Matrix and Tensor Decompositions in Signal Processing. Gérard Favier

CPD decompositions for the tensor modeling of multidimensional harmonics, the problem of source separation in an instantaneous linear mixture and the modeling and estimation of a finite impulse response (FIR) linear system, using a tensor of fourth-order cumulants of the system output.

      High-order cumulants of random signals that can be viewed as tensors play a central role in various signal processing applications, as illustrated in Chapter 5. This motivated us to include an Appendix to present a brief overview of some basic results concerning the higher order statistics (HOS) of random signals, with two applications to the HOS-based estimation of a linear time-invariant system and a homogeneous quadratic system.

      1 1 Scalars, vectors, and matrices are written in lowercase, bold lowercase, and bold uppercase letters, respectively: a, a, A.

      2 2 A decomposition satisfies the essential uniqueness property if it is unique up to permutation and scaling factors in the columns of its factor matrices.

      3 3 Big data is characterized by 3Vs (Volume, Variety, Velocity) linked to the size of the data set, the heterogeneity of the data and the rate at which it is captured, stored and processed.

      4 4 Electroencephalography (EEG), magnetoencephalography (MEG), electrocardiography (ECG) and electrooculography (EOG).

      5 5 PARAFAC for parallel factors; CANDECOMP for canonical decomposition; CPD for canonical polyadic decomposition; TD for Tucker decomposition; TT for tensor train.