MeixnerNet: Adaptive and Robust Spectral Graph Neural Networks with Discrete Orthogonal Polynomials

TL;DR

MeixnerNet introduces a discrete, adaptive spectral GNN based on Meixner polynomials, with learnable parameters $β$ and $c$ that tailor the polynomial basis to a graph's spectrum. A two-fold stabilization, combining Laplacian scaling and per-basis LayerNorm, enables stable training of polynomial filters with varying degree $K$. Empirically, it achieves competitive-to-superior results against ChebyNet at $K=2$ and shows markedly higher robustness to $K$ than the baseline, indicating practical advantages in hyperparameter sensitivity. The approach highlights the benefit of aligning spectral filtering with the discrete nature of graphs and motivates exploration of additional discrete polynomial families for robust graph representation learning.

Abstract

Spectral Graph Neural Networks (GNNs) have achieved state-of-the-art results by defining graph convolutions in the spectral domain. A common approach, popularized by ChebyNet, is to use polynomial filters based on continuous orthogonal polynomials (e.g., Chebyshev). This creates a theoretical disconnect, as these continuous-domain filters are applied to inherently discrete graph structures. We hypothesize this mismatch can lead to suboptimal performance and fragility to hyperparameter settings. In this paper, we introduce MeixnerNet, a novel spectral GNN architecture that employs discrete orthogonal polynomials -- specifically, the Meixner polynomials $M_k(x; β, c)$. Our model makes the two key shape parameters of the polynomial, beta and c, learnable, allowing the filter to adapt its polynomial basis to the specific spectral properties of a given graph. We overcome the significant numerical instability of these polynomials by introducing a novel stabilization technique that combines Laplacian scaling with per-basis LayerNorm. We demonstrate experimentally that MeixnerNet achieves competitive-to-superior performance against the strong ChebyNet baseline at the optimal K = 2 setting (winning on 2 out of 3 benchmarks). More critically, we show that MeixnerNet is exceptionally robust to variations in the polynomial degree K, a hyperparameter to which ChebyNet proves to be highly fragile, collapsing in performance where MeixnerNet remains stable.

Figures (3)

• Figure 1: Training loss (left column) and validation accuracy (right column) curves for 'MeixnerNet' (orange) and 'ChebyNet' (blue) at $K=2$. (Ensure this figure is regenerated for the K=2 experiment.)
• Figure 2: The effect of $K$ on test accuracy on the 'PubMed' dataset. 'ChebyNet' (blue) performance collapses at $K=3$, while 'MeixnerNet' (orange) remains robust.
• Figure 3: The effect of model capacity (hidden dimension) on test accuracy on 'PubMed' (at $K=2$).