# Spectral Graph Neural Networks with Hermite Polynomials: A Comprehensive Study

arXiv: https://arxiv.org/abs/2609.28979 · HTML: https://mlsanity.com/paper/2609.28979/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 33 references that are in the map.
- First posted on arXiv 24 Sep 2026.

## Builds on (most cited first, 23 of 23)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Simple and Deep Graph Convolutional Networks](https://mlsanity.com/paper/2007.02133/) (2020)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [Beyond Low-frequency Information in Graph Convolutional Networks](https://mlsanity.com/paper/2101.00797/) (2021)
- [A critical look at the evaluation of GNNs under heterophily: Are we really making progress?](https://mlsanity.com/paper/2302.11640/) (2023)
- [BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation](https://mlsanity.com/paper/2106.10994/) (2021)
- [How Powerful are Spectral Graph Neural Networks](https://mlsanity.com/paper/2205.11172/) (2022)
- [Stability Properties of Graph Neural Networks](https://mlsanity.com/paper/1905.04497/) (2019)
- [Convolutional Neural Networks on Graphs with Chebyshev Approximation, Revisited](https://mlsanity.com/paper/2202.03580/) (2022)
- [Graph Neural Networks with Learnable and Optimal Polynomial Bases](https://mlsanity.com/paper/2302.12432/) (2023)
- [How Universal Polynomial Bases Enhance Spectral Graph Neural Networks: Heterophily, Over-smoothing, and Over-squashing](https://mlsanity.com/paper/2405.12474/) (2024)
- [ChebNet: Efficient and Stable Constructions of Deep Neural Networks with Rectified Power Units via Chebyshev Approximations](https://mlsanity.com/paper/1911.05467/) (2019)
- [PolyCF: Towards the Optimal Spectral Graph Filters for Collaborative Filtering](https://mlsanity.com/paper/2401.12590/) (2024)
- [Revisiting convolutional neural network on graphs with polynomial approximations of Laplace-Beltrami spectral filtering](https://mlsanity.com/paper/2010.13269/) (2020)
- [Shape-aware Graph Spectral Learning](https://mlsanity.com/paper/2310.10064/) (2023)
- [Learning to Approximate Adaptive Kernel Convolution on Graphs](https://mlsanity.com/paper/2401.11840/) (2024)
- [A Comprehensive Benchmark on Spectral GNNs: The Impact on Efficiency, Memory, and Effectiveness](https://mlsanity.com/paper/2406.09675/) (2024)
- [Clenshaw Graph Neural Networks](https://mlsanity.com/paper/2210.16508/) (2022)
- [Optimizing Polynomial Graph Filters: A Novel Adaptive Krylov Subspace Approach](https://mlsanity.com/paper/2403.07954/) (2024)
- [Piecewise Constant Spectral Graph Neural Network](https://mlsanity.com/paper/2505.04808/) (2025)

## Abstract (arXiv)

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.