Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features, such as cycles of arbitrary length, in combination with multi-scale topological descriptors, has improved predictive performance for data sets with prominent topological structures, such as molecules. At the same time, the theoretical properties of persistent homology have not been formally assessed in this context. This paper intends to bridge the gap between computational topology and graph machine learning by providing a brief introduction to persistent homology in the context of graphs, as well as a theoretical discussion and empirical analysis of its expressivity for graph learning tasks.

翻译：持久同调，一种计算拓扑技术，近年来在图分类任务中展现出强大的实证性能。通过捕捉长程图属性（例如任意长度的环）这一高阶拓扑特征，结合多尺度拓扑描述符，该方法显著提升了具有显著拓扑结构的数据集（如分子）的预测性能。然而，在此背景下，持久同调的理论特性尚未得到正式评估。本文旨在通过简要介绍图表环境中的持久同调，并对其在图表学习任务中的表达性进行理论探讨与实证分析，从而弥合计算拓扑与图表机器学习之间的鸿沟。