Random Matrices, Random Graphs, and Deep Neural Networks

随机矩阵理论在很多学科中都被证明是有用的，包括凝聚态物理、高维统计、数论和网络理论。随机矩阵的效用在于其特征值和特征向量统计量对于非常大的矩阵的普适性。这种现象只取决于基本的对称性，而与个别条目的规律无关。本研究项目旨在扩大对随机矩阵普适性现象的理解，并为随机矩阵理论的更多应用开发新的工具和技术。该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，该项目旨在了解随机d-正则图的邻接矩阵的谱性质。条目之间的稀疏性和依赖性使得对这类模型的分析比对Wigner型随机矩阵的分析更具挑战性。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。该项目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-正则图的极值特征值的涨落。这将为极稀疏系统的普遍性现象提供洞察，并有望在理论计算机科学中得到应用。在第二个方向上，研究者的目标是研究统计学习模型，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的，但由于非线性，随机矩阵理论的强大机制迄今在研究它们方面取得的成功有限。该项目旨在开发随机矩阵理论，以纳入非线性，并为将随机矩阵理论更多地应用于统计学习打开大门。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.