Geometry of Nonhomogeneous Random Matrices, Vectors, and Processes

高维随机结构出现在数学及其应用中。概率论的许多经典问题都致力于理解高维系统中由随机噪声产生的模式。例如，假设一个人有一个大矩阵，它的每一个条目都是一个独立的公平抛硬币的结果，也就是说，这个矩阵充满了噪声。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在应用数学中，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。这类问题的挑战在于捕捉结构和噪声之间的相互作用，人们对这类问题的理解远不及经典问题。这个项目的目的是发展数学技术，使它有可能在相当普遍的情况下解决这些问题。除了它们的基本兴趣之外，这些技术还有望应用于统计学和计算机科学等高维随机结构发挥重要作用的领域。一些教育和推广活动进一步构成了这个项目的关键部分。该项目的总体目标是在高维概率中开发新的机制和技术，以解决可能被广泛描述为具有非均匀结构的问题。这类问题的根本问题是：底层几何结构如何反映在模型的概率行为中？这些问题在各种纯数学和应用数学问题中具有重要意义。本项目旨在系统地研究在三个相互关联的主题中出现的问题：(i)非齐次随机矩阵的规范（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，与测量浓度或凸性有关）。该项目将利用这些主题的最新进展来推动概率及其应用领域的最新技术。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。

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.