Graph Neural Networks (GNNs) have established themselves as the leading models for learning on graph-structured data, generally categorized into spatial and spectral approaches. Central to these architectures is the Graph Shift Operator (GSO), a matrix representation of the graph structure used to filter node signals. However, selecting the optimal GSO, whether fixed or learnable, remains largely empirical. In this paper, we introduce a novel alignment gain metric that quantifies the geometric distortion between the input signal and label subspaces. Crucially, our theoretical analysis connects this alignment directly to generalization bounds via a spectral proxy for the Lipschitz constant. This yields a principled, computation-efficient criterion to rank and select the optimal GSO for any prediction task prior to training, eliminating the need for extensive search.

翻译：图神经网络（GNN）已成为处理图结构数据的主流学习模型，通常分为空间方法和谱方法两大类。这些架构的核心是图移位算子（GSO），一种用于滤波节点信号的图结构矩阵表示。然而，如何选择最优的GSO（无论是固定的还是可学习的）在很大程度上仍依赖于经验。本文提出了一种新颖的对齐增益度量，用于量化输入信号与标签子空间之间的几何失真。关键的是，我们的理论分析通过对Lipschitz常数的谱代理，将这种对齐直接与泛化界联系起来。这产生了一种原则性的、计算高效的准则，可在训练前针对任何预测任务对最优GSO进行排序和选择，从而无需进行大量搜索。