Non-asymptotic problems on random operators in geometric functional analysis and applications

这项建议将推进和应用几何泛函分析技术的理论和计算问题有关的随机算子。经典的随机矩阵理论传统上集中在渐近制度，当矩阵的维数增加到无穷大。然而，今天的许多应用程序在非渐近制度，固定，但大尺寸，他们需要明确的概率界。将进行非渐近随机矩阵理论的系统发展。将特别注意推进随机算子的可逆性理论。在估计具有独立项的随机矩阵的条件数和最小奇异值的最新进展的基础上，该程序现在将扩展到包括一些更困难的类或随机运算符：随机厄米特矩阵，随机扰动的确定性矩阵，确定性矩阵的随机子矩阵，分解随机矩阵的算子，独立随机矩阵与列独立随机矩阵的和。这些进展将基于概率、分析和几何工具的并行发展，包括小球概率的Littlewood-Offord理论及其与加法组合学的联系。在高维空间中，随机算子/矩阵被广泛用于模拟太复杂或太一般而无法确定性或显式理解的变换。这种随机化方法的更经典的应用，包括数学物理中的应用，都是在渐近状态下进行的。这意味着随机算子作用于维数无限增加的空间，人们试图捕捉维数接近无穷大时的极限图像。相比之下，许多今天的应用程序在非渐近制度，作用于空间的大，但固定的尺寸。这尤其发生在函数分析中，其中随机算子对典型算子进行建模;算法的数值分析，其中随机矩阵对典型输入进行建模;压缩感知的新领域，其中随机算子是最知名的测量系统;统计学，其中随机矩阵捕获人口样本的各种参数之间的依赖关系。这一建议将系统地推进和统一的随机矩阵理论在非渐近制度，以期在上述领域的应用。该计划是基于与几何功能分析，统计，电气工程和计算机科学研究人员的合作。一名研究生预计将加入PI的努力开始第二年的计划。

This proposal will advance and apply the techniques of geometric functional analysis for theoretical and computational problems related to random operators. The classical random matrix theory traditionally focuses on the asymptotic regime, when the dimensions of the matrices increase to infinity. However, many of today's applications operate in the non-asymptotic regime, with fixed but large dimensions, and they require explicit probability bounds. A systematic development of the non-asymptotic random matrix theory will be carried out. Special attention will be paid to advancing the invertibility theory of random operators. Building on the recent progress on estimating the condition numbers and the smallest singular values of random matrices with independent entries, the program will now be expanded to include some more difficult classes or random operators: random Hermitian matrices, randomly perturbed deterministic matrices, random submatrices of deterministic matrices, operators that factor through random matrices, sums of independent random matrices and random matrices with independent columns. These advances will be based on on parallel development of probabilistic, analytical and geometric tools, including the Littlewood-Offord theory of small ball probabilities and its connections with additive combinatorics.In high dimensional spaces, random operators/matrices are widely used to model transformations that are too complicated or too general to be understood deterministically or explicitely. The more classical applications of this randomized approach, including those in mathematical physics, operate in the asymptotic regime. This means that the random operators act on spaces whose dimension increases indefinitely, and one seeks to capture the limiting picture as the dimension approaches infinity. In contrast, many of today's applications operate in the non-asymptotic regime, acting on spaces of large but fixed dimensions. This happens in particular in functional analysis where random operators model typical operators; numerical analysis of algorithms where random matrices model typical inputs; the new area of compressed sensing where random operators are the best known measurement systems; statistics where random matrices capture the dependencies among the various parameters of a sample of a population. This proposal will systematically advance and unite the random matrix theory in the non-asymptotic regime, with a view toward applications in the areas mentioned above. The program is based on collaboration with researchers in geometric functional analysis, statistics, electrical engineering, and computer science. A graduate student is expected to join the PI's efforts starting the second year of the program.