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

Huseyin Goksu

TL;DR

This paper tackles two core limitations of spectral GNNs—poor performance on heterophilic graphs and over-smoothing at high polynomial degrees—by introducing LaguerreNet, an adaptive continuous filter built from learnable Laguerre polynomials. Leveraging a LayerNorm-based stabilization, the model tames unbounded $O(k^2)$ coefficient growth, enabling deep, adaptive filtering. Empirical results show state-of-the-art performance on heterophilic benchmarks and robust, improving performance with larger $K$, contrasting with static filters that collapse. The approach offers a simple, principled alternative to architectural or IIR-based solutions and extends adaptive filtering into the continuous domain within the Askey scheme.

Abstract

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.

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

TL;DR

This paper tackles two core limitations of spectral GNNs—poor performance on heterophilic graphs and over-smoothing at high polynomial degrees—by introducing LaguerreNet, an adaptive continuous filter built from learnable Laguerre polynomials. Leveraging a LayerNorm-based stabilization, the model tames unbounded coefficient growth, enabling deep, adaptive filtering. Empirical results show state-of-the-art performance on heterophilic benchmarks and robust, improving performance with larger , contrasting with static filters that collapse. The approach offers a simple, principled alternative to architectural or IIR-based solutions and extends adaptive filtering into the continuous domain within the Askey scheme.

Abstract

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.

Paper Structure

This paper contains 17 sections, 4 equations, 3 figures, 5 tables.

Table of Contents

  1. INTRODUCTION
  2. RELATED WORK
  3. Spectral Filter Design in GNNs
  4. Solutions for Heterophily
  5. Solutions for Over-smoothing
  6. PROPOSED METHOD: ADAPTIVE POLYNOMIAL FILTERS
  7. Prior Work: Adaptive Discrete Filters
  8. Proposed: LaguerreNet (Adaptive Continuous Filter)
  9. The LaguerreConv Layer and Stabilization
  10. EXPERIMENTS
  11. Experimental Setup
  12. Performance on Homophilic Graphs
  13. Hypothesis 1: Performance on Heterophilic Graphs
  14. Analysis of Adaptive Parameters
  15. Hypothesis 2: Robustness to Over-smoothing (Varying $K$)
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Training dynamics comparison (K=3, H=16). Top 3 rows (homophilic): All models are stable. Bottom 2 rows (heterophilic): 'GAT' and 'APPNP' fail to converge, while our adaptive polynomial filters ('MeixnerNet', 'LaguerreNet', 'KrawtchoukNet') converge quickly to a high, stable accuracy.
  • Figure 2: $K$ (Polynomial Degree) vs. Test Accuracy (PubMed Dataset). 'ChebyNet' (blue) collapses. 'KrawtchoukNet' (green) is stable (by design). 'LaguerreNet' (purple) is also stable *and* improves, despite having unbounded coefficients.
  • Figure 3: Hidden Dimension (Capacity) vs. Test Accuracy (PubMed, K=3). 'LaguerreNet' (purple) and 'MeixnerNet' (orange) consistently outperform other polynomial models across capacities.