Uniqueness Result and Algebraic Algorithm for Decomposition into Multilinear Rank- $(M_{r},N_{r},\cdot)$ Terms and Joint Block Diagonalization

The block term decomposition (BTD) is a highly flexible tensor decomposition and can be viewed as a unification of the canonical polyadic and multilinear singular value decompositions. The BTD expresses a tensor as a sum of rank-one and block (CPD/MLSVD-like) terms. As a subproblem, the BTD captures joint block diagonalization (JBD) of a collection of matrices. This makes the BTD well-suited to a large number of signal processing tasks including blind source separation, hyperspectral unmixing, and source localization. We present novel, mild uniqueness conditions for the BTD and JBD, and give a new algebraic algorithm. A key feature of the algorithm is that it does not require knowledge of the block sizes appearing in the BTD. The efficacy of the algorithm is demonstrated through a blind source separation experiment.