Geometry-based solution of joint diagonalisation in blind source separation

Joint diagonalisation (JD) of a set of target matrices is a common approach to solve the blind source separation (BSS) problem. In fact, the separating matrix (the inverse of the mixing matrix) of the sources is the joint diagonaliser of the target matrices. In this study, the authors show that each row of the separating matrix maps the target matrices using a linear mapping into the new vectors which are located on a direct line along the corresponding column of the mixing matrix. Based on this geometrical interpretation, the authors propose a method for solving JD problem, which let us estimate the rows of the separating matrix and the columns of the mixing matrix, (i) independently and in parallel, (ii) consecutively or (iii) simultaneously. Simulation results in different scenarios such as additive noise, ill-posed mixing matrix, and high-dimensional data demonstrate the effectiveness of the proposed method relative to state-of-the-art JD methods. The authors also propose an approach to omit the effect of outlier data, which severely degrades parameters estimation. The proposed approach can be used in JD algorithms to improve their performance in the presence of outlier data.