基于Hermite多项式的谱图神经网络：综合研究

中文摘要 AI 辅助

我们研究了基于Hermite多项式构建的谱图神经网络，并提出了HermNet，这是一个结合节点级预测器与归一化Hermite传播的简单模型。其稀疏递推既不需要特征分解，也不需要学习基。我们将基本模型与可选的坐标校准、响应归一化和高斯导数正则化区分开来。Hermite和其他完备多项式基张成相同的度有界滤波器空间，但它们的坐标在有限训练预算下可能产生不同的优化行为。我们通过谱信号能量、标签采样、学习特征的变化以及正则化的偏差-方差权衡来分析这种行为。受控的合成实验确定了一个区间，在该区间内，普通HermNet优于匹配的多项式基替代方案，包括联合训练的非线性预测器。当每个比较器都可用相同的功能惩罚时，曲率正则化进一步改善了HermNet。固定预测器控制支持短训练预算下的优势，但更长的训练消除了普通模型的领先优势。匹配的真实数据比较显示准确性缺陷，而架构和数值研究确定了进一步的限制。总之，分析和实验阐明了Hermite传播何时有用，以及校准和正则化如何影响其性能。

英文摘要

We study spectral graph neural networks built from Hermite polynomials and propose HermNet, a simple model that combines a nodewise predictor with normalized Hermite propagation. Its sparse recurrence requires neither eigendecomposition nor a learned basis. We distinguish the basic model from optional coordinate calibration, response normalization and Gaussian derivative regularization. Hermite and other complete polynomial bases span the same degree-bounded filter space, but their coordinates can produce different optimization behavior under limited training budgets. We analyze this behavior through spectral signal energy, label sampling, changes in learned features and the bias--variance trade-off of regularization. Controlled synthetic experiments identify a regime in which plain HermNet outperforms matched polynomial-basis alternatives, including with a jointly trained nonlinear predictor. Curvature regularization further improves HermNet when the same functional penalty is available to every comparator. Fixed-predictor controls support the advantage under short training budgets, but longer training removes the plain-model lead. Matched real-data comparisons show accuracy deficits, and architectural and numerical studies identify further limits. Together, the analysis and experiments clarify when Hermite propagation is useful and how calibration and regularization affect its performance.

发表机构

• UCLA（加州大学洛杉矶分校）

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