Spectral Graph Neural Network is a kind of Graph Neural Network (GNN) based on graph signal filters. Some models able to learn arbitrary spectral filters have emerged recently. However, few works analyze the expressive power of spectral GNNs. This paper studies spectral GNNs' expressive power theoretically. We first prove that even spectral GNNs without nonlinearity can produce arbitrary graph signals and give two conditions for reaching universality. They are: 1) no multiple eigenvalues of graph Laplacian, and 2) no missing frequency components in node features. We also establish a connection between the expressive power of spectral GNNs and Graph Isomorphism (GI) testing, the latter of which is often used to characterize spatial GNNs' expressive power. Moreover, we study the difference in empirical performance among different spectral GNNs with the same expressive power from an optimization perspective, and motivate the use of an orthogonal basis whose weight function corresponds to the graph signal density in the spectrum. Inspired by the analysis, we propose JacobiConv, which uses Jacobi basis due to its orthogonality and flexibility to adapt to a wide range of weight functions. JacobiConv deserts nonlinearity while outperforming all baselines on both synthetic and real-world datasets.
翻译:光谱 GNNs 的表达力。 我们首先证明光谱 GNNs 没有非线性, 也能产生任意的图形信号, 并给出实现普遍性的两个条件 。 它们是:(1) 图形 Laplacecian 的不多个电子值, 2 节点特性中没有缺失的频率组件。 我们还在光谱 GNNS 和图形地貌测试的表达力之间建立起了联系, 后者通常用来描述空间GNNs 的表达力。 此外, 我们从优化角度研究不同光谱GNNNNs 的实验性能差异, 这些光谱GNNs 具有相同的表达力, 并激励使用其重量功能与光谱中图形信号密度相匹配的正数基础。 根据分析, 我们建议 Jacobi Convil, 后者使用Jacobi Convol, 其真实性基础是用于空间GNNNS的显示力, 以及所有模拟性、 和感官化水平的模型上的灵活性。