BernNet Revisited: Adaptive Spectral Warping and Decomposed Regularisation for Heterophilic Graphs

Spectral graph neural networks on heterophilic graphs face two structural limitations: fixed polynomial bandwidth, which allocates capacity to uninformative frequency regions, and homogeneous regularization, which cannot simultaneously capture smooth trends and sharp spectral transitions. This issue is further compounded by self-information dilution under high-order neighborhood aggregation. We present BernNetII, a framework that resolves each limitation through principled modifications to the Bernstein spectral filter: a learnable spectral warp that adaptively concentrates polynomial resolution on the graph's effective frequency band; and a morphological decomposition of filter coefficients into smooth and step components with individually tailored regularisation, augmented by a residual bypass that preserves node-level features under high-order propagation. We establish a Rademacher complexity bound with separate capacity control for each component and prove convergence of the joint proximal gradient optimisation. Experiments on eight node classification benchmarks demonstrate that BernNetII consistently outperforms or matches state-of-the-art methods, validating its effectiveness across diverse settings. Our code is available at https://github.com/AI-FZY/BernNetII.