Advances in Topological Data Analysis

该项目的总体目标是扩展拓扑数据分析（TDA）在数理统计和应用概率方面的基础，以了解TDA方法的优势，以及它是否能够使数据科学家做出更明智的决策。从统计学的角度来看，TDA方法可以被认为是聚类分析的推广，其目的是检测数据中的拓扑结构。该项目的第一部分针对Betti曲线的渐近行为和从后者获得的泛函，如持久性景观。建立开创性的成果将为该项目的第二部分奠定基础，该项目的第二部分是研究基于TDA的对象的恢复程序，例如Betti曲线的引导和TDA在经典统计方法中的进一步整合，例如变点检测。

The overarching aim of this project is to extend the foundations of Topological Data Analysis (TDA) in mathematical statistics and applied probability in order to understand the strengths of the TDA methodology and whether it can enable data scientists to make a more informed decision. From the statistical perspective the TDA methodology can be considered as a generalization of cluster analysis which aims at detecting topological structure in data.The first part of the project targets the asymptotic behavior of Betti curves and of functionals obtained from the latter such as persistence landscapes. Establishing pioneering results will lay the groundwork for the second part of the project which is the study of resampling procedures of TDA based objects such as the bootstrap of Betti curves and a further integration of TDA in classical statistical methods such as change point detection.