About Notations in Multiway Array Processing

This paper gives an overview of notations used in multiway array processing. We redefine the vectorization and matricization operators to comply with some properties of the Kronecker product. The tensor product and Kronecker product are also represented with two different symbols, and it is shown how these notations lead to clearer expressions for multiway array operations. Finally, the paper recalls the useful yet widely unknown properties of the array normal law with suggested notations. INTRODUCTION Since tensors have become a popular topic in data science and signal processing, a large number of papers have been published to describe at best their different properties [1]–[7]. Yet a consensus on notations has not been found. Some authors refer to the Tucker and Kruskal operators [2], [8] while others never use these notations but make use of the n-way product [3], [9]. There is at least three different unfolding methods in the literature [1], [3], [10]. The Kronecker product alone has two notations in the community [1], [10], [11]. The tensor product itself is almost never used, even though it is the very foundation of tensor algebra. Instead authors sometimes refer to the outer product ̋, which may be seen as a tensor product of vectors in the canonical basis of each vector space. But the tensor product allows for wider generalization since it applies to tensors of any order and without referring to any basis. Some authors suggest the use of two different symbols for the Kronecker product ⊠ and the general tensor product b [11]. But the same symbol has been used historically and through the literature [12]. The Kronecker product is the expression of the tensor product for matrices when a basis has been given. Yet using the same symbol leads to confusion when manipulating arrays and multilinear operators at the same time. Moreover, manipulating the arrays with matricizations and vectorizations exactly means that the difference between a general tensor product and a basis-dependent Kronecker product is of crucial importance. This leads also to redefining some well known operations, namely the matricization and the vectorization. The main goal of this paper is therefore to set notations for array and manipulations on arrays that are consistent with one another, and with notations used in quantum physics and algebraic geometry where tensors have been used for decades [13]. We recommend that the vectorization and matricization operators are computed so that the tensor products and Kronecker products do not have to be swapped versions of each others, which is compatible with definitions from algebraic geometry [10]. In other words, the suggested notations will lead to vec pab bq “ a⊠ b, where b is the outer product, and ⊠ is the Kronecker product. This differs from the usual equality vec pab bq “ b⊠a, which leads to unnecessarily complicated equations. The first section introduces some notations for all operators used extensively in multiway array processing. Then some useful properties of tensors are given in Section II using the suggested notations. Finally Section III exposes the array normal law as defined by Hoff [9]. It is shown that the array normal law can be a handy tool for multiway array analysis. I. A FEW DEFINITIONS A. About tensor products First let us define the tensor product as given in [14]. Definition 1: Let E , F be two vector spaces on a field K. There is a vector space E bF , called the tensor product space, and a bilinear mapping b : E ˆ F Ñ E bF so that for every vector space G and for all bilinear mapping g from E ˆF to G, there is one and only one linear mapping h from E bF to G defined by gpx, yq “ hpxb yq. This extends to multilinear mappings. In other words, the tensor product of vector spaces pEiqiďN builds one (linear) vector space  i Ei. This definition is not basis dependent, which means that the tensor product applies not solely to arrays, and that real-valued tensors are items from a tensor space  iďN R ni . When a basis is given for the tensor space, this represents a tensor by an array of coordinates. Two well known basis-dependent representation for the tensor product are the outer product and the Kronecker product [3], [12]. First let us define the Kronecker product of two arrays : Definition 2: The Kronecker product of two arrays A P R p1ˆq1 and B P R22 is denoted by A⊠B P R1212 and is defined by: A⊠B :“ »