Efficient Computation of Generalized Persistence Diagrams

拓扑数据分析（TDA）将代数拓扑学的技术引入到应用领域。它强调稳定的方法，因此对数据中的噪声具有弹性，以及计算效率高的方法，因此在一系列应用中具有实用性。在过去的二十年中，TDA技术已经在许多领域中得到应用，包括生物化学，宇宙学，材料科学，神经科学，气候研究等。从业者遇到的一个重要限制是TDA中的主要方法之一，持久同源性，仅限于研究单参数数据，即，测量一个单一的数量。这个项目的重点是，在实践中，多种测量是可用的，正是它们之间的相关性将被用来揭示问题的重要特征。 将开发用于使用这种方法的从业者的软件，该项目将包括拓扑数据分析方面的研究生培训。最近，引入了广义持续同源性。它将持久性解释为某个函数的Moebius反演，该函数来自数据在参数之间的拓扑变化。该构造概括了应用程序中所需的单参数持久性的所有属性，包括稳定性以及机器学习和统计管道中使用的图的特定结构。 该项目将开发一个计算广义持久性图的软件实现，这是该研究计划中的一个关键差距，也是理论和应用之间缺失的桥梁。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Topological data analysis (TDA) brings techniques from algebraic topology to the applied domains. It emphasizes methods that are stable, and as such resilient to noise in the data, as well as methods that are computationally efficient, and as such practical across a range of applications. Over the last two decades, TDA techniques have found applications in many domains, including biochemistry, cosmology, materials science, neuroscience, climate research, and many others. One significant limitation that practitioners encounter is that one of the main methods in TDA, persistent homology, is limited to studying one-parameter data, i.e., measurements of a single quantity. The focus of this project is that in practice, multiple measurements are available, and it is precisely the correlation between them that will be used to reveal important features of the problem. Software for practitioners using this methodology will be developed and the project will include graduate student training in topological data analysis.Recently, generalized persistent homology was introduced. It interprets persistence as a Moebius inversion of a certain function derived from the changes in topology of the data across parameters. The construction generalizes all the properties of 1-parameter persistence needed in applications, including stability and the particular structure of the diagrams used in machine learning and statistical pipelines. This project will develop a software implementation for computing generalized persistence diagrams, a crucial gap in this research program and the missing bridge between theory and applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.