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L-JacobiNet and S-JacobiNet: An Analysis of Adaptive Generalization, Stabilization, and Spectral Domain Trade-offs in GNNs

Huseyin Goksu

TL;DR

This work challenges the notion of a single best spectral filter for graphs by examining adaptive orthogonal polynomial filters (AOPF) across two spectral domains. It introduces $L$-JacobiNet$ (adaptive Jacobi basis in $[-1,1]$) and $S$-JacobiNet$ (LayerNorm-stabilized baseline) and compares them to $[0, \infty)$-domain AOPFs, revealing a domain-dependent trade-off: ${[0,\infty)}$ domain excels at modeling heterophily, while the ${[-1,1]}$ domain offers superior numerical stability at high polynomial degrees. Surprisingly, stabilization, not adaptation, is the principal limitation of ChebyNet, as $S$-JacobiNet$ outperforms the adaptive $L$-JacobiNet$ on most datasets. The findings refram e design choices from seeking a single best filter to balancing spectral domain, adaptation, and stabilization, and suggest a dual-branch or split-spectrum GNN as a promising future direction.

Abstract

Spectral GNNs, like ChebyNet, are limited by heterophily and over-smoothing due to their static, low-pass filter design. This work investigates the "Adaptive Orthogonal Polynomial Filter" (AOPF) class as a solution. We introduce two models operating in the [-1, 1] domain: 1) `L-JacobiNet`, the adaptive generalization of `ChebyNet` with learnable alpha, beta shape parameters, and 2) `S-JacobiNet`, a novel baseline representing a LayerNorm-stabilized static `ChebyNet`. Our analysis, comparing these models against AOPFs in the [0, infty) domain (e.g., `LaguerreNet`), reveals critical, previously unknown trade-offs. We find that the [0, infty) domain is superior for modeling heterophily, while the [-1, 1] domain (Jacobi) provides superior numerical stability at high K (K>20). Most significantly, we discover that `ChebyNet`'s main flaw is stabilization, not its static nature. Our static `S-JacobiNet` (ChebyNet+LayerNorm) outperforms the adaptive `L-JacobiNet` on 4 out of 5 benchmark datasets, identifying `S-JacobiNet` as a powerful, overlooked baseline and suggesting that adaptation in the [-1, 1] domain can lead to overfitting.

L-JacobiNet and S-JacobiNet: An Analysis of Adaptive Generalization, Stabilization, and Spectral Domain Trade-offs in GNNs

TL;DR

This work challenges the notion of a single best spectral filter for graphs by examining adaptive orthogonal polynomial filters (AOPF) across two spectral domains. It introduces -JacobiNet[-1,1]S (LayerNorm-stabilized baseline) and compares them to -domain AOPFs, revealing a domain-dependent trade-off: domain excels at modeling heterophily, while the domain offers superior numerical stability at high polynomial degrees. Surprisingly, stabilization, not adaptation, is the principal limitation of ChebyNet, as -JacobiNetL on most datasets. The findings refram e design choices from seeking a single best filter to balancing spectral domain, adaptation, and stabilization, and suggest a dual-branch or split-spectrum GNN as a promising future direction.

Abstract

Spectral GNNs, like ChebyNet, are limited by heterophily and over-smoothing due to their static, low-pass filter design. This work investigates the "Adaptive Orthogonal Polynomial Filter" (AOPF) class as a solution. We introduce two models operating in the [-1, 1] domain: 1) `L-JacobiNet`, the adaptive generalization of `ChebyNet` with learnable alpha, beta shape parameters, and 2) `S-JacobiNet`, a novel baseline representing a LayerNorm-stabilized static `ChebyNet`. Our analysis, comparing these models against AOPFs in the [0, infty) domain (e.g., `LaguerreNet`), reveals critical, previously unknown trade-offs. We find that the [0, infty) domain is superior for modeling heterophily, while the [-1, 1] domain (Jacobi) provides superior numerical stability at high K (K>20). Most significantly, we discover that `ChebyNet`'s main flaw is stabilization, not its static nature. Our static `S-JacobiNet` (ChebyNet+LayerNorm) outperforms the adaptive `L-JacobiNet` on 4 out of 5 benchmark datasets, identifying `S-JacobiNet` as a powerful, overlooked baseline and suggesting that adaptation in the [-1, 1] domain can lead to overfitting.

Paper Structure

This paper contains 19 sections, 2 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 and Stabilization
  6. The AOPF Framework and Domain Analysis
  7. Domain 1: The $[0, \infty)$ (Semi-Infinite) Domain
  8. Domain 2: The $[-1, 1]$ (Finite) Domain
  9. EXPERIMENTAL ANALYSIS
  10. Experimental Setup
  11. Trade-off 1: Heterophily vs. Spectral Domain
  12. Trade-off 2: Stability vs. Spectral Domain
  13. Trade-off 3: Adaptation vs. Stabilization in the $[-1, 1]$ Domain
  14. Discussion and Future Work
  15. The Heterophily vs. Stability Trade-off
  16. ...and 4 more sections

Figures (2)

  • Figure 1: Figure 1: Training dynamics comparison (K=3, H=16). On heterophilic datasets (Texas, Cornell), the $[-1, 1]$ domain filters ('ChebyNet', 'LJacobiNet', 'S-JacobiNet') and standard baselines ('GAT', 'APPNP') struggle, while the $[0, \infty)$ domain filters ('MeixnerNet', 'KrawtchoukNet', 'LaguerreNet') are visibly more stable and accurate.
  • Figure 2: Figure 2: $K$ (Polynomial Degree) vs. Test Accuracy (PubMed). 'ChebyNet' (blue) collapses at $K=5$. 'LaguerreNet' (magenta) is stable until $K=25$, where it catastrophically collapses. 'L-JacobiNet' (purple) remains perfectly stable up to $K=30$.