Spectral Graph Neural Networks (GNNs), alternatively known as graph filters, have gained increasing prevalence for heterophily graphs. Optimal graph filters rely on Laplacian eigendecomposition for Fourier transform. In an attempt to avert prohibitive computations, numerous polynomial filters have been proposed. However, polynomials in the majority of these filters are predefined and remain fixed across different graphs, failing to accommodate the varying degrees of heterophily. Addressing this gap, we demystify the intrinsic correlation between the spectral property of desired polynomial bases and the heterophily degrees via thorough theoretical analyses. Subsequently, we develop a novel adaptive heterophily basis wherein the basis vectors mutually form angles reflecting the heterophily degree of the graph. We integrate this heterophily basis with the homophily basis to construct a universal polynomial basis UniBasis, which devises a polynomial filter based graph neural network - UniFilter. It optimizes the convolution and propagation in GNN, thus effectively limiting over-smoothing and alleviating over-squashing. Our extensive experiments, conducted on a diverse range of real-world and synthetic datasets with varying degrees of heterophily, support the superiority of UniFilter. These results not only demonstrate the universality of UniBasis but also highlight its proficiency in graph explanation.

基于谱图神经网络和多项式滤波器，本研究提出了一种新颖的自适应异质基础，通过理论分析探索了期望多项式基与异质性程度之间的内在关联，并将其与同质性基础相结合，构建了一个多项式滤波器的图神经网络UniFilter，成功优化了卷积和传播过程，在各种异质性程度的真实和合成数据集上进行的大量实验证明了UniFilter的优越性，同时突出了UniBasis在图解释方面的能力。

如何通过通用多项式基加强光谱图神经网络：异质性、过度平滑和过度压缩