AF: Small: Analyzing Complex Data with a Topological Lens

AF：小：用拓扑透镜分析复杂数据

• 批准号: 1526513
• 负责人: Yusu Wang
• 金额: $ 40万
• 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: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1526513&HistoricalAwards=false
• 关键词: AF; Small; Analyzing; Complex; Data

In the modern data-centric era, one is constantly faced with the task of extracting intelligent summaries out of diverse, complex data. This task is becoming increasingly challenging as the data becomes more complex. Recent work has demonstrated that topological ideas and concepts can be powerful in extracting essential structures/features that are hidden in data. Although existing topological methods are promising and powerful, they are limited when analyzing data that is laced with complex maps (e.g, non-real valued functions) and temporal components. This project aims to broaden the scope of topological techniques and methodologies for analyzing such complex data. Specifically, the PIs will investigate novel methodologies and computational issues to address key challenges caused by complexity in modern data: the diverse properties/information associated with data, the dynamic/time-varying behavior of data, and the sheer volume of the data. The project will provide a theoretical understanding of a recently proposed framework, called Mapper, and its extension to a multiscale formulation. It will explore the use of persistence methodologies, including zigzag constructions, for understanding the time-varying aspects of data. The geometric and topological ideas behind this project will bring new perspectives to the important field of computational data analysis. A successful algorithmic theory for summarizing and characterizing complex and dynamic data with topological techniques can provide a powerful tool for data exploration and analysis in various fields of science and engineering. The educational impact of this project is in a large synergy between mathematics and computer science motivated by real applications. The findings from the project are planned to be part of the course materials that the PIs develop. This project will train graduate students who will develop skills in mathematics and theoretical computer science, most notably in algorithms and topology, in writing efficient software, and its application to analyzing data sets. The combination of such skills is becoming increasingly essential in modern data science.

在现代以数据为中心的时代，人们不断面临着从多样化、复杂的数据中提取智能摘要的任务。随着数据变得越来越复杂，这项任务变得越来越具有挑战性。 最近的工作表明，拓扑思想和概念在提取隐藏在数据中的基本结构/特征方面非常强大。 尽管现有的拓扑方法很有前景且功能强大，但在分析包含复杂映射（例如非实值函数）和时间分量的数据时，它们受到限制。该项目旨在扩大用于分析此类复杂数据的拓扑技术和方法的范围。 具体来说，PI 将研究新的方法和计算问题，以解决现代数据复杂性带来的关键挑战：与数据相关的不同属性/信息、数据的动态/随时间变化的行为以及数据的庞大数量。该项目将为最近提出的名为 Mapper 的框架及其对多尺度公式的扩展提供理论理解。它将探索使用持久性方法（包括锯齿形结构）来理解数据随时间变化的方面。该项目背后的几何和拓扑思想将为计算数据分析的重要领域带来新的视角。用拓扑技术总结和表征复杂动态数据的成功算法理论可以为科学和工程各个领域的数据探索和分析提供强大的工具。该项目的教育影响在于由实际应用推动的数学和计算机科学之间的巨大协同作用。该项目的研究结果计划成为 PI 开发的课程材料的一部分。该项目将培训研究生，培养他们在数学和理论计算机科学方面的技能，尤其是算法和拓扑、编写高效软件及其在分析数据集方面的应用。这些技能的结合在现代数据科学中变得越来越重要。

项目成果

Efficient Algorithms for Computing a Minimal Homology Basis

计算最小同调基础的有效算法

• DOI: 978-3-319-77404-6_28
• 发表时间: 2018-03
• 期刊: Latin American Symposium on Theoretical Informatics
• 作者: Dey, Tamal;Li, Tianqi;Wang, Yusu
• 通讯作者: Wang, Yusu