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CAREER: Topological Descriptors

职业：拓扑描述符

基本信息

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

项目摘要

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，特别是由噪声数据产生的拓扑空间是困难的。拓扑数据分析（TDA）用描述符（例如，持久性图和Reeb图）捕获数据的“形状”。单独来看，这些拓扑描述符已被证明是强大的数据分析工具；然而，单个拓扑描述符通常不够丰富，无法捕捉实际应用程序中出现的大型复杂数据的复杂性。因此，该项目开发了一个框架，通过将拓扑空间和数据转换为描述符族来捕获数据的不同“视图”的拓扑，从而研究拓扑空间和数据。研究这一描述符族为总结和比较拓扑空间提供了新的方法，并为它们在统计设置中的使用创造了途径。因此，该项目将开发可用的、基于理论的数据分析技术，使TDA能够用于大型复杂数据，包括网络、图像和点云。通过研究活动，研究者将培养研究生和本科生进行跨学科研究，并将特别努力招募第一代和代表性不足的少数民族学生。拟议的教育活动将促进第一代和代表性不足的少数民族学生作为STEM研究生的归属感，更广泛地说，作为跨学科领域的未来学者。更具体地说，该项目通过参数化的拓扑描述符族为表示和比较大型复杂拓扑空间和数据集奠定了基础，其中拓扑描述符是拓扑空间的任何摘要。这些拓扑描述符族称为拓扑变换。实现这一目标的两个目标是：(1)研究拓扑变换并量化其表示拓扑空间的能力。项目组将研究现有的转换，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。通过允许一组不同的拓扑描述符（从持久性图到欧拉特征曲线到小图）和参数化集的选择（例如，子空间或环境方向），拓扑转换框架是灵活的。(2)开发使用拓扑变换进行数据汇总和比较所需的统计工具。在此过程中，项目团队将定义转换之间的距离，并将转换作为拓扑描述符空间上的分布来研究。在整个过程中，理论发展将以应用为基础，从而建立数学基础和算法发展，解决分析现实世界应用数据的核心问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

期刊论文数量（10）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Path-Connectivity of Fréchet Spaces of Graphs

图的 Fréchet 空间的路径连通性

DOI：
发表时间：
2022
期刊：
Young Researcher's Forum (CG Week
影响因子：
0
作者：
Chambers, Erin;Fasy, Brittany Terese;Holmgren, Benjamin;Majhi, Sushovan;Wenk, Carola
通讯作者：
Wenk, Carola

Combinatorial Conditions for Directed Collapsing

定向塌陷的组合条件

DOI：
10.1007/978-3-030-95519-9
发表时间：
2022
期刊：
Association for Women in Mathematics series
影响因子：
0
作者：
Belton, Robin;Brooks, Robyn;Ebli, Stefania;Fajstrup, Lisbeth;Fasy, Brittany Terese;Sanderson, Nicole;Vidaurre, Elizabeth
通讯作者：
Vidaurre, Elizabeth

DBSpan: Density-Based Clustering Using a Spanner, With an Application to Persistence Diagrams

DBSpan：使用 Spanner 的基于密度的集群以及持久性图的应用