Geometry of Nonhomogeneous Random Matrices, Vectors, and Processes

高维随机结构出现在整个数学及其应用。概率论的许多经典问题致力于理解高维系统中随机噪声产生的模式。例如，假设有一个大矩阵，其每个条目都是独立的公平硬币投掷的结果，即，矩阵中充满了噪声。这样一个矩阵的特征值是什么样的？这些问题都有美丽的答案，现在已经很好地理解了。然而，在许多情况下，无论是在纯数学还是应用数学中，在噪声之下都存在着非平凡的结构。例如，在随机矩阵问题中，我们可以允许每个条目具有反映相应变量强度的任意方差。这些问题的挑战是捕捉结构和噪声之间的相互作用，这些问题比经典问题更难理解。该项目的目的是开发数学技术，使之能够在相当大的一般性解决这些问题。除了他们的基本利益，这些技术预计将适用于统计和计算机科学等领域的高维随机结构发挥着重要作用。几个教育和推广活动进一步形成了这个项目的一个重要组成部分。这个项目的总体目标是在高维概率中开发新的机制和技术，以解决可以广泛地描述为具有非均匀结构的问题。这些问题的基本问题是：如何反映在模型的概率行为的基本几何结构？这类问题在各种纯数学问题和应用数学问题中具有重要意义。本计画旨在系统地探讨三个相关主题中的问题：（i）非齐次随机矩阵的范数（例如，具有任意方差或稀疏模式）;（ii）非齐次随机过程的几何（例如在随机矩阵理论或泛函分析中出现的那些）;（iii）几何不等式和无量纲现象（例如，与测度集中或凸性有关）。该项目将利用这些主题的最新进展，推动概率及其应用领域的最新技术。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。

High-dimensional random structures arise throughout mathematics and its applications. Many classical problems of probability theory are devoted to understanding the patterns that are generated by random noise in high-dimensional systems. For example, suppose one has a large matrix each of whose entries is the outcome of an independent fair coin flip, i.e., the matrix is filled with noise. What do the eigenvalues of such a matrix look like? Such questions have beautiful answers that are now well understood. There are however many situations, both in pure and in applied mathematics, in which there is nontrivial structure underneath the noise. For example, in the random matrix problem, we could allow each entry to have an arbitrary variance that reflects the intensity of the corresponding variable. The challenge in such problems, which are much less well understood than their classical counterparts, is to capture the interplay between the structure and the noise. The aim of this project is to develop mathematical techniques that make it possible to address such questions in considerable generality. Beside their fundamental interest, such techniques are expected to be applicable to areas such as statistics and computer science where high-dimensional random structures play an important role. Several education and outreach activities further form a key part of this project.The overarching goal of this project is to develop new mechanisms and techniques in high-dimensional probability to address problems that could be broadly characterized as having nonhomogeneous structure. The fundamental question in such problems is: how is the underlying geometric structure reflected in the probabilistic behavior of the model? Such questions are of significant importance in various pure and applied mathematical problems. This project aims to systematically investigate such problems that arise in three interrelated themes: (i) the norms of nonhomogeneous random matrices (for example, with arbitrary variance or sparsity pattern); (ii) the geometry of nonhomogeneous random processes (such as those that arise, for example, in random matrix theory or in functional analysis); (iii) geometric inequalities and dimension-free phenomena (for example, in connection with measure concentration or convexity). The project will leverage recent advances on these topics to push forward the state-of-the-art in this area of probability and its applications.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.