Collaborative Research: Topological Invariants for Enhanced Data Analysis

拓扑数据分析（TDA）在过去十年中作为主流数据分析工具的强大替代品出现。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，这个形状很重要，它描述了应该使用哪些类型的模型。数据的形状通常揭示了隐藏的模式，这些模式描述了潜在的过程，从而为我们提供了对正在调查的现象的更多见解。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，渗透到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，可以提供对理解底层过程至关重要的大量附加信息。建议的工作旨在创建一套工具，连同理论保证和算法工具，允许定义和计算数据集的局部拓扑结构。主要研究人员旨在证明的理论结果将涉及对来自底层空间、非均匀或不规则分布的随机噪声样本的正确局部拓扑恢复概率的估计。在本提案中开发的算法和软件工具将在来自现代科学或工程前沿主题的数据分析的几个问题上进行测试和磨练。更具体地说，pi打算(1)开发用于描述数据中的局部拓扑结构的数学工具；(2)建立理解局部拓扑不变量跨尺度行为的方法；(3)研究随机函数的局部拓扑不变量，这是量化纯噪声局部拓扑结构的重要一步；(4)开发基于局部拓扑的鲁棒方法，用于捕获动态系统中的瞬态行为（例如相变）。为了说明和验证所获得的结果，它们将用于(1)捕获电网中的瞬态行为，例如危险振荡的开始；(2)描述复杂网络，特别是互联网和植物根系；(3)研究大脑活动，重点研究各种耳鸣相关疾病的神经特征。

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.