Random Matrices, Random Graphs, and Deep Neural Networks

随机矩阵理论已被证明在广泛的学科中是有用的，包括凝聚态物理，高维统计，数论和网络理论。随机矩阵的实用性在于其特征值和特征向量统计量对于非常大的矩阵具有普适性。这种现象只依赖于潜在的对称性，而不依赖于单个入口的定律。本研究项目旨在扩大对随机矩阵的普遍性现象的认识，并为随机矩阵理论的更多应用开发新的工具和技术。本项目将探索与随机矩阵理论相关的两个研究方向。在第一个方向上，该项目旨在了解随机d正则图邻接矩阵的谱性质。条目之间的稀疏性和依赖性使得分析此类模型比分析wigner型随机矩阵更具挑战性。研究者和合作者先前证明了具有增长度的稀疏随机图的局部谱统计量的普适性。本课题的目标是了解固定度随机d-正则图的局部统计量，特别是随机d-正则图的极端特征值的波动。这将为极稀疏系统的普适性现象提供见解，并有望在理论计算机科学中得到应用。在第二个方向，研究者致力于研究统计学习模型，如深度神经网络和张量模型。尽管这些模型是由随机矩阵和随机张量建立起来的，但由于非线性，随机矩阵理论的强大机制迄今为止在研究它们方面取得的成功有限。本项目旨在发展随机矩阵理论，以纳入非线性，并为随机矩阵理论在统计学习中的更多应用打开大门。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。

Random matrix theory has proven useful in a wide range of disciplines, including condensed matter physics, high dimensional statistics, number theory, and network theory. The utility of random matrices lies in the universality of their eigenvalue and eigenvector statistics for very large matrices. This phenomenon depends only on the underlying symmetry and is independent of the law of individual entries. This research project aims to broaden understanding of the universality phenomenon of random matrices and to develop new tools and techniques for more applications of random matrix theory. The project will explore two research directions related to random matrix theory. In the first direction, the project aims to understand the spectral properties of adjacency matrices of random d-regular graphs. The sparsity and dependency among entries make the analysis of such models more challenging than that for Wigner-type random matrices. The investigator and collaborators previously proved the universality of the local spectral statistics for sparse random graphs with growing degrees. The project's goal is to understand the local statistics of random d-regular graphs with fixed degree, particularly the fluctuations of extreme eigenvalues of random d-regular graphs. This will provide insights for the universality phenomenon for extremely sparse systems and is expected to have applications in theoretical computer science. In the second direction, the investigator aims to study statistical learning models, such as deep neural networks and tensor models. Even though these models are built out of random matrices and random tensors, the powerful machinery of random matrix theory has so far found limited success in studying them, due to the nonlinearity. This project seeks to develop random matrix theory to incorporate nonlinearity and open the door for more applications of random matrix theory to statistical learning.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.