Whilst spectral Graph Neural Networks (GNNs) are theoretically well-founded in the spectral domain, their practical reliance on polynomial approximation implies a profound linkage to the spatial domain. As previous studies rarely examine spectral GNNs from the spatial perspective, their spatial-domain interpretability remains elusive, e.g., what information is essentially encoded by spectral GNNs in the spatial domain? In this paper, to answer this question, we establish a theoretical connection between spectral filtering and spatial aggregation, unveiling an intrinsic interaction that spectral filtering implicitly leads the original graph to an adapted new graph, explicitly computed for spatial aggregation. Both theoretical and empirical investigations reveal that the adapted new graph not only exhibits non-locality but also accommodates signed edge weights to reflect label consistency between nodes. These findings thus highlight the interpretable role of spectral GNNs in the spatial domain and inspire us to rethink graph spectral filters beyond the fixed-order polynomials, which neglect global information. Built upon the theoretical findings, we revisit the state-of-the-art spectral GNNs and propose a novel Spatially Adaptive Filtering (SAF) framework, which leverages the adapted new graph by spectral filtering for an auxiliary non-local aggregation. Notably, our proposed SAF comprehensively models both node similarity and dissimilarity from a global perspective, therefore alleviating persistent deficiencies of GNNs related to long-range dependencies and graph heterophily. Extensive experiments over 13 node classification benchmarks demonstrate the superiority of our proposed framework to the state-of-the-art models.

翻译：尽管谱图神经网络（GNN）在谱域中具有坚实的理论基础，但其在实际中对多项式近似的依赖暗示了其与空间域的深刻关联。由于以往研究鲜少从空间视角审视谱GNN，其在空间域的可解释性仍不明确，例如谱GNN在空间域中本质上编码了何种信息？本文为回答这一问题，建立了谱滤波与空间聚合之间的理论联系，揭示了一种内在交互机制：谱滤波隐式地将原始图导向一个经自适应调整的新图，并显式地用于空间聚合。理论与实证研究均表明，该自适应新图不仅呈现非局部性，还包含带符号的边权重以反映节点间的标签一致性。这些发现凸显了谱GNN在空间域中的可解释性角色，并启发我们重新审视忽视全局信息的固定阶多项式之外的图谱滤波器。基于理论发现，我们重新评估了当前最优的谱GNN，并提出了一种新型空间自适应滤波（SAF）框架，该框架利用谱滤波得到的自适应新图进行辅助非局部聚合。值得注意的是，所提出的SAF从全局视角综合建模了节点相似性与相异性，从而缓解了GNN在长程依赖与图异质性方面的持续缺陷。在13个节点分类基准上的大量实验表明，所提框架性能优于现有最优模型。