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

TL;DR

MeixnerNet introduces a spectral GNN using discrete orthogonal polynomials with learnable parameters, improving robustness and performance over traditional polynomial filters like Chebyshev, especially across different hyperparameter settings.

Contribution

The paper proposes MeixnerNet, a novel spectral GNN architecture employing discrete Meixner polynomials with learnable shape parameters, addressing numerical instability and hyperparameter fragility.

Findings

MeixnerNet outperforms ChebyNet on 2 out of 3 benchmarks.

It maintains stable performance across polynomial degree variations.

The model effectively adapts its spectral filter to graph properties.

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 $M_k(x; \beta, c)$. 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.