Uniqueness Analysis of Non-Unitary Matrix Joint Diagonalization

TL;DR

This paper investigates the conditions under which matrix joint diagonalization is unique, providing a comprehensive theoretical framework that unifies existing results and offers a closed-form solution for complex blind source separation.

Contribution

It offers a complete characterization of the identifiability conditions for matrix joint diagonalization in BSS, unifying various prior results and deriving a new closed-form solution for complex cases.

Findings

Unified conditions for joint diagonalizer uniqueness

Generalized results encompassing non-circularity, non-stationarity, non-whiteness, non-Gaussianity

Closed-form eigenvalue and singular value decomposition solution for complex BSS

Abstract

Matrix Joint Diagonalization (MJD) is a powerful approach for solving the Blind Source Separation (BSS) problem. It relies on the construction of matrices which are diagonalized by the unknown demixing matrix. Their joint diagonalizer serves as a correct estimate of this demixing matrix only if it is uniquely determined. Thus, a critical question is under what conditions a joint diagonalizer is unique. In the present work we fully answer this question about the identifiability of MJD based BSS approaches and provide a general result on uniqueness conditions of matrix joint diagonalization. It unifies all existing results which exploit the concepts of non-circularity, non-stationarity, non-whiteness, and non-Gaussianity. As a corollary, we propose a solution for complex BSS, which can be formulated in a closed form in terms of an eigenvalue and a singular value decomposition of two matrices.