This work considers the notion of random tensors and reviews some fundamental concepts in statistics when applied to a tensor based data or signal. In several engineering fields such as Communications, Signal Processing, Machine learning, and Control systems, the concepts of linear algebra combined with random variables have been indispensable tools. With the evolution of these subjects to multi-domain communication systems, multi-way signal processing, high dimensional data analysis, and multi-linear systems theory, there is a need to bring in multi-linear algebra equipped with the notion of random tensors. Also, since several such application areas deal with complex-valued entities, it is imperative to study this subject from a complex random tensor perspective, which is the focus of this paper. Using tools from multi-linear algebra, we characterize statistical properties of complex random tensors, both proper and improper, study various correlation structures, and fundamentals of tensor valued random processes. Furthermore, the asymptotic distribution of various tensor eigenvalue and singular value definitions is also considered, which is used for the study of spiked real tensor models that deals with recovery of low rank tensor signals perturbed by noise. This paper aims to provide an overview of the state of the art in random tensor theory of both complex and real valued tensors, for the purpose of enabling its application in engineering and applied science.

翻译：本文探讨随机张量的概念，并评述统计学中应用于基于张量的数据或信号的基本原理。在通信、信号处理、机器学习、控制系统等工程领域中，线性代数与随机变量相结合的概念一直不可或缺。随着这些学科向多域通信系统、多路信号处理、高维数据分析及多线性系统理论演进，亟需引入配备随机张量概念的多线性代数。此外，由于多个此类应用领域涉及复值实体，从复随机张量视角研究该主题势在必行——这正是本文的核心。借助多线性代数工具，我们刻画了复随机张量（包括适定与不适定情形）的统计特性，研究了多种相关结构及张量值随机过程的基本原理。进一步地，本文还探讨了各类张量特征值与奇异值定义的渐近分布，将其用于研究加性噪声扰动下低秩张量信号恢复的尖峰实张量模型。本文旨在综述实值与复值张量随机张量理论的研究现状，以推动其在工程与应用科学中的实践应用。