Exact algebraic blind source separation using side information

Classical Blind Source Separation (BSS) methods rarely attain exact separation, even under noiseless conditions. In addition, they often rely on particular structural or statistical assumptions (e.g., mutual independence) regarding the sources. In this work we consider a (realistic) “twist” in the classical linear BSS plot, which, quite surprisingly, not only enables perfect separation (under noiseless conditions), but is also free of any assumptions (except for regularity assumptions) regarding the sources or the mixing matrix. In particular, we consider the standard linear mixture model, augmented by a single ancillary, unknown linear mixture of some known linear transformations of the sources. We derive a closed-form expression for an exact algebraic solution, free of any statistical considerations whatsoever, attaining perfect separation in the noiseless case. In addition, we propose a well-behaved solution for the same model in the presence of noise or other measurement inaccuracies. Our derivations are corroborated by several simulation results.