNN滤波器。`LaguerreNet`通过使其核心参数α可训练来学习滤波器的谱形状，从而推进了自适应多项式方法。我们采用基于`LayerNorm`的稳定化技术，解决了这些无界多项式严重的O(k^2)数值不稳定性问题。实验证明该方法极为有效：1）`LaguerreNet`在具有挑战性的异质性基准测试中取得了最先进的结果；2）其对过平滑具有卓越的鲁棒性，性能在K=10时达到峰值，比ChebyNet崩溃时的阶数高出一个数量级。