Topological data analysis is a powerful tool for describing topological signatures in real world data. An important challenge in topological data analysis is matching significant topological signals across distinct systems. In geometry and probability theory, optimal transport formalises notions of distance and matchings between distributions and structured objects. We propose to combine these approaches, constructing a mathematical framework for optimal transport-based matchings of topological features. Building upon recent advances in the domains of persistent homology and optimal transport for hypergraphs, we develop a transport-based methodology for topological data processing. We define measure topological networks, which integrate both geometric and topological information about a system, introduce a distance on the space of these objects, and study its metric properties, showing that it induces a geodesic metric space of non-negative curvature. The resulting Topological Optimal Transport (TpOT) framework provides a transport model on point clouds that minimises topological distortion while simultaneously yielding a geometrically informed matching between persistent homology cycles.

翻译：拓扑数据分析是描述真实数据中拓扑特征的有力工具。拓扑数据分析中的一个重要挑战是匹配不同系统中的显著拓扑信号。在几何学和概率论中，最优传输形式化了分布与结构化对象之间距离和匹配的概念。我们提出将这两种方法结合，构建一个基于最优传输的拓扑特征匹配数学框架。基于持续同调与超图最优传输领域的最新进展，我们开发了一种基于传输的拓扑数据处理方法。我们定义了度量拓扑网络，该网络整合了系统的几何与拓扑信息，引入了此类对象空间上的距离，并研究了其度量性质，表明其诱导出一个非负曲率的测地线度量空间。由此得到的拓扑最优传输（TpOT）框架为点云提供了一种传输模型，该模型在最小化拓扑畸变的同时，还能在持续同调圈之间产生具有几何信息的匹配。