Non-Asymptotic Random Matrix Theory and Random Graphs

这项研究的目的是在数学的两个领域，概率和功能分析之间提供新的联系。研究的主要对象之一是随机矩阵，即随机数据的大矩形数组。首席研究员努力理解这种具有高概率的数组的属性，以及这些属性对随机条目的性质和矩阵结构的依赖。这项研究将有超越纯数学领域的潜在应用，因为随机矩阵用于统计学，计算机算法和无线通信。我们将特别强调稀疏矩阵的研究，因为这些矩阵自然地出现在信号重构和大数据分析中。另一个研究方向是随机图的研究，随机图是由道路（边）连接的随机节点网络。除了表示真实的交通网络，图形还可以用来模拟材料中原子的相互作用，互联网社区等。本项目为研究生提供研究训练机会。本研究的主要方向是随机矩阵的非渐近理论，这是一个新兴的快速发展的研究领域，分析一个大而固定大小的随机矩阵的谱特征，并努力获得高概率有效的界。本课题拟研究大尺寸随机矩阵不同集合的奇异值、特征值和特征向量。在这个方向上得到的结果在证明随机矩阵谱特征的极限律方面具有重要的应用。另一个研究领域将是这些矩阵的几何性质，这些矩阵被认为是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，其中随机矩阵被广泛用于信号编码和解码。本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，重点研究随机图的生长过程。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.