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

TL;DR

This paper analyzes spectral GNNs, introducing adaptive and stabilized models in the [-1, 1] domain, revealing trade-offs in modeling heterophily and stability, and highlighting the effectiveness of static models like S-JacobiNet.

Contribution

It introduces L-JacobiNet and S-JacobiNet models, providing a comparative analysis of adaptive versus static spectral filters in GNNs, and uncovers key trade-offs and stabilization issues.

Findings

S-JacobiNet outperforms L-JacobiNet on 4/5 datasets.

The [-1, 1] domain offers better numerical stability at high K.

The main flaw of ChebyNet is stabilization, not static nature.

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.