On approximate joint "biagonalization" - a tool for noisy blind source separation

Quite a few algorithms for blind source separation (BSS) rely of approximate joint diagonalization (AJD) of a set of matrices. These matrices are usually estimates of some underlying matrices which admit exact joint diagonalization (EJD) in a noiseless scenario. When additive noise is present, the underlying set no longer admits EJD, since an unknown noise-related matrix is usually added to the diagonalizable form. Often this noise-related matrix is known to be diagonal. Hence, we define the "approximate joint biagonalization" (AJB) problem, aimed at fitting the noisy model to the estimated set of matrices. AJB differs from AJD in the presence of an additional unknown diagonal matrix in the model. We provide an iterative algorithm for minimizing the AJB least-squares (LS) criterion, based on an extension of an existing AJD algorithm. In addition, we provide some analytical results on exact and approximate biagonalization, applicable only to the special cases of two- and three-dimensional BSS problems