Non-Asymptotic Random Matrix Theory and Random Graphs

非渐近随机矩阵理论和随机图

基本信息

批准号：
2054408
负责人：
Mark Rudelson
金额：
$ 23.4万
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：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054408&HistoricalAwards=false
关键词：
Non Asymptotic Random Matrix Theory

项目摘要

This 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 Principal Investigator 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. A special emphasis will be placed on the study of sparse matrices as these matrices naturally appear in signal reconstruction and big data analysis. Another direction of the research is the study of random graphs, which are random networks of nodes connected by roads (edges). Besides representing real transportation networks, graphs can be used to model interaction of atoms in a material, internet communities, etc. The project provides research training opportunities for graduate students. The main direction 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 Principal Investigator 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 area of study will be the geometric properties of such matrices considered as linear operators between certain normed spaces. Such results can find applications in computer science and signal reconstruction where random matrices are widely used for signal encoding and decoding. Another part of this research will address the problems arising in geometry of random graphs. The Principal Investigator will also concentrate on studying the process of growth of random graphs by analyzing the evolution of corresponding random matrices.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这项研究旨在提供数学、概率和泛函分析这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。首席研究员致力于了解此类数组具有高概率的属性以及这些属性对随机条目的性质和矩阵结构的依赖性。这项研究将具有超出纯数学领域的潜在应用，因为随机矩阵可用于统计、计算机算法和无线通信。将特别强调稀疏矩阵的研究，因为这些矩阵自然出现在信号重建和大数据分析中。研究的另一个方向是随机图的研究，随机图是由道路（边）连接的节点的随机网络。除了代表真实的交通网络之外，图还可以用于模拟材料中原子的相互作用、互联网社区等。该项目为研究生提供研究培训机会。本研究的主要方向是随机矩阵的非渐近理论，这是一个快速发展的新兴研究领域，分析大而固定大小的随机矩阵的谱特征，并力求获得高概率有效的界限。首席研究员打算研究大尺寸随机矩阵的不同集合的奇异值、特征值和特征向量。在这个方向上获得的结果将在随机矩阵理论中具有重要的应用，以证明随机矩阵谱特性的极限定律。另一个研究领域是此类矩阵的几何性质，被视为某些赋范空间之间的线性算子。这些结果可以在计算机科学和信号重建中找到应用，其中随机矩阵广泛用于信号编码和解码。这项研究的另一部分将解决随机图几何中出现的问题。首席研究员还将通过分析相应随机矩阵的演化来集中研究随机图的增长过程。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，认为值得支持。