Non-orthogonal tensor diagonalization based on successive rotations and LU decomposition

Abstract

Canonical polyadic decomposition (CPD) has been extensively studied and used in solving blind source separation (BSS) problems, mainly due to its nice identifiability property in mild conditions. In over-determined BSS and joint BSS (J-BSS), CPD is shown to be equivalent to tensor diagonalization (TD). In this study, we propose an algorithm for non-orthogonal TD (NTD) based on LU decomposition and successive rotations, and examine its applications in BSS and J-BSS. We use LU decomposition to convert the overall optimization into L and U stages, and then the factor matrices in these stages can be appropriately parameterized by a sequence of simple elementary triangular matrices, which can be solved analytically. We compared the proposed algorithm with orthogonal TD,tensor DIAgonalization (TEDIA) and CPD with simulations, the results show that in the over-determined case, NTD generates improved accuracy over TEDIA, CPD and orthogonal TD, and faster convergence than TEDIA.