Random Matrices, Random Graphs, and Deep Neural Networks

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.