Random matrices and geometric functional analysis

随机矩阵和几何泛函分析

• 批准号: 1161372
• 负责人: Mark Rudelson
• 金额: $ 19.8万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1161372&HistoricalAwards=false
• 关键词: Random; matrices; geometric; functional; analysis

This mathematics research project belongs to the interface of probability, convex geometry, and functional analysis. One of the main areas of investigation is the non-asymptotic theory of random matrices, a new and rapidly developing area of research, which analyzes spectral characteristics of a random matrix of large, but fixed size, and strives to obtain bounds, which are valid with high probability. The requirements for such bounds arise naturally in convex geometry and geometric functional analysis, where random matrices are used to study typical sections of a high-dimensional convex body. Another direction of this research is the geometry of non-symmetric convex bodies, where the probabilistic methods also play a crucial role.Several components of this mathematics research project are motivated by problems in computer science and engineering. These include signal reconstruction problems arising in computer tomography, as well as questions related to protecting privacy while releasing statistical information. Rudelson and his collaborators will establish new connections between probability and geometric functional analysis with the objective to study the spectral properties of random matrices. Random matrices have become the main model for signal reconstruction in wireless communication and can lead to a better understanding of the convergence of many computer science algorithms.

这个数学研究项目属于概率、凸几何和泛函分析的接口。研究的主要领域之一是随机矩阵的非渐近理论，这是一个新的和迅速发展的研究领域，它分析了一个大的，但固定大小的随机矩阵的谱特性，并努力获得边界，这是有效的高概率。这种界限的要求自然出现在凸几何和几何泛函分析中，其中随机矩阵用于研究高维凸体的典型截面。本研究的另一个方向是非对称凸体的几何，其中概率方法也发挥了至关重要的作用。这个数学研究项目的几个组成部分是由计算机科学和工程中的问题所激发的。这些问题包括计算机断层扫描中出现的信号重建问题，以及在发布统计信息的同时保护隐私的问题。Rudelson和他的合作者将建立概率和几何泛函分析之间的新联系，目的是研究随机矩阵的谱特性。 随机矩阵已经成为无线通信中信号重构的主要模型，并且可以更好地理解许多计算机科学算法的收敛性。