Collaborative Research: Multiparameter Topological Data Analysis

复杂的数据集出现在许多科学和工程学科中，它们的解释需要多参数数据分析，广义地说，多参数数据分析研究一个现象或空间对多个参数的依赖关系。例如，在气候模拟中，科学家们对识别、验证和评估与雷暴和飓风等高影响天气事件相关的天气模式的检测、跟踪和特征方面的趋势感兴趣。近年来，拓扑数据分析(TDA)已成为数据科学中的一个新兴领域。到目前为止，它的大多数应用都局限于单参数的情况，即表示单变量行为的数据。随着其应用范围的扩大，从各种复杂数据中提取智能摘要的任务要求研究多参数相关性。该项目将帮助满足这一需求，方法是发展一套完善的数学理论，并以有效的算法工具加以支持，从而为科学和工程应用中的数据探索和分析提供一个强大的平台。数学和计算机科学以及综合应用之间的协同作用将加速教育的影响。该项目支持的研究生将接受培训，以发展数学和理论计算机科学方面的技能，尤其是算法和拓扑学方面的技能，并分析一些现实世界的数据集。调查人员将遵循最佳做法，从代表人数不足的群体中招募和指导将参与该项目的学生。研究人员还计划通过在计算拓扑学和TDA场所举办研讨会或教程来扩大研究参与度。虽然涉及单参数的TDA已经得到了很好的研究和发展，但对于多参数情况还不是这样。在目前的初级阶段，多参数TDA还没有开发出实用的工具来处理复杂、多样化和高维的数据。为了迎接这一挑战，该项目将在多参数TDA的数学和算法方面取得进展。为了有效地扩大范围，重点将主要放在三个研究上：(I)探索广义特征的多参数持久性并开发算法来计算它们；(Ii)利用Z字形持久性与多参数设置的联系来支持动态数据分析；以及(Iii)推广图形拓扑描述符。从方法论的角度来看，这项工作背后的几何和拓扑思想为计算数据分析这一重要领域注入了新的视角和方向。特别是，项目组将结合算法研究几个新的数学概念，以应对上述推力中出现的各种挑战。由此产生的TDA方法有可能补充和加强机器学习和统计数据分析等领域的传统数据分析方法。研究人员汇集了理论计算机科学、算法设计、数学，特别是拓扑数据分析方面的专业知识来进行这项研究。这一奖项反映了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。

Complex datasets arise in many disciplines of science and engineering and their interpretation requires Multiparameter Data Analysis, which broadly speaking, studies the dependency of a phenomenon or a space on multiple parameters. For instance, in climate simulations, scientists are interested in identifying, verifying, and evaluating trends in detecting, tracking, and characterizing weather patterns associated with high impact weather events such as thunderstorms and hurricanes. In recent years, topological data analysis (TDA) has evolved as an emerging area in data science. So far, most of its applications have been limited to the single parameter case, that is, to data expressing the behavior of a single variable. As its reach to applications expands, the task of extracting intelligent summaries out of diverse, complex data demands the study of multiparameter dependencies. This project will help address this demand by developing a sound mathematical theory supported by efficient algorithmic tools thus providing a powerful platform for data exploration and analysis in scientific and engineering applications. The educational impact will be accelerated by the synergy between mathematics and computer science and integrated applications. Graduate students supported by the project will be trained to develop skills in mathematics and theoretical computer science, most notably in algorithms and topology, and analyze some real-world data sets. The investigators will follow best practice to recruit and mentor students from underrepresented groups who will participate in the project. The investigators also plan to broaden research engagement via workshops or tutorials at computational topology and TDA venues. Although TDA involving a single parameter has been well researched and developed, the same is not yet true for the multiparameter case. At its current nascent stage, multiparameter TDA is yet to develop tools to practically handle complex, diverse, and high-dimensional data. To meet this challenge, this project will make both mathematical and algorithmic advances for multiparameter TDA. To scope effectively, focus will be mainly on three research thrusts to: (I) explore multiparameter persistence for generalized features and develop algorithms to compute them; (II) exploit the connections of zigzag persistence to multiparameter settings to support dynamic data analysis, and (III) generalize graphical topological descriptors. From a methodological point of view, the geometric and topological ideas behind the proposed work inject novel perspectives and directions to the important field of computational data analysis. In particular, the project team will investigate several novel mathematical concepts in conjunction with algorithms to address various challenges appearing in the aforementioned thrusts. The resulting TDA methodologies have the potential to complement and augment traditional data analysis approaches in fields such as machine learning and statistical data analysis. The investigators bring together expertise in theoretical computer science, algorithms design, mathematics, and in particular topological data analysis to conduct this research.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.