拓展图信号与图论分析框架，提升图神经网络理论的实用性与适用范围。
A Note on Graphon-Signal Analysis of Graph Neural Networks
- 将图信号分析从一维扩展到多维，支持更复杂的图数据建模
- 改进泛化界并引入鲁棒性分析，增强模型可靠性
- 适用于非对称图结构与核函数，扩大实际应用场景
近期论文《图论-信号分析视角下的图神经网络》通过将带属性图（图信号）嵌入到带属性图论（图论信号）空间，对消息传递图神经网络（MPNNs）进行了理论分析，推导了泛化界与采样引理。但该研究在实际图机器学习场景中存在若干局限。本文针对这些问题提出多项改进：1）将主结果推广至多维图信号；2）将李普希茨连续性分析扩展至含读出层的MPNNs，并以切距离为度量；3）利用鲁棒性泛化界进一步优化泛化性能；4）将分析框架拓展至非对称图论与核函数。这些改进显著提升了理论分析的实用性和覆盖范围。
原文摘要 · Abstract (English)
A recent paper, ``A Graphon-Signal Analysis of Graph Neural Networks'', by Levie, analyzed message passing graph neural networks (MPNNs) by embedding the input space of MPNNs, i.e., attributed graphs (graph-signals), to a space of attributed graphons (graphon-signals). Based on extensions of standard results in graphon analysis to graphon-signals, the paper proved a generalization bound and a sampling lemma for MPNNs. However, there are some missing ingredients in that paper, limiting its applicability in practical settings of graph machine learning. In the current paper, we introduce several refinements and extensions to existing results that address these shortcomings. In detail, 1) we extend the main results in the paper to graphon-signals with multidimensional signals (rather than 1D signals), 2) we extend the Lipschitz continuity to MPNNs with readout with respect to cut distance (rather than MPNNs without readout with respect to cut metric), 3) we improve the generalization bound by utilizing robustness-type generalization bounds, and 4) we extend the analysis to non-symmetric graphons and kernels.
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