Formalization of the Two-Step Approach to Overcomplete BSS

Abstract

We discuss overcomplete blind source separation (BSS), that is separation with more sources than sensors. Finding the mixing matrix solves the linear quadratic BSS problem; in overcomplete BSS however, it is then still not clear and not even unique how to get the sources from the mixtures and the mixing matrix. We therefore follow Bofill and Zibulevsky and many others and take a two-step approach to overcomplete BSS: In the first so called blind mixing model recovery (BMMR) step, the mixing model has to be reconstructed from the mixtures - in the linear case this would mean finding the mixing matrix. Then, in the blind source recovery (BSR) step, the sources have to be reconstructed given the mixing matrix and the mixtures. We furthermore introduce some notation and sayings and describe the usual BSR step in order to enable forthcoming overcomplete BSS papers to concentrate on one of the two steps, mainly on the BMMR step. Finally, we prove that the shortest-path algorithm as proposed by Bofill and Zibulevsky indeed solves the maximum-likelihood conditions in the BSR step. weight matrices.