# Tensor-Based Approaches for Nonlinear and Multilinear Systems Modeling and Identification
## 1 Introduction  
The continuous development of mathematical knowledge, together with a constantly renewed and growing need to study, represent and analyze ever more complex physical phenomena and systems, are at the origin of new mathematical objects and models. Inparticular, the notion of matrices introduced by [PERSON] in 1810 to solve systems of linear algebraic equations, with the foundations of matrix computation developed during the 19 th century by [PERSON] (1814-1897) and [PERSON] (1821-1895), among several other mathematicians, has later given rise to the notion of tensors. Tensors of order higher than two, i.e., mathematical objects indexed by more than two indices, are multidimensional generalizations of vectors and matrices which are tensors of orders one and two, respectively. Such objects are well suited to represent and process multidimensional and multimodal signals and data, like in computer vision [1], pattern recognition [2], array processing [3], machine learning [4], recommender systems [5], ECG applications [6], bioinformatics [7], and wireless communications [8], among many other fields of application. Today, with ever constantly growing big data (texts, images, audio, and videos) to manage in multimedia applications and social networks, tensor tools are well adapted to fuse, classify, analyze and process digital information [9]. The purpose of this paper is to present an overview of tensor-based methods for modeling and identifying nonlinear and multilinear systems using input-output data, as encountered in signal processing applications, with a focus on truncated Volterra models and block-oriented nonlinear ones and an introduction to memoryless input-output tensor systems. With a detailed reminder of the tensor tools useful to make the presentation as self-contained as possible and a review of main nonlinear models and their applications, this paper should be of interest to researchers and engineers concerned with signal processing applications. First of all, developed as computational and representation tools in physics and geometry, tensors were the subject of mathematical developments related to polyadic decomposition [10], aiming to generalize dyadic decompositions, i.e., matrix decompositions such as the singular value decomposition (SVD), discovered independently by [PERSON] (1835-1900) and [PERSON] (1838-1922) in 1873 and 1874, respectively. Then, tensors were used for the analysis of three-dimensional data generalizing matrix analysis to sets of matrices, seen as arrays of data characterized by three indices, in the fields of psychometrics and chemometrics [11; 12; 13; 14]. This explains the other name given to tensors as multiway arrays in the context of data analysis and data mining [15]. Matrix decompositions, such as the SVD, have thus been generalized into tensor decompositions, such as the PARAFAC decomposition [13], also called canonical polyadic decomposition (CPD), and the Tucker decomposition (TD) [12]. Tensor decompositions consist in representing a high-order tensor by means of factor matrices and lower-order tensors, called core tensors. In the context of data analysis, such decompositions make it possible to highlight hidden structures of the data while preserving their multilinear structure, which is not the case when stacking the data in the form of vectors or matrices. Tensor decompositions can be used to reduce data dimensionality [16], merge coupled data tensors [17], handle missing data through the application of tensor completion methods [18; 19], and design semi-blind receivers for tensor-based wireless communication systems [8]. In Table 1, we present basic and very useful matrix and third-order tensor decompositions, namely the reduced SVD, also known as the compact SVD, PARAFAC/CPD and TD, in a comparative way. A detailed presentation of PARAFAC and Tucker decompositions is given in Section 4.2.