Ye Xue, Sun Fang, Xiang Shiming
National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China.
School of Artificial Intelligence, University of Chinese Academy of Sciences, Beijing 101408, China.
Entropy (Basel). 2023 Feb 10;25(2):331. doi: 10.3390/e25020331.
Topological Data Analysis (TDA) is an approach to analyzing the shape of data using techniques from algebraic topology. The staple of TDA is Persistent Homology (PH). Recent years have seen a trend of combining PH and Graph Neural Networks (GNNs) in an end-to-end manner to capture topological features from graph data. Though effective, these methods are limited by the shortcomings of PH: incomplete topological information and irregular output format. Extended Persistent Homology (EPH), as a variant of PH, addresses these problems elegantly. In this paper, we propose a plug-in topological layer for GNNs, termed Topological Representation with Extended Persistent Homology (TREPH). Taking advantage of the uniformity of EPH, a novel aggregation mechanism is designed to collate topological features of different dimensions to the local positions determining their living processes. The proposed layer is provably differentiable and more expressive than PH-based representations, which in turn is strictly stronger than message-passing GNNs in expressive power. Experiments on real-world graph classification tasks demonstrate the competitiveness of TREPH compared with the state-of-the-art approaches.
拓扑数据分析（TDA）是一种利用代数拓扑技术来分析数据形状的方法。TDA的主要内容是持久同调（PH）。近年来，出现了一种将PH和图神经网络（GNN）以端到端方式相结合的趋势，以便从图数据中捕捉拓扑特征。尽管这些方法很有效，但它们受到PH缺点的限制：拓扑信息不完整和输出格式不规则。扩展持久同调（EPH）作为PH的一种变体，优雅地解决了这些问题。在本文中，我们为GNN提出了一个插件式拓扑层，称为具有扩展持久同调的拓扑表示（TREPH）。利用EPH的一致性，设计了一种新颖的聚合机制，将不同维度的拓扑特征整理到确定其生存过程的局部位置。所提出的层被证明是可微的，并且比基于PH的表示更具表现力，而基于PH的表示在表达能力上又严格强于消息传递GNN。在实际图分类任务上的实验证明了TREPH与现有最先进方法相比的竞争力。
