An Introduction to Complex Random Tensors

TL;DR

This paper reviews the fundamental concepts and statistical properties of complex random tensors, emphasizing their applications in engineering fields like signal processing and machine learning, and discusses tensor eigenvalues, singular values, and low-rank recovery.

Contribution

It introduces the theory of complex random tensors, characterizes their statistical properties, and explores asymptotic distributions relevant for signal recovery in high-dimensional data.

Findings

Characterization of statistical properties of complex random tensors

Analysis of correlation structures and tensor-valued random processes

Asymptotic distribution of tensor eigenvalues and singular values

Abstract

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.