AF: Small: Analyzing Complex Data with a Topological Lens

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。最近的研究表明，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。虽然现有的拓扑方法是有前途的和强大的，但它们在分析带有复杂映射（例如，非实值函数）和时间组件的数据时受到限制。该项目旨在扩大拓扑技术和方法的范围，以分析如此复杂的数据。具体来说，pi将研究新的方法和计算问题，以解决由现代数据复杂性引起的关键挑战：与数据相关的不同属性/信息，数据的动态/时变行为以及数据的绝对量。该项目将提供对最近提出的Mapper框架的理论理解，并将其扩展到多尺度公式。它将探索持久性方法的使用，包括之字形结构，以理解数据的时变方面。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。利用拓扑技术成功地总结和表征复杂动态数据的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。这个项目的教育影响是数学和计算机科学之间的巨大协同作用，这些协同作用是由实际应用驱动的。项目的研究结果将被纳入pi开发的课程材料中。该项目将培养研究生在数学和理论计算机科学方面的技能，尤其是在算法和拓扑方面，编写高效的软件，并将其应用于分析数据集。这些技能的结合在现代数据科学中变得越来越重要。

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.