AF: Small: Analyzing Complex Data with a Topological Lens

在以数据为中心的现代时代，人们经常面临从多样化、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，这项任务正变得越来越具有挑战性。最近的工作表明，拓扑思想和概念可以强大地提取隐藏在数据中的基本结构/特征。尽管现有的拓扑方法前景看好且功能强大，但它们在分析含有复杂地图(如非实值函数)和时间分量的数据时存在局限性。该项目旨在扩大分析这种复杂数据的拓扑学技术和方法的范围。具体地说，PIS将调查新的方法和计算问题，以应对现代数据复杂性造成的关键挑战：与数据相关的各种属性/信息、数据的动态/时变行为以及数据的绝对数量。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上的理解。它将探索持久化方法的使用，包括Z字形构造，以了解数据的时变方面。这个项目背后的几何和拓扑思想将为计算数据分析这一重要领域带来新的视角。一个成功的用拓扑技术概括和刻画复杂动态数据的算法理论可以为科学和工程的各个领域的数据探索和分析提供强有力的工具。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，受到实际应用的推动。该项目的调查结果计划作为私人投资促进机构开发的课程材料的一部分。这个项目将培养研究生在数学和理论计算机科学方面的技能，特别是在算法和拓扑学方面，编写高效的软件，以及它在分析数据集方面的应用。在现代数据科学中，这些技能的结合正变得越来越重要。

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.