AF: Small: Geometric Data Processing and Analysis via Light-weight Structures

AF：小型：通过轻量结构进行几何数据处理和分析

• 批准号: 1319406
• 负责人: Yusu Wang
• 金额: $ 48.14万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-06-01 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1319406&HistoricalAwards=false
• 关键词: AF; Small; Geometric; Data; Processing

In the current era of information, large amounts of complex data are routinely generated across science and engineering. Geometric and topological data analysis is a major component involved in organizing, analyzing, as well as utilizing the huge amount of data generated. As data becomes more complex and high dimensional, it can be computationally formidable to compute and maintain even local structures from data. Hence it has become crucial to identify and use light-weight structures that can be efficiently computed, maintained and/or manipulated; while at the same time are also sufficiently informative. This project will focus on two such light-weight structures: the Reeb graph and the Gaussian-weighted graph Laplacian. Both of them have already been extensively used for various data analysis applications. Nevertheless, there are many theoretical and algorithmic questions about them that need to be addressed to better understand their behavior, their potential, as well as to further their practical usage. This project aims to address these issues, develop a solid theoretical foundation as well as efficient algorithms for their practical usage, and investigate novel geometric processing applications based on these two objects. Data analysis is a fundamental problem in computational science, ubiquitous in a broad range of application fields in science and engineering. The intellectual value of this project lies not only in developing novel computational geometry algorithms and techniques, but also in transferring the results and insights to practical applications. Results from this project, especially algorithms and software, will be disseminated to a large scientific community so as to potentially produce a broader impact in scientific and engineering endeavors.

在当今的信息时代，科学和工程领域经常产生大量复杂的数据。几何和拓扑数据分析是组织、分析和利用所生成的大量数据的主要组成部分。随着数据变得更加复杂和高维，计算和维护数据的局部结构在计算上可能会非常困难。因此，识别和使用可以有效计算、维护和/或操纵的轻量级结构变得至关重要；同时也提供了足够的信息。该项目将重点关注两种这样的轻量级结构：Reeb 图和高斯加权图拉普拉斯。它们都已广泛用于各种数据分析应用。尽管如此，仍然存在许多关于它们的理论和算法问题需要解决，以便更好地理解它们的行为、潜力并进一步促进它们的实际应用。该项目旨在解决这些问题，为其实际应用开发坚实的理论基础和有效的算法，并基于这两个对象研究新颖的几何处理应用。数据分析是计算科学的一个基本问题，在科学和工程的广泛应用领域中普遍存在。该项目的智力价值不仅在于开发新颖的计算几何算法和技术，而且在于将结果和见解转化为实际应用。该项目的成果，特别是算法和软件，将被传播到大型科学界，从而可能对科学和工程事业产生更广泛的影响。