Non-Asymptotic Approach in Random Matrix Theory

拟议的研究旨在提供数学，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。PI计划特别强调稀疏矩阵的研究，因为这些矩阵自然会出现在信号重建和大数据分析中。所提出的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，图形可以用来模拟原子在材料，互联网社区等的相互作用。本研究的主要方向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的频谱特性的随机矩阵的一个大的，但固定的大小，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。它们在计算机科学中也很有用，因为奇异值控制着许多数值算法的收敛速度。另一部分的拟议研究将解决随机图的几何中出现的问题，这个方向是密切相关的随机矩阵理论以及。PI将研究随机图的特征向量的离域性和节点域.该奖项反映了NSF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。

The proposed 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. The PI plans to put a special emphasis on the study of sparse matrices as these matrices naturally appear in signal reconstruction and big data analysis. Another direction of the proposed 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 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 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. They would be also useful in computer science, as the singular values control the rate of convergence of many numerical algorithms. Another part of the proposed research will address the problems arising in geometry of random graphs This direction is strongly related to random matrix theory as well. The PI will study delocalization and nodal domains of eigenvectors a random graph. The information about them would be valuable in mathematical analysis of congestion in transportation networks.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.