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

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.