Collaborative Research: Multiparameter Topological Data Analysis

合作研究：多参数拓扑数据分析

• 批准号: 2301359
• 负责人: Facundo Memoli
• 金额: $ 20万
• 依托单位: Ohio State University
• 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=2301359&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.

复杂的数据集在科学和工程的许多学科中都产生，其解释需要多参数数据分析，从广义上讲，这是一种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，验证和评估检测，跟踪和表征与高影响天气事件（如雷暴和飓风）相关的天气模式的趋势。近年来，拓扑数据分析（TDA）已演变为数据科学的新兴领域。到目前为止，其大多数应用程序都限于单个参数案例，即表达单个变量行为的数据。随着应用程序的覆盖范围的扩展，从各种各样的数据中提取智能摘要的任务，复杂的数据需要研究多参数依赖性。该项目将通过开发一个有效的算法工具支持的合理数学理论来帮助解决这一需求，从而为科学和工程应用中的数据探索和分析提供了强大的平台。数学与计算机科学与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的技能，最著名的是算法和拓扑，并分析一些现实世界中的数据集。调查人员将遵循最佳实践，以招募和导师的学生来自代表性不足的团体，他们将参加该项目。研究人员还计划通过计算拓扑和TDA场地的研讨会或教程扩大研究参与。尽管涉及单个参数的TDA经过了很好的研究和开发，但对于多参数案例而言尚不正确。在当前的新生阶段，多参数TDA尚未开发工具来实际处理复杂，多样和高维数据。为了应对这一挑战，该项目将使多参数TDA的数学和算法进步同时实现。为了有效地范围，重点将主要放在三个研究推力上：（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）利用锯齿形持久性与多参数设置的连接以支持动态数据分析，并且（iii）概括图形拓扑描述符。从方法论的角度来看，拟议的工作背后的几何和拓扑思想将新颖的观点和方向注入了重要的计算数据分析领域。特别是，项目团队将与算法一起研究几个新颖的数学概念，以解决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。研究人员将理论计算机科学，算法设计，数学以及尤其是拓扑数据分析的专业知识汇集在一起​​，以进行这项研究。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛的影响来通过评估来支持的。