Collaborative Research: The Topology of Functional Data on Random Metric Spaces, Graphs and Graphons

协作研究：随机度量空间、图和图子上函数数据的拓扑

• 批准号: 1722995
• 负责人: Washington Mio
• 金额: $ 35.49万
• 依托单位: Florida State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-15 至 2021-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1722995&HistoricalAwards=false
• 关键词: Collaborative; Research; Topology; Functional; Data

Extracting, modeling, conceptualizing, and visualizing information residing in complex datasets in actionable form are core themes in modern data analysis. This project will take up these challenges in the broad realm of functional data, as they permeate the sciences and practical applications. The project will develop mathematical methods and computational tools that help us execute these tasks and establish a pipeline from functional data to actionable knowledge. There is a vast landscape of potential uses of these methods. Examples include analyses of dynamical social networks, understanding variation in the spatial profile of gene expression to enable studies of associations with health and developmental outcomes, and exploratory studies that interrogate microbiome and metabolomic networks. The project will leverage techniques from topological data analysis (TDA) to further develop and extend persistent homology and integrate TDA with probabilistic methods. This will enable development of methods and tools for probing structural variation in functions defined on random domains. The project will develop topological methods, computational tools, and their foundations to summarize, analyze, and visualize functional data on random compact metric spaces, networks and graphons, and establish a pipeline from such data to actionable knowledge. The advances resulting from the project will open many new perspectives in practical applications and research in such disciplines as developmental and evolutionary biology, medicine, social sciences, and engineering. Though powered by state-of-the-art methods, the resulting tools will be simple to use so as to attain broad utility. To enhance this important aspect of the project, by way of interdisciplinary collaboration with domain scientists, the methods will be tested on case studies that involve analyses of plant shape, rock micro-structure, and human microbiome data. The methods and software tools resulting from this project will be documented and distributed online for use by the research community further broadening the impact on other disciplines. The project also will generate resources to support several outreach activities.

以可操作的形式提取、建模、概念化和可视化复杂数据集中的信息是现代数据分析的核心主题。该项目将在功能数据的广泛领域中应对这些挑战，因为它们渗透到科学和实际应用中。 该项目将开发数学方法和计算工具，帮助我们执行这些任务，并建立从功能数据到可操作知识的管道。 这些方法有着广阔的潜在用途。例子包括动态社交网络的分析，了解基因表达的空间分布的变化，使研究与健康和发育结果的关联，以及探索性研究，询问微生物组和代谢网络。该项目将利用拓扑数据分析（TDA）技术进一步开发和扩展持久同源性，并将TDA与概率方法相结合。这将使开发的方法和工具，用于探测随机域上定义的功能的结构变化。该项目将开发拓扑方法，计算工具及其基础，以总结，分析和可视化随机紧凑度量空间，网络和图子上的功能数据，并建立从这些数据到可操作知识的管道。该项目所取得的进展将在发展和进化生物学、医学、社会科学和工程等学科的实际应用和研究中开辟许多新的视角。虽然由最先进的方法提供动力，但由此产生的工具将易于使用，以获得广泛的实用性。为了加强该项目的这一重要方面，通过与领域科学家的跨学科合作，将在涉及植物形状，岩石微结构和人类微生物组数据分析的案例研究中测试这些方法。该项目产生的方法和软件工具将记录在案，并在网上分发，供研究界使用，进一步扩大对其他学科的影响。该项目还将产生资源来支持几项外展活动。

项目成果

A topological study of functional data and Frechet functions of metric measure spaces

度量测度空间函数数据和Frechet函数的拓扑研究

• DOI: 10.1007/s41468-019-00037-8
• 发表时间: 2019
• 期刊: Journal of applied and computational topology
• 作者: Hang, H;Memoli, F;Mio, W.
• 通讯作者: Mio, W.