Tensor Decompositions and Algorithms for Efficient Multidimensional Signal Processing

Due to the extensive growth of big data applications, the widespread use of multisensor technologies, and the need for efficient data representations, multidimensional techniques are a primary tool for many signal processing applications. Multidimensional arrays or tensors allow a natural representation of high-dimensional data. Therefore, they are particularly suited for tasks involving multi-modal data sources such as biomedical sensor readings or multiple-input and multiple-output (MIMO) antenna arrays. While tensor-based techniques were still in their infancy several decades ago, nowadays, they have already proven their effectiveness in various applications. There are many different tensor decompositions in the literature, and each finds use in diverse signal processing fields. In this thesis, we focus on two tensor factorization models: the rank-(Lr,Lr,1) Block-Term Decomposition (BTD) and the Multilinear Generalized Singular Value Decomposition (ML-GSVD) that we propose in this thesis. The ML-GSVD is an extension of the Generalized Singular Value Decomposition (GSVD) of two matrices to the tensor case. The properties of the original matrix GSVD render it an attractive tool for different applications, including genomic signal processing, MIMO relaying, coordinated beamforming, physical layer security, and multiuser MIMO systems. Yet, since the GSVD is restricted to two matrices, its use in wireless communications is limited to two users. Furthermore, this also ties into the fact that the literature lacked the extension of the GSVD for more than two matrices that would also inherit the properties of the original decomposition. Therefore, in this thesis, we extend the GSVD of two matrices to the tensor case while preserving its orthogonality properties and demonstrate its efficient application to multi-user MIMO communication systems. We provide a detailed discussion of the ML-GSVD subspace structure and propose an algorithm to compute it. Furthermore, we present three applications of the ML-GSVD in MIMO communication systems: multiuser downlink MIMO systems with joint unicast and multicast transmissions; non-orthogonal multiple access (NOMA); and multi-user MIMO broadcast systems with rate splitting at the transmitter. For these applications, we exploit the structure of the ML-GSVD with common and private subspaces and show how the factors of the ML-GSVD can be used for the design of the precoders and decoders. In the other part of the thesis, we focus on the rank-(Lr,Lr,1) Block-Term Decomposition. In contrast to the more common Canonical Polyadic (CP) decomposition, the rank-(Lr,Lr,1) decomposition has not yet been investigated as extensively and still has unexplored areas, such as its efficient computation. This thesis provides the algorithms to calculate both single and coupled rank-(Lr,Lr,1) decompositions by exploiting the connections of the BTD to CP decompositions. The proposed SECSI-BTD (SEmi-algebraic framework for approximate Canonical polyadic decompositions via SImultaneous Matrix Diagonalizations) algorithm includes the initial calculation of the factor estimates, followed by clustering and refinement procedures that return the appropriate rank-(Lr,Lr,1) BTD terms. Moreover, we introduce a new approach to estimate the multilinear rank structure of the tensor based on the higher-order singular value decomposition (HOSVD) and k-means clustering. Since the proposed SECSI-BTD algorithm does not require a known rank structure but can still take advantage of the known ranks when available, it is more flexible than the existing techniques in the literature. As an application of the coupled rank-(Lr,Lr,1) decomposition, we consider near-field localization in multi-static MIMO radar systems. We show how the BTD can be employed for parameter estimation in 3D space based on the exact spherical wavefront model. Finally, we consider the application of the coupled rank-(Lr,Lr,1) BTD to the Electroencephalogram (EEG) and Magnetoencephalogram (MEG) recordings of somatosensory evoked electrical potentials (SEPs) and somatosensory evoked magnetic fields (SEFs) to separate the signal components related to the 200 Hz band activity. In contrast to state-of-the-art works on the EEG and MEG recordings, we perform the fusion of the complete data set, including the gradiometer measurements, i.e., yielding a coupled rank-(Lr,Lr,1) BTD of four tensors (EEG, MEG-MAG, MEG-GRAD1, and MEG-GRAD2). Additionally, this thesis provides the background material on the fundamentals of multilinear algebra, reviews the basic matrix and tensor decompositions, and identifies future research directions.