Matrix and Tensor Decompositions in Signal Processing, Volume 2

2Hadamard, Kronecker and Khatri–Rao Products

2.1. Introduction

In this chapter, we consider three matrix products that play a very important role in matrix computation: the Hadamard, Kronecker and Khatri–Rao products. The Kronecker product, also known as the tensor product, is widely used in many signal and image processing applications, such as compressed sampling using Kronecker dictionaries (Duarte and Baraniuk 2012), and for image restoration (Nagy et al. 2004). It is also widely used in systems theory (Brewer 1978) and in linear algebra to express and solve matrix equations such as Sylvester and Lyapunov equations (Bartels and Stewart 1972), as well as generalized Lyapunov equations (Lev Ari 2005). Additionally, it plays a key role in simplifying and implementing rapid transform algorithms, such as Fourier, Walsh-Hadamard, and Haar transforms (Regalia and Mitra 1989; Pitsianis 1997; Van Loan 2000). Recently, the identification of bilinear filters decomposed into Kronecker products was proposed in Paleologu et al. (2018), and so-called Kronecker receivers allowed various wireless communication systems based on different tensor models to be presented with a single, unified approach (da Costa et al. 2018). The Hadamard product has been applied in statistics (Styan 1973; Neudecker et al. 1995; Neudecker and Liu 2001). The Khatri–Rao product has been used to define space-time codes (Sidiropoulos and Budampati 2002) and space-time-frequency codes (de Almeida and Favier 2013) ...