Geometric functional analysis, random matrices and applications

几何泛函分析、随机矩阵及其应用

• 批准号: 1265782
• 负责人: Roman Vershynin
• 金额: $ 21.9万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-15 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1265782&HistoricalAwards=false
• 关键词: Geometric; functional; analysis; random; matrices

This project seeks to advance methods of geometric functional analysis and to apply them for problems of non-asymptotic random matrix theory and high-dimensional data. One of the main goals of this project is to expand our understanding of non-asymptotic properties of random matrices, in particular, their quantitative invertibility. A related goal is to develop a direct, nonspectral approach to delocalization of eigenvectors of random matrices. Methods of geometric functional analysis may succeed even where spectral methods fail due to an unknown (or nonexisting) limiting spectral distribution. Next, functional analytic and probabilistic methodology will be applied to high-dimensional data. This may result in new geometric approaches to compressed sensing, community detection in networks, and robust principal component analysis. Geometric functional analysis studies fundamental properties of high-dimensional structures. Such structures are ubiquitous in modern applications. The unprecedented volume of data described by large number of parameters prevents many traditional statistical approaches from working. This project will look for new ways to understand, represent, and analyze high-dimensional structures using methods of geometric functional analysis. Specific structures for which the new methodology can be applied include large matrices (e.g., consumer data), networks (e.g., food chains, social networks), signals (e.g., images, audio, video). Conversely, the fundamental challenges of high-dimensional data are likely to open up some new directions of basic research at the intersection of functional analysis, high-dimensional geometry, and probability.

本项目旨在推进几何泛函分析方法，并将其应用于非渐近随机矩阵理论和高维数据问题。这个项目的主要目标之一是扩展我们对随机矩阵的非渐近性质的理解，特别是它们的定量可逆性。一个相关的目标是发展一种直接的、非谱的方法来求解随机矩阵的特征向量的离域。几何泛函分析方法可能成功，即使光谱方法失败，由于未知的（或不存在的）限制光谱分布。接下来，函数分析和概率方法将应用于高维数据。这可能会产生新的几何方法来压缩感知、网络中的社区检测和鲁棒主成分分析。几何泛函分析研究高维结构的基本性质。这种结构在现代应用中无处不在。由大量参数描述的空前的数据量使许多传统的统计方法无法工作。这个项目将寻找新的方法来理解，表示和分析高维结构使用几何功能分析的方法。新方法可以应用的具体结构包括大型矩阵（例如，消费者数据）、网络（例如，食物链、社会网络）、信号（例如，图像、音频、视频）。相反，高维数据的基本挑战可能会在功能分析、高维几何和概率的交叉领域开辟一些新的基础研究方向。