AF: Small: Topological Data Analysis for Big and High Dimensional Data

AF：小：大维和高维数据的拓扑数据分析

• 批准号: 1318595
• 负责人: Tamal Dey
• 金额: $ 49.63万
• 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: 2013-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1318595&HistoricalAwards=false
• 关键词: AF; Small; Topological; Data; Analysis

The computing community is increasingly facing the challenge of processing and interpreting huge amounts of data that is being generated all around us by personal devices, digital sensors, data centers, scientific studies, and social networks. The nature of this data is usually high dimensional, that is, when interpreted as points, they reside in an high dimensional Euclidean space. The curse of `dimensionality' together with `size' puts up a formidable challenge to decipher knowledge from them in a principled way. Among the various approaches proposed to `mine' this data, topological approaches are emerging as robust and global methods that could complement the other approaches, be it statistical or geometrical. How can one deal with the difficulty of `big' data by developing new algorithms in computational topology and geometry is the focus of this proposal. The proposed research aims to investigate three techniques, namely, subsampling, localization, and simplicial collapse to handle the menace of `size' and `dimension'. This requires developing innovative computational tools grounded in algorithmic theory and mathematics from algebraic topology, analysis, and discrete geometry.The proposed topological methods would enhance the understanding of `big' data in general which could originate from images in the medical fields, videos of some events, trends in finance, or connectivities in social networks. The goal is to produce practical algorithms that can be turned into usable software which would be useful for both academia and industry. Other than standard academic forums, there are plans to disseminate the results from this project through course notes, tutorials, and web-pages to reach wider audience. The PI also plans to develop course materials on topological data analysis that will include results obtained in the project. The support from the project will train graduate students. Efforts will be made to recruit students from under-represented groups.

计算社区越来越多地面临着处理和解释大量数据的挑战，这些数据是由个人设备、数字传感器、数据中心、科学研究和社交网络在我们周围产生的。这些数据的性质通常是高维的，也就是说，当被解释为点时，它们位于高维欧几里得空间中。“维度”和“大小”的诅咒对以有原则的方式从它们中解读知识提出了巨大的挑战。在为“挖掘”这一数据而提出的各种方法中，拓扑方法正在成为一种强有力的全球方法，可以补充其他方法，无论是统计方法还是几何方法。如何通过开发计算拓扑学和几何学的新算法来处理“大”数据的困难，是这项建议的重点。 建议的研究旨在研究三种技术，即，子采样，本地化，单纯形崩溃处理的威胁的“大小”和“尺寸”。这就需要开发基于算法理论和数学的创新计算工具，包括代数拓扑、分析和离散几何，拟议的拓扑方法将提高对一般“大”数据的理解，这些数据可能来自医疗领域的图像、某些事件的视频、金融趋势或社交网络中的连接。其目标是产生实用的算法，可以转化为可用的软件，这将是有用的学术界和工业界。除了标准的学术论坛外，还计划通过课程笔记、教程和网页传播这一项目的成果，以接触更广泛的受众。PI还计划开发拓扑数据分析课程材料，其中将包括项目中获得的结果。该项目的支持将培养研究生。将努力从代表性不足的群体中招收学生。