Random Matrices, Random Graphs, and Deep Neural Networks

随机矩阵、随机图和深度神经网络

• 批准号: 2054835
• 负责人: Jiaoyang Huang
• 金额: $ 15.37万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-04-01 至 2023-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054835&HistoricalAwards=false
• 关键词: Random; Matrices; Graphs; Deep; Neural

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.

事实证明，随机矩阵理论在广泛的学科中有用，包括凝结物理，高维统计，数量理论和网络理论。随机矩阵的效用在于其特征值和特征向量统计的普遍性，用于非常大的矩阵。这种现象仅取决于潜在的对称性，并且独立于单个条目的定律。该研究项目旨在扩大对随机矩阵的普遍性现象的了解，并开发新的工具和技术，以应用随机矩阵理论的更多应用。 该项目将探讨与随机矩阵理论有关的两个研究方向。在第一个方向上，该项目旨在了解随机D期权图的邻接矩阵的光谱特性。条目之间的稀疏性和依赖性使得对此类模型的分析比Wigner型随机矩阵更具挑战性。研究人员和合作者先前证明了局部频谱统计的普遍性，用于稀疏的随机图，该图的增长程度。该项目的目标是了解具有固定程度的随机D期权图的局部统计数据，尤其是随机d期限图的极端特征值的波动。这将为极稀疏的系统提供通用现象的见解，并有望在理论计算机科学中应用。在第二个方向上，研究者旨在研究统计学习模型，例如深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在研究它们方面的成功有限。 该项目旨在开发随机矩阵理论，以纳入非线性，并为更多的随机矩阵理论应用于统计学习而打开大门。该奖项反映了NSF的法定任务，并被认为是通过基金会的智力优点和更广泛影响的评估标准来评估值得通过评估来支持的。