Matrix and Tensor Decompositions in Signal Processing. Gérard Favier

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       – blind source separation and blind system identification. These problems play a fundamental role in signal processing. They involve separating the input signals (also called sources) and identifying a system from the knowledge of only the output signals and certain hypotheses about the input signals, such as statistical independence in the case of independent component analysis (Comon 1994), or the assumption of a finite alphabet in the context of digital communications. This type of processing is, in particular, used to jointly estimate communication channels and information symbols emitted by a transmitter. It can also be used for speech or music separation, or to process seismic signals;

       – use of tensor decompositions to analyze biomedical signals (EEG, MEG, ECG, EOG4) in the space, time and frequency domains, in order to provide a medical diagnostic aid; for instance, Acar et al. (2007) used a PARAFAC model of EEG signals to analyze epileptic seizures; Becker et al. (2014) used the same type of decomposition to locate sources within EEG signals;

       – 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.