Multilinear Generalized Singular Value Decomposition (ML-GSVD) and Its Application to Multiuser MIMO Systems

In this paper, we introduce a Multilinear Generalized Singular Value Decomposition (ML-GSVD) for two or more matrices with one common dimension. The ML-GSVD extends the Generalized Singular Value decomposition (GSVD) of two matrices to higher orders. The proposed decomposition allows us to jointly factorize a set of matrices with one common dimension. In comparison with other approaches that extend the GSVD, the ML-GSVD preserves the essential properties of the original (matrix-based) GSVD, such as orthogonality of the second-mode factor matrices as well as the subspace structure of the third-mode factor matrices. We introduce an ALS-based algorithm to compute the ML-GSVD, which has been inspired by PARAFAC2 decomposition algorithms. In addition, we present an application of the ML-GSVD for transceiver optimization in multicast and unicast MIMO-OFDM systems. Our numerical results show that the proposed ML-GSVD multicast and unicast beamforming outperforms existing state-of-the-art schemes in terms of the sum rate.