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

计算社区正日益面临着处理和解释大量数据的挑战，这些数据是由我们周围的个人设备、数字传感器、数据中心、科学研究和社交网络产生的。这些数据的性质通常是高维的，也就是说，当被解释为点时，它们驻留在高维欧几里德空间中。“维度”和“尺寸”的诅咒，给从它们中以原则性的方式解读知识带来了巨大的挑战。在提出的“挖掘”这些数据的各种方法中，拓扑方法正在成为强大的全局方法，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和几何中的新算法来处理“大”数据的困难是本提案的重点。提出的研究旨在探讨三种技术，即子采样、局部化和简单崩溃来处理“尺寸”和“维度”的威胁。这需要开发基于算法理论和代数拓扑、分析和离散几何数学的创新计算工具。所提出的拓扑方法将增强对“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的连接。目标是产生实用的算法，这些算法可以转化为对学术界和工业界都有用的可用软件。除了标准的学术论坛，我们还计划通过课程笔记、教程和网页来传播这个项目的成果，以吸引更多的受众。PI还计划开发拓扑数据分析的课程材料，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。

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.