– analysis of brain activity by merging imaging data (fMRI) and biomedical signals (EEG and MEG) with the goal of enabling non-invasive medical tests (see Table I.4);
– analysis and classification of hyperspectral images used in many fields (medicine, environment, agriculture, monitoring, astrophysics, etc.). To improve the spatial resolution of hyperspectral images, Li et al. (2018) merged hyperspectral and multispectral images using a coupled Tucker decomposition with a sparse core (coupled sparse tensor factorization (CSTF)) (see Table I.4);
– design of semi-blind receivers for point-to-point or cooperative MIMO communication systems based on tensor models; see the overviews by de Almeida et al. (2016) and da Costa et al. (2018);
– modeling and identification of nonlinear systems via a tensor representation of Volterra kernels or Wiener–Hammerstein systems (see, for example, Kibangou and Favier 2009a, 2010; Favier and Kibangou 2009; Favier and Bouilloc 2009, 2010; Favier et al. 2012a);
– identification of tensor-based separable trilinear systems that are linear with respect to (w.r.t.) the input signal and trilinear w.r.t. the coefficients of the global impulse response, modeled as a Kronecker product of three individual impulse responses (Elisei-Iliescu et al. 2020). Note that such systems are to be compared with third-order Volterra filters that are linear w.r.t. the Volterra kernel coefficients and trilinear w.r.t. the input signal;
– facial recognition, based on face tensors, for purposes of authentication and identification in surveillance systems. For facial recognition, photos of people to recognize are stored in a database with different lighting conditions, different facial expressions, from multiple angles, for each individual. In Vasilescu and Terzopoulos (2002), the tensor of facial images is of order five, with dimensions: 28 × 5 × 3 × 7943, corresponding to the modes: people × views × illumination × expressions × pixels per image. For an overview of various facial recognition systems, see Arachchilage and Izquierdo (2020);
– tensor-based anomaly detection used in monitoring and surveillance systems.
Table I.1 presents a few examples of signal and image tensors, specifying the nature of the modes in each case.
Table I.1. Signal and image tensors
| Signals | Modes | References |
| Antenna processing | space (antennas) × time × sensor subnetwork space × time × polarization | (Sidiropoulos et al. 2000a) (Raimondi et al. 2017) |
| Digital communications | space (antennas) × time × code antennas × blocks × symbol periods × code × frequencies | (Sidiropoulos et al. 2000b) (Favier and de Almeida 2014b) |
| ECG | space (electrodes) × time × frequencies | (Acar et al. 2007; Padhy et al. 2019) |
| EEG | space (electrodes) × time × frequencies × subjects or trials | (Becker et al. 2014; Cong et al. 2015) |
| EEG + fMRI | subjects × electrodes × time + subjects × voxels (model with matrix and tensor factorizations coupled via the “subjects” mode) | (Acar et al. 2017) |
| Images | Modes | References |
| Color images | space (width) × space (height) × channel (colors) | |
| Videos in grayscale | space (width) × space (height) × time | |
| Videos in color | space × space × channel × time | |
| Hyperspectral images | space × space × spectral bands | (Makantasis et al. 2018) |
| Computer vision | people × views × illumination × expressions × pixels | (Vasilescu and Terzopoulos 2002) |
Other fields of application are considered in Table I.2.
Below, we give some details about the application concerning recommendation systems, which play an important role in various websites. The goal of these systems is to help users to select items from tags that have been assigned to each item by users. These items could, for example, be movies, books, musical recordings, webpages, products for sale on an e-commerce site, etc. A standard recommendation system is based on the three following modes: users × items × tags.
Collaborative filtering techniques use the opinions of a set of people, or assessments from these people based on a rating system, to generate a list of recommendations for a specific user. This type of filtering is, for example, used by websites like Netflix for renting DVDs. Collaborative filtering methods are classified into three categories, depending on whether the filtering is based on (a) history and a similarity metric; (b) a model based on matrix factorization using algorithms like SVD or non-negative matrix factorization (NMF); (c) some combination of both, known as hybrid collaborative filtering techniques. See Luo et al. (2014) and Bokde et al. (2015) for approaches based on matrix factorization.
Other so-called passive filtering techniques exploit the data of a matrix of relations between items to deduce recommendations for a user from correlations between items and the user’s previous choices, without using any kind of rating system. This is known as a content-based approach.
Table I.2. Other fields of application
| Domains | Modes | References |
| Phonetics | subjects × vowels × formants | (Harshman 1970) |
| Chemometrics (fluorescence) | excitation × emission × samples (excitation/emission wavelengths) | (Bro 1997, 2006; Smilde et al. 2004) |
| Contextual recommendation systems | users × items × tags × context 1 × • • • × context N | (Rendle and Schmidt-Thieme 2010) (Symeonidis and Zioupos 2016) (Frolov and Oseledets 2017) |
| Transportation (speed measurements) |
Space (sensors) × time (days) × time (weeks)
|