Spectral graph networks with Hermite polynomials improve training behavior
Spectral Graph Neural Networks with Hermite Polynomials: A Comprehensive Study
Machine Learning
Summary
The paper looks at a way to build neural networks that work on graph data using Hermite polynomials. These polynomials help process signals on graphs without needing complex computations like eigendecomposition. The authors made a simple model called HermNet and studied how it compares with other methods, especially when training time or data is limited. They found that HermNet can perform better in certain controlled cases, but its advantages reduce with longer training or real-world data.
What this means in practice
- •For graph machine learning engineers: Implement graph neural network layers without eigendecomposition for faster training on graphs with limited labeled data.
- •For biomedical data analysts: Use polynomial-based spectral graph models to improve prediction tasks on biological networks under constraints of small training sets.
Authors
Shuang Wu
Abstract
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.