Efficient Computation of Generalized Persistence Diagrams

拓扑数据分析（TDA）将代数拓扑技术引入应用领域。它强调的方法是稳定的，并且对数据中的噪声具有弹性，以及计算效率高的方法，并且在一系列应用中具有实用性。在过去的二十年里，TDA技术已经在许多领域得到了应用，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的主要方法之一，持续同源性，仅限于研究单参数数据，即单个量的测量。这个项目的重点是，在实践中，多种测量方法是可用的，而正是它们之间的相关性将被用来揭示问题的重要特征。将为使用这种方法的实践者开发软件，该项目将包括拓扑数据分析方面的研究生培训。最近引入了广义持久同调。它将持久性解释为某个函数的莫比乌斯反演，该函数是由跨参数的数据拓扑变化派生的。该构造概括了应用程序中所需的1参数持久性的所有属性，包括稳定性和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，这是本研究项目的关键空白，也是理论与应用之间缺失的桥梁。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.