Bounded Component Analysis of Noisy Underdetermined and Overdetermined Mixtures

Our paper addresses the problem of the bounded component analysis of the observations in noisy mixtures. We present an original set of assumptions that guarantee the identifiability of the mixture in underdetermined mixtures and the separability of the sources in overdetermined mixtures. These assumptions are especially well suited for the blind identification of communication channels. Our proof of the identifiability of the mixing system is non-constructive. Thus, we develop a novel blind identification criterion for underdetermined and overdetermined mixtures, which is based on the least square fit of the perimeter of a set of projections of the observations. For the optimization of this criterion we propose the BCA-PM algorithm, which implements the natural gradient descent, along with an acceleration of the convergence designed for the neighbourhood of the solution. In situations of isotropic Gaussian noise and for reasonable signal to noise ratio, BCA-PM compares favorably with respect to other state-of-the-art methods, such as the ICA simultaneous diagonalization algorithms for underdetermined and overdetermined mixtures. The simulations also corroborate the advantages of using bounded component analysis for the blind identification of the channel with small datasets or when the transmitters cannot be regarded as statistically independent.