MeixnerNet: Adaptive and Robust Spectral Graph Neural Networks with Discrete Orthogonal Polynomials
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| Format: | Preprint |
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2025
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| author | Goksu, Huseyin |
| author_facet | Goksu, Huseyin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_00113 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | MeixnerNet: Adaptive and Robust Spectral Graph Neural Networks with Discrete Orthogonal Polynomials Goksu, Huseyin Machine Learning Signal Processing 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. |
| title | MeixnerNet: Adaptive and Robust Spectral Graph Neural Networks with Discrete Orthogonal Polynomials |
| topic | Machine Learning Signal Processing |
| url | https://arxiv.org/abs/2511.00113 |