Non-Asymptotic Random Matrix Theory and Geometric Functional Analysis

非渐近随机矩阵理论与几何泛函分析

• 批准号: 1464514
• 负责人: Mark Rudelson
• 金额: $ 37.5万
• 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: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1464514&HistoricalAwards=false
• 关键词: Non; Asymptotic; Random; Matrix; Theory

The research is intended to provide new connections between two areas of mathematics, probability and functional analysis. One of the main objects of investigation is a random matrix, a large rectangular array of random data. The PI strives to understand the properties of such arrays which hold with high probability and the dependence of those properties on the nature of random entries and the structure of the matrix. This study will have potential applications beyond the realm of pure mathematics, as random matrices are used in statistics, computer algorithms, and wireless communication. Another direction of the proposed research is the study of high dimensional convex sets with the emphasis on their complexity and approximation. This research will also have computer science applications including rate estimates for various high-dimensional algorithms.One of the main directions of this research is the non-asymptotic theory of random matrices, a new and rapidly developing area of research analyzing spectral characteristics of a random matrix of a large but fixed size and striving to obtain bounds valid with high probability. The PI intends to study singular values, eigenvalues, and eigenvectors of different ensembles of random matrices of a large size. The results obtained in this direction would have important applications within the random matrix theory in proving limit laws for the spectral characteristics of random matrices. Another group of problems comes from geometric functional analysis, an area of mathematics concerned with the study of high-dimensional convex bodies and normed spaces. A progress in this direction would lead to better understanding of the structure of sections and projections of such bodies, as well as possibility of approximation of a general convex body by a body with certain nice properties. Both directions have a significant component related to the theoretical computer science.

该研究旨在提供数学，概率和函数分析两个领域之间的新联系。研究的主要对象之一是随机矩阵，即随机数据的大型矩形阵列。PI致力于理解这些数组的属性，这些属性具有高概率，以及这些属性对随机条目的性质和矩阵结构的依赖性。这项研究将具有超越纯数学领域的潜在应用，因为随机矩阵用于统计、计算机算法和无线通信。另一个研究方向是高维凸集的研究，重点是它们的复杂性和近似性。这项研究也将有计算机科学的应用，包括各种高维算法的速率估计。这项研究的主要方向之一是随机矩阵的非渐近理论，一个新的和迅速发展的研究领域，分析一个大的，但固定大小的随机矩阵的谱特性，并努力获得高概率有效的界限。PI的目的是研究奇异值，特征值和特征向量的不同合奏的随机矩阵的大规模。在这个方向上得到的结果将有重要的应用范围内的随机矩阵理论在证明极限法律的谱特征的随机矩阵。另一组问题来自几何泛函分析，这是一个与研究高维凸体和赋范空间有关的数学领域。在这个方向上的进展将导致更好地理解的结构的部分和投影等机构，以及一个一般的凸体的近似的可能性，具有一定的良好性能的机构。这两个方向都与理论计算机科学有关。