# LaguerreNet: Advancing a Unified Solution for Heterophily and Over-smoothing with Adaptive Continuous Polynomials

arXiv: https://arxiv.org/abs/2511.15328 · HTML: https://mlsanity.com/paper/2511.15328/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 29 references that are in the map.
- First posted on arXiv 19 Nov 2025.

## Builds on (most cited first, 17 of 17)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [Representation Learning on Graphs with Jumping Knowledge Networks](https://mlsanity.com/paper/1806.03536/) (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)
- [Beyond Low-frequency Information in Graph Convolutional Networks](https://mlsanity.com/paper/2101.00797/) (2021)
- [CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters](https://mlsanity.com/paper/1705.07664/) (2017)
- [Graph Neural Networks with convolutional ARMA filters](https://mlsanity.com/paper/1901.01343/) (2019)
- [Graph Filters for Signal Processing and Machine Learning on Graphs](https://mlsanity.com/paper/2211.08854/) (2022)
- [ERGNN: Spectral Graph Neural Network With Explicitly-Optimized Rational Graph Filters](https://mlsanity.com/paper/2412.19106/) (2024)
- [MeixnerNet: Adaptive and Robust Spectral Graph Neural Networks with Discrete Orthogonal Polynomials](https://mlsanity.com/paper/2511.00113/) (2025)

## Built on by (most cited first, 2 of 2)

- [L-JacobiNet and S-JacobiNet: An Analysis of Adaptive Generalization, Stabilization, and Spectral Domain Trade-offs in GNNs](https://mlsanity.com/paper/2511.16081/) (2025)
- [HybSpecNet: A Critical Analysis of Architectural Instability in Hybrid-Domain Spectral GNNs](https://mlsanity.com/paper/2511.16101/) (2025)

## Abstract (arXiv)

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.