Collaborative Research: Multiparameter Topological Data Analysis

合作研究：多参数拓扑数据分析

• 批准号: 2301361
• 负责人: Bei Phillips
• 金额: $ 13万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2301361&HistoricalAwards=false
• 关键词: Collaborative; Research; Multiparameter; Topological; Data

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.

复杂的数据集出现在科学和工程的许多学科中，它们的解释需要多参数数据分析，从广义上讲，多参数数据分析研究现象或空间对多个参数的依赖性。例如，在气候模拟中，科学家有兴趣识别、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、跟踪和特征化趋势。近年来，拓扑数据分析（TDA）已发展成为数据科学的一个新兴领域。到目前为止，它的大多数应用仅限于单参数情况，即表达单个变量行为的数据。随着其应用范围的扩大，从多样化、复杂的数据中提取智能摘要的任务需要研究多参数依赖性。该项目将通过开发由高效算法工具支持的完善的数学理论来帮助满足这一需求，从而为科学和工程应用中的数据探索和分析提供强大的平台。数学和计算机科学以及集成应用之间的协同作用将加速教育影响。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，尤其是算法和拓扑方面的技能，并分析一些现实世界的数据集。研究人员将遵循最佳实践，招募和指导来自代表性不足群体的学生来参与该项目。研究人员还计划通过计算拓扑和 TDA 场所的研讨会或教程来扩大研究参与度。尽管涉及单参数的 TDA 已经得到了很好的研究和开发，但对于多参数情况来说还不是这样。在目前的初级阶段，多参数 TDA 尚未开发出能够实际处理复杂、多样化和高维数据的工具。为了应对这一挑战，该项目将在多参数 TDA 的数学和算法上取得进展。为了有效地确定范围，重点将主要集中在三个研究重点上：（I）探索通用特征的多参数持久性并开发计算它们的算法； （II）利用锯齿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）概括图形拓扑描述符。从方法论的角度来看，该工作背后的几何和拓扑思想为计算数据分析的重要领域注入了新的视角和方向。特别是，项目团队将结合算法研究几个新颖的数学概念，以解决上述主旨中出现的各种挑战。由此产生的 TDA 方法有可能补充和增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。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查进行评估，被认为值得支持标准。