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 among 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：尽管谱图神经网络在谱域中具有坚实的理论基础，但其实际应用中依赖多项式逼近的特性，意味着它们与空间域存在深刻联系。由于以往研究鲜少从空间视角审视谱图神经网络，其在空间域的可解释性仍不明确，例如：谱图神经网络在空间域中本质上编码了哪些信息？为解答这一问题，本文在谱滤波与空间聚合之间建立了理论关联，揭示了一种内在交互机制：谱滤波隐式地将原始图引导至一个经过修正的新图，该新图可显式地用于空间聚合。理论与实证研究均表明，这一修正后的新图不仅展现出非局部性，还引入了带符号的边权重以反映节点间的标签一致性。这些发现凸显了谱图神经网络在空间域中的可解释性作用，并启发我们重新审视那些忽略全局信息且受限于固定阶多项式的图谱滤波器。基于上述理论发现，我们重新评估了当前最先进的谱图神经网络，并提出了一种新颖的空间自适应滤波框架，该框架利用谱滤波产生的修正新图实现辅助性的非局部聚合。值得注意的是，我们提出的SAF框架从全局视角综合建模了节点的相似性与差异性，从而缓解了图神经网络在处理长程依赖和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，我们提出的框架优于现有最先进模型。