Large-Dimensional Random Matrix Theory and Its Applications in Deep Learning and Wireless Communications

TL;DR

This paper reviews how large-dimensional random matrix theory (RMT) provides valuable insights and tools for analyzing and designing complex systems in wireless communications and deep learning, especially as data and system sizes grow.

Contribution

It summarizes key RMT results and demonstrates its applications in spectrum sensing, system analysis, and neural network optimization, highlighting new opportunities.

Findings

RMT aids in spectrum sensing for cognitive radio.

RMT helps analyze neural network Hessians and Jacobians.

RMT offers asymptotic analysis tools for large systems.

Abstract

Large-dimensional random matrix theory, RMT for short, which originates from the research field of quantum physics, has shown tremendous capability in providing deep insights into large dimensional systems. With the fact that we have entered an unprecedented era full of massive amounts of data and large complex systems, RMT is expected to play more important roles in the analysis and design of modern systems. In this paper, we review the key results of RMT and its applications in two emerging fields: wireless communications and deep learning. In wireless communications, we show that RMT can be exploited to design the spectrum sensing algorithms for cognitive radio systems and to perform the design and asymptotic analysis for large communication systems. In deep learning, RMT can be utilized to analyze the Hessian, input-output Jacobian and data covariance matrix of the deep neural networks, thereby to understand and improve the convergence and the learning speed of the neural networks. Finally, we highlight some challenges and opportunities in applying RMT to the practical large dimensional systems.