LaguerreNet: Advancing a Unified Solution for Heterophily and Over-smoothing with Adaptive Continuous Polynomials

Abstract: Spectral Graph Neural Networks (GNNs) suffer from two critical limitations: poor performance on "heterophilic" graphs and performance collapse at high polynomial degrees (K), known as over-smoothing. Both issues stem from the static, low-pass nature of standard filters (e.g., ChebyNet). While adaptive polynomial filters, such as the discrete MeixnerNet, have emerged as a potential unified solution, their extension to the continuous domain and stability with unbounded coefficients remain open questions. In this work, we propose LaguerreNet, a novel GNN filter based on continuous Laguerre polynomials. LaguerreNet learns the filter's spectral shape by making its core alpha parameter trainable, thereby advancing the adaptive polynomial approach. We solve the severe O(k^2) numerical instability of these unbounded polynomials using a LayerNorm-based stabilization technique. We demonstrate experimentally that this approach is highly effective: 1) LaguerreNet achieves state-of-the-art results on challenging heterophilic benchmarks. 2) It is exceptionally robust to over-smoothing, with performance peaking at K=10, an order of magnitude beyond where ChebyNet collapses.