Advances in Topological Data Analysis

拓扑数据分析的进展

• 批准号: 439304438
• 负责人: Professor Dr. Johannes Theodor Nikolaus Krebs
• 金额: --
• 依托单位: Department of Statistics
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2020
• 资助国家: 德国
• 起止时间: 2019-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/439304438?language=en
• 关键词: Advances; Topological; Data; Analysis

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.

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让数据科学家做出更明智的决策。从统计角度来看，TDA 方法可以被视为聚类分析的推广，旨在检测数据中的拓扑结构。该项目的第一部分针对 Betti 曲线的渐近行为以及从后者获得的函数（例如持久性景观）。开创性成果将为该项目的第二部分奠定基础，即研究基于 TDA 的对象的重采样程序（例如 Betti 曲线的引导）以及将 TDA 进一步集成到经典统计方法（例如变化点检测）中。