Collaborative Research: Topological Methods for Parsing Shapes and Networks and Modeling Variation in Structure and Function

合作研究：解析形状和网络以及建模结构和功能变化的拓扑方法

• 批准号: 1418007
• 负责人: Washington Mio
• 金额: $ 31.29万
• 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: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1418007&HistoricalAwards=false
• 关键词: Collaborative; Research; Topological; Methods; Parsing

Critical to advancing data-enabled science is our ability to probe, conceptualize, interpret, and visualize information residing in complex datasets in order to transform data into knowledge. This project develops computational methods and tools for investigation and visualization of structural variation in networks and 3D shapes, as well as inference of functional outcomes of such variation. These problems arise in areas of strategic interest such as health and medicine, where it is important to understand patterns of morphological variation in biological shapes and structural changes in biological networks, and their roles in behavior, health and disease. To emphasize this important interdisciplinary facet, the project is supported by several case studies that investigate: (i) mechanisms underlying the development of facial shape; (ii) interactions between brain shape, skull morphology, and behavior; (iii) organization of microbial communities in the digestive tract; and (iv) connections between social networks and microbiome networks. We envision many long-term ramifications of the project. Potential applications include analyses of dynamical social networks, exploratory discovery of associations between biological networks and phenotypic traits or diseases, quantitative studies of evolution, development, and inheritance of morphological traits, and challenges such as indexing and organizing databases of networks or 3D shapes for efficient data management, search, and retrieval. The project integrates techniques from topological data analysis, integral geometry, spectral geometry, and statistics to develop computational methods and tools for modeling and visualizing structural variation in diverse collections of shapes and networks, and exploring associations between variation in structure and function. The project addresses theoretical foundations, computational methods, implementation of tools for statistical analysis and visualization, and validation of methodology. A shape or network is represented by a Borel probability measure on a Hilbert space whose elements represent Euler characteristic curves that encode rich geometric and topological properties. Dimension reduction and discretization of the probability measure lead to representations that allow organization of complex datasets into compact dictionaries that facilitate data analytics, processing, visualization, and search. This enables integration of the new methods with an array of existing techniques of multivariate statistical analysis to address such problems as development of regression-based models over networks.

对支持数据的科学至关重要的是我们能够探究，概念化，解释和可视化驻留在复杂数据集中的信息以将数据转换为知识的信息。 该项目开发了用于研究和可视化网络和3D形状结构变化的计算方法和工具，以及这种变化功能结果的推断。这些问题出现在战略兴趣领域，例如健康和医学，其中重要的是要了解生物形状的形态变化模式和生物网络的结构变化及其在行为，健康和疾病中的作用。为了强调这一重要的跨学科方面，该项目得到了一些研究的案例研究的支持：（i）面部形状发展的机制； （ii）大脑形状，颅骨形态和行为之间的相互作用； （iii）消化道中微生物群落的组织； （iv）社交网络与微生物组网络之间的联系。 我们设想了该项目的许多长期影响。潜在的应用包括对动态社交网络的分析，生物网络与表型性状或疾病之间关联的探索性发现，对形态特征的进化，发展和遗传的定量研究以及形态学性状的遗传以及索引和组织数据库或3D塑造的索引和组织数据库的挑战，以高效的数据管理，搜索，搜索和检索。该项目从拓扑数据分析，积分几何形状，光谱几何学和统计数据中整合了技术，以开发计算方法和工具，以建模和可视化形状和网络集合中的结构变化，并探索结构和功能的变化之间的关联。该项目解决了理论基础，计算方法，实施用于统计分析和可视化的工具以及方法论的验证。形状或网络由希尔伯特空间上的Borel概率度量表示，其元素代表编码丰富的几何和拓扑特性的Euler特征曲线。概率度量的尺寸降低和离散化导致表示复杂的数据集中的表示形式，以促进数据分析，处理，可视化和搜索的紧凑字典。这使得新方法将新方法与一系列多元统计分析的技术集成，以解决诸如网络上基于回归的模型的开发问题。