Direct Update of Back-Projected Demixing Matrices in Blind Source Separation

Blind source separation (BSS) is a widely used technique for separating mixed signals originating from multiple sources. Traditionally, BSS is formulated as an optimization problem with respect to the demixing matrix. In order to resolve the scale ambiguity inherent in BSS, a post-processing step called “projection back” is employed, yielding a rescaled demixing matrix which we call a “back-projected” matrix. However, incorporating prior information or constraints directly on the back-projected matrix has not been possible, as the optimization problem is defined with respect to the original demixing matrix and the back-projected matrix is obtained only after the optimization and post-processing. In this paper, we address this limitation by reformulating the BSS optimization problem in terms of the back-projected matrices instead of the original matrices. Specifically, we derive simple conditions that back-projected matrices must satisfy and reformulate the BSS problem under these constraints. We propose a novel algorithm for solving the reformulated problem, which is shown to monotonically decrease the objective function while satisfying the constraints. Through experiments, we demonstrate that our reformulation enables the direct imposition of constraints on the back-projected matrices, providing a more flexible and effective approach to BSS with prior information on the rescaled demixing matrix.