将谱信息融入拓扑描述，提升图神经网络对结构的表达能力。
Graph Persistence goes Spectral
- 提出SpectRe：融合谱信息与持久同调的图描述符
- 理论证明其表达力强于传统同调与谱方法
- 在合成与真实数据集上验证有效，适合图学习任务
包含复杂拓扑信息（如环）可严格提升消息传递图神经网络（GNN）的表达能力，超越Weisfeiler-Leman（WL）层级。因此，持久同调（PH）方法被越来越多用于图表示学习。近期工作尝试在经典PH图中加入顶点和边特征以增强表达力，但仍未捕捉基本图结构信息。本文提出SpectRe——一种将谱信息融入PH图的新拓扑描述符。SpectRe在理论上严格优于仅用PH或谱信息的方法。我们还引入全局与局部稳定性概念分析现有描述符，并证明SpectRe具有局部稳定性。在合成与真实世界数据集上的实验表明，SpectRe能有效提升图模型在相关学习任务中的性能。代码已开源：https://github.com/Aalto-QuML/SpectRe/
原文摘要 · Abstract (English)
Including intricate topological information (e.g., cycles) provably enhances the expressivity of message-passing graph neural networks (GNNs) beyond the Weisfeiler-Leman (WL) hierarchy. Consequently, Persistent Homology (PH) methods are increasingly employed for graph representation learning. In this context, recent works have proposed decorating classical PH diagrams with vertex and edge features for improved expressivity. However, these methods still fail to capture basic graph structural information. In this paper, we propose SpectRe -- a new topological descriptor for graphs that integrates spectral information into PH diagrams. Notably, SpectRe is strictly more expressive than PH and spectral information on graphs alone. We also introduce notions of global and local stability to analyze existing descriptors and establish that SpectRe is locally stable. Finally, experiments on synthetic and real-world datasets demonstrate the effectiveness of SpectRe and its potential to enhance the capabilities of graph models in relevant learning tasks. Code is available at https://github.com/Aalto-QuML/SpectRe/.
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