Large-dimensional (LD) 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.

中文翻译：

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大维随机矩阵理论及其在深度学习和无线通信中的应用

大维（LD）随机矩阵理论，简称RMT，起源于量子物理学研究领域，在深入洞察大维系统方面表现出巨大的能力。随着我们进入前所未有的海量数据和大型复杂系统时代，RMT有望在现代系统的分析和设计中发挥更重要的作用。在本文中，我们回顾了 RMT 的主要成果及其在两个新兴领域的应用：无线通信和深度学习。在无线通信中，我们展示了 RMT 可用于设计认知无线电系统的频谱感知算法，并执行大型通信系统的设计和渐近分析。在深度学习中，RMT 可用于分析 Hessian，深度神经网络的输入-输出雅可比矩阵和数据协方差矩阵，从而了解和提高神经网络的收敛性和学习速度。最后，我们强调了将 RMT 应用于实际大维系统的一些挑战和机遇。