Collaborative Research: Topological Invariants for Enhanced Data Analysis

点击翻译按钮获取中文摘要

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.