Postnonlinear overcomplete blind source separation using sparse sources

Abstract

We present an approach for blindly decomposing an observed random vector x into As where f is a diagonal function i.e. f=f_1 x ... x f_m with one-dimensional functions f_i and A an (m x n)-matrix. This postnonlinear model is allowed to be overcomplete, which means that less observations than sources (mn) are given. In contrast to Independent Component Analysis (ICA) we do not assume the sources s to be independent but to be sparse in the sense that at each time instant they have at most (m-1) non-zero components (Sparse Component Analysis or SCA). Identifiability of the model is shown, and an algorithm for model and source recovery is proposed. It first detects the postnonlinearities in each component, and then identifies the now linearized model using previous results.