The Shape of Data: Using Topology and Geometry in Statistics

在我们的现代社会，技术进步不断催生新的数据收集机制，使科学家能够提出日益复杂的问题。统计领域迫切需要开发可用于应对此类科学挑战的方法。这些科学挑战中的许多可以通过首先尝试通过所谓的特征提取来降低数据的大小和复杂性来解决。在第二步中，这些特征被用来设计合理的统计方法。该项目考虑提取有关数据形状的信息，并正在开发和研究基于这些特征的统计和机器学习方法。该项目的成果预计将对统计、机器学习和各种应用领域产生影响。PI与生物学家、动物科学家和材料科学家的现有合作也将促进向这些领域的传播。该项目还将为研究生和本科生提供培训机会。这个项目的贡献将在于统计学和机器学习的交叉。它们将通过派生新的方法，通过进行相关的理论分析，加深对这些方法的理解，并提供对实际目的有用的方法的实施，来增强工具箱和增进知识。更具体地说，这个项目将(I)研究霍奇-拉普拉斯在分析更高关系数据的背景下的作用，包括它在聚类中的作用；(Ii)推导关于稳定统计量的理论结果，例如贝蒂曲线，它允许为持久贝蒂曲线的泛函构造渐近精确的基于自举的置信度区间；(Iii)通过使用多重测试开发一种新颖的、统计增强的持久性图表，以及(Iv)实施将允许探索新方法的有限样本属性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

In our modern society, technical advances constantly give rise to new data collection mechanisms that allow scientists to ask questions of ever-increasing complexities. The field of statistics is pressed to develop methodologies that can be used for answering such scientific challenges. Many of these scientific challenges can be addressed by first attempting to decrease size and complexity of the data through what is called feature extraction. In a second step these features are used to devise sound statistical methodologies. This project considers extracting information about the shape of the data, and is developing and studying statistical and machine learning methodologies based on these features. The outcomes of this project are expected to impact the fields of statistics, machine learning and various fields of application. Existing collaborations of the PI with biologists, animal scientists and material scientists, will also facilitate dissemination to these fields. This project will also provide training opportunities for both graduate and undergraduate students. The contributions of this project will lie in the intersection of statistics and machine learning. They will enhance toolboxes and advance knowledge through the derivation of novel methodologies, through developing a deep understanding of these methodologies by performing relevant theoretical analyses, and by providing implementations of the methodologies that are useful for practical purposes. More specifically, this project will (i) study the role of the Hodge Laplacian in the context of analyzing higher-relational data, including its role in clustering; (ii) derive theoretical results about stabilizing statistics such as Betti-curves that allow the construction of asymptotically exact bootstrap based confidence intervals for functionals of persistent Betti-curves; (iii) develop a novel, statistically enhanced persistence diagram thought the use of multiple testing, and (iv) Implementations will allow the exploration of finite sample properties of the new methods.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.