Matrix and Tensor Decompositions in Signal Processing. Gérard Favier

using certain transforms (wavelets, Fourier, etc.), and finally, in some cases, the calculation of statistics of signals to be processed.

      Preprocessing is fundamental, both to improve the quality of the estimated models and, therefore, of the subsequent processing operations, and to avoid numerical problems with optimization algorithms, such as conditioning problems that may cause the algorithms to fail to converge. Centering and scaling preprocessing operations are potentially problematic because they are interdependent and can be combined in several different ways. If data are missing, centering can also reduce the rank of the tensor model. For a more detailed description of these preprocessing operations, see Smilde et al. (2004).

      For the processing operations themselves, we can distinguish between several different classes:

       – supervised/non-supervised (blind or semi-blind), i.e. with or without training data, for example, to solve classification problems, or when a priori information, called a pilot sequence, is transmitted to the receiver for channel estimation;

       – real-time (online)/batch (offline) processing;

       – centralized/distributed;

       – adaptive/blockwise (with respect to the data);

       – with/without coupling of tensor and/or matrix models;

       – with/without missing data.

      It is important to distinguish batch processing, which is performed to analyze data recorded as signal and image sets, from the real-time processing required by wireless communication systems, recommendation systems, web searches and social networks. In real-time applications, the dimensionality of the model and the algorithmic complexity are predominant factors. The signals received by receiving antennas, the information exchanged between a website and the users and the messages exchanged between the users of a social network are time-dependent. For instance, a recommendation system interacts with the users in real-time, via a possible extension of an existing database by means of machine learning techniques. For a description of various applications of tensors for data mining and machine learning, see Anandkumar et al. (2014) and Sidiropoulos et al. (2017).

      Tensor-based processings lead to various types of optimization algorithm as follows:

       – constrained/unconstrained optimization;

       – iterative/non-iterative, or closed-form;

       – alternating/global;

       – sequential/parallel.

      Furthermore, depending on the information that is available a priori, different types of constraints can be taken into account in the cost function to be optimized: low rank, sparseness, non-negativity, orthogonality and differentiability/smoothness. In the case of constrained optimization, weights need to be chosen in the cost function according to the relative importance of each constraint and the quality of the a priori information that is available.

      Table I.4 presents a few examples of cost functions that can be minimized for the parameter estimation of certain third-order tensor models (CPD, Tucker, coupled matrix Tucker (CMTucker) and coupled sparse tensor factorization (CSTF)), for the imputation of missing data in a tensor and for the estimation of a sparse data tensor with a low-rank constraint expressed in the form of the nuclear norm of the tensor.

      REMARK I.1.– We can make the following remarks:

       – the cost functions presented in Table I.4 correspond to data fitting criteria. These criteria, expressed in terms of tensor and matrix Frobenius norms (||.||F), are quadratic in the difference between the data tensor χ and the output of CPD and TD models, as well as between the data matrix Y and a matrix factorization model, in the case of the CMTucker model. They are trilinear and quadrilinear, respectively, with respect to the parameters of the CPD and TD models to be estimated, and bilinear with respect to the parameters of the matrix factorization model;

       – for the missing data imputation problem using a CPD or TD model, the binary tensor , which has the same size as χ, is defined as:

      [I.2]

      The purpose of the Hadamard product (denoted ) of , with the difference between χ and the output of the CPD and TD models, is to fit the model to the available data only, ignoring any missing data for model estimation. This imputation problem, known as the tensor completion problem, was originally dealt with by Tomasi and Bro (2005) and Acar et al. (2011a) using a CPD model, followed by Filipovic and Jukic (2015) using a TD model. Various articles have discussed this problem in the context of different applications. An overview of the literature will be given in the next volume;

Table I.4. Cost functions for model estimation and recovery of missing data

ProblemsData
Estimation	Cost functions
CPD
TD
CMTucker
CSTF
Imputation	Cost functions
CPD
TD
Imputation with low-rank constraint	Cost functions
CPD
TD

       – for the imputation problem with the low-rank constraint, the term χ in the cost function replaces the low-rank constraint with the nuclear norm of χ, since the function rank (χ) is not convex, and the nuclear norm is the closest convex approximation of the rank. In Liu et al.