Matrix factorization for bilinear blind source separation: Methods, separability and conditioning

This paper deals with a general class of blind source separation methods for bilinear mixtures, using a structure based on matrix factorization, which models the direct, i.e. mixing, function, thus not requiring the analytical form of the inverse model. This approach also initially does not set restrictions on e.g. statistical independence, nonnegativity or sparsity, but on linear independence of sources and some source products. The separation principle used for adapting the parameters of the above structure consists in fitting the observations with the above direct model. We prove (for two sources at this stage) that this principle ensures separability, i.e. unique decomposition. Associated criteria and algorithms are also described. Performance is illustrated with preprocessed hyperspectral remote sensing data. This also allows us to highlight potential conditioning issues of some practical bilinear matrix factorization (BMF) methods and to suggest how to extend them.