A new concept for separability problems in blind source separation

Abstract

The goal of blind source separation (BSS) lies in recovering the original independent sources of a mixed random vector without knowing the mixing structure. A key ingredient to successfully perform BSS is to know the indeterminacies of the problem that is to know how the separating model relates to the original mixing model (separability). For linear BSS Comon showed using the Darmois-Skitovitch theorem that the linear mixing matrix can be found except for permutation and scaling. In this work, a much simpler, direct proof for linear separability is given. The idea is based on the fact that a random vector is independent if and only if the Hessian of its logarithmic density respectively characteristic function is diagonal everywhere. This property is then exploited to propose a new algorithm for performing BSS. Furthermore first ideas of how to generalize separability results based on Hessian diagonalization to more complicated nonlinear models are studied in the setting of postnonlinear BSS.