论文标题
光谱图神经网络的强大程度
How Powerful are Spectral Graph Neural Networks
论文作者
论文摘要
光谱图神经网络是基于图信号过滤器的一种图神经网络（GNN）。一些能够学习任意光谱过滤器的模型最近出现了。但是，很少有作品分析光谱GNN的表达能力。本文理论上研究了光谱GNNS的表现力。我们首先证明，即使没有非线性的光谱GNN也可以产生任意的图形信号，并给出了两个条件以达到普遍性。它们是：1）图laplacian的多个特征值和2）节点特征中没有缺少频率组件。我们还建立了光谱GNN的表达能力与图形同构（GI）测试之间的联系，后者通常用于表征空间GNNS的表达能力。此外，我们从优化的角度研究了具有相同表达能力的不同光谱GNN之间的经验性能差异，并激发了其重量函数对应于光谱中图信号密度的正交基础的使用。受分析的启发，我们提出了Jacobiconv，该雅各比（Jacobiconv）由于其正交性和灵活性而使用雅各比（Jacobi）的基础来适应广泛的重量功能。 Jacobiconv抛弃了非线性，同时在合成和现实世界中的所有基线都超过了所有基准。
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.
