CAREER: Topological Descriptors

职业：拓扑描述符

• 批准号: 2046730
• 负责人: Brittany Fasy
• 金额: $ 59.93万
• 依托单位: Montana State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2046730&HistoricalAwards=false
• 关键词: CAREER; Topological; Descriptors; CAREER; Topological; Descriptors

Comparing topological spaces, especially those arising from noisy data, is difficult. Topological data analysis (TDA) captures the 'shape' of data with descriptors (e.g., persistence diagrams and Reeb graphs). Individually, these topological descriptors have proven to be powerful data-analysis tools; however, a single topological descriptor is often not rich enough to capture the intricacies of large, complex data that arises in real applications. For that reason, this project develops a framework for studying topological spaces and data by transforming them into families of descriptors that capture the topology of different 'views' of the data. Studying this family of descriptors enables new methods for summarizing and comparing topological spaces, and creates a pathway for their use in statistical settings. As a result, this project will develop usable, theoretically-grounded data-analysis techniques that will enable TDA for large complex data, including networks, images, and point clouds. Through the research activities, the investigator will train graduate and undergraduate students in interdisciplinary research, and special efforts will be made to recruit first-generation and underrepresented minority students. The proposed educational activities will promote a sense of belonging of first-generation and underrepresented minority students as graduate students in STEM, and, more generally, as future academics in interdisciplinary domains.More specifically, this project builds the foundations for representing and comparing large, complex topological spaces and data sets through parameterized families of topological descriptors, where a topological descriptor is any summary of a topological space. These families of topological descriptors are called topological transforms. The two objectives to accomplish this are: (1) Studying topological transforms and quantifying their ability to represent topological spaces. The project team will study existing transforms, reframe other topological and data analysis concepts as a topological transform, and propose new transforms. By allowing a diverse set of topological descriptors (from persistence diagrams to Euler characteristic curves to small graphs) and a choice of parameterization set (e.g., subspaces or ambient directions), the topological transform framework is flexible. (2) Developing the statistical tools necessary for using topological transforms for data summarization and comparison. In doing so, the project team will define distances between transforms and study the transforms as distributions over spaces of topological descriptors. Throughout, theoretical developments will be grounded in applications, thus establishing mathematical foundations and algorithmic developments that address core issues in analyzing data from real-world 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.

比较拓扑空间，尤其是由嘈杂数据引起的拓扑空间。 拓扑数据分析（TDA）用描述符（例如持续图和REEB图）捕获了数据的“形状”。 单独地，这些拓扑描述符已被证明是强大的数据分析工具。但是，单个拓扑描述符通常不足以捕获实际应用中出现的大型复杂数据的复杂性。 因此，该项目通过将其转换为描述符的家族来捕获数据的不同“观点”的拓扑结构来开发一个用于研究拓扑空间和数据的框架。 研究这个描述符家族可以为总结和比较拓扑空间提供新的方法，并为它们在统计环境中的使用创造途径。 结果，该项目将开发可用的，理论上的数据分析技术，该技术将为大型复杂数据（包括网络，图像和点云）提供TDA。 通过研究活动，研究人员将培训跨学科研究的毕业生和本科生，并将为招募第一代和代表性不足的少数族裔学生做出特别的努力。 拟议的教育活动将促进第一代和代表性不足的少数族裔学生作为STEM的研究生的归属感，更普遍地是跨学科领域的未来学者。更具体地说，该项目为代表大型，复杂的拓扑空间和数据集建立了基础，以通过拓扑来源的拓扑分数来代表大型，复杂的拓扑空间和数据集。 这些拓扑描述符家族称为拓扑转换。 实现这一目标的两个目标是：（1）研究拓扑变换并量化其代表拓扑空间的能力。 项目团队将研究现有的转换，将其他拓扑和数据分析概念重新构建为拓扑转换，并提出新的转换。 通过允许一组拓扑描述符（从持久图到Euler特征曲线到小图），并选择参数化集合集（例如子空间或环境方向），拓扑转换框架是灵活的。 （2）开发使用拓扑转换进行数据汇总和比较所需的统计工具。 这样一来，项目团队将定义转换之间的距离，并研究转换为拓扑描述符空间的分布。 在整个过程中，理论发展将基于应用程序，从而建立了数学基础和算法发展，这些发展解决了分析现实世界应用程序数据中的核心问题。该奖项反映了NSF的法定任务，并被视为值得通过基金会的知识分子和更广泛影响的评估来通过评估来获得支持的审查标准。