Non-Asymptotic Random Matrix Theory and Random Graphs

这项研究旨在为概率分析和泛函分析这两个数学领域提供新的联系。调查的主要对象之一是随机矩阵，即随机数据的大型矩形阵列。首席调查员努力了解这种高概率保持的数组的属性，以及这些属性对随机条目的性质和矩阵结构的依赖。这项研究将具有超越纯数学领域的潜在应用，因为随机矩阵被用于统计学、计算机算法和无线通信。由于稀疏矩阵自然出现在信号重建和大数据分析中，因此将特别强调对稀疏矩阵的研究。研究的另一个方向是研究随机图，它是由道路(边)连接的节点的随机网络。除了表示真实的交通网络外，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。该项目为研究生提供了研究培训机会。本文的主要研究方向是随机矩阵的非渐近理论，这是一个新的、发展迅速的研究领域，它分析大而固定大的随机矩阵的谱特性，努力获得高概率有效的界。首席调查员打算研究大型随机矩阵的不同集合的奇异值、特征值和特征向量。在这个方向上得到的结果将在随机矩阵理论中在证明随机矩阵的谱特征的极限定律方面有重要的应用。另一个研究领域将是某些赋范空间之间被视为线性算子的这种矩阵的几何性质。这样的结果可以在计算机科学和信号重构中得到应用，在这些领域中，随机矩阵被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。首席调查员还将通过分析相应随机矩阵的演变来集中研究随机图的增长过程。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.