Collaborative Research: Topological Invariants for Enhanced Data Analysis

拓扑数据分析(TDA)是近十年来兴起的一种替代主流数据分析工具的强大工具。非正式地说，TDA的主要前提是任何给定问题中的数据都有一个形状，而这个形状很重要，描述了应该使用什么类别的模型。数据的形状通常揭示了隐藏的模式，这些模式表征了潜在的过程，从而为我们提供了对正在调查的现象的更多洞察。拓扑数据分析的大部分工作都集中在数据的全局属性上。然而，渗透在复杂数据集中的局部结构，特别是数据中不同局部结构的分布，可以提供对理解基本过程至关重要的丰富的附加信息。这项工作旨在创建一系列工具，以及理论保证和算法工具，使人们能够定义和计算数据集的局部拓扑结构。主要研究人员致力于证明的理论结果将涉及从底层空间、非均匀或不规则分布的随机噪声样本进行正确局部拓扑恢复的概率估计。在这项提议中开发的算法和软件工具将在数据分析的几个问题上进行测试和磨练，这些问题源于现代科学或工程的前沿课题。更具体地说，PI的目的是(1)开发描述数据中局部拓扑结构的数学工具；(2)开发跨尺度理解局部拓扑不变量的方法；(3)研究随机函数的局部拓扑不变量，这是量化纯噪声的局部拓扑结构的重要步骤；(4)发展稳健的、基于局部拓扑的方法来捕获动态系统中的瞬时行为(例如相变)。为了说明和验证所获得的结果，它们将被用于(1)捕获电力网络中的瞬时行为，例如危险振荡的开始；(2)表征复杂网络，特别是互联网和植物根系；(3)调查大脑活动，重点是各种耳鸣相关情况的神经特征。

Topological Data Analysis (TDA) emerged over the past decade as a powerful alternative to the mainstream tools of data analysis. Informally stated, the main premise of the TDA is that the data in any given problem have a shape, and this shape matters, describing what classes of models should be used. The shapes of data often reveal otherwise hidden patterns which characterize the underlying process, thus providing us with more insight into the phenomenon under investigation. Most of the work in topological data analysis has been focused on global properties of data. However, local structures permeating complex data sets, and in particular the distribution of different local structures within the data, can provide a wealth of additional information crucial to understanding of the underlying process. The proposed work aims at creating a collection of tools, together with theoretical guarantees and algorithmic instruments, allowing one to define and compute local topological structure of the datasets.The theoretical results which the principal investigators aim to prove would involve the estimates on the probability of correct local topology recovery for random noisy samples from the underlying space, from non-uniform or irregular distributions. The algorithms and software tools developed within this proposal will be tested and honed on several problems of data analysis stemming from topics at the forefront of modern science or engineering. More specifically, the PIs intend to (1) develop mathematical tools for describing local topological structures in data; (2) develop methods for understanding behavior of local topological invariants across scales; (3) study local topological invariants of random functions, which is an important step towards quantifying local topological structure of pure noise; (4) develop robust, local topology based methods for capturing transient behavior (e.g. phase transitions) in dynamical systems. To illustrate and validate the obtained results, they will be employed to (1) capture transient behavior in power networks, such as onset of dangerous oscillations; (2) characterize complex networks, in particular, the Internet and plant root systems; (3) investigate brain activity, with the focus on neural characteristics of various tinnitus related conditions.