Sparse coding and NMF

Non-negative matrix factorization (NMF) is a very efficient parameter-free method for decomposing multivariate data into strictly positive activations and basis vectors. However, the method is not suited for overcomplete representations, where usually sparse coding paradigms apply. We show how to merge the concepts of non-negative factorization with sparsity conditions. The result is a multiplicative algorithm that is comparable in efficiency to standard NMF, but that can be used to gain sensible solutions in the overcomplete cases. This is of interest e.g. for the case of learning and modeling of arrays of receptive fields arranged in a visual processing map, where an overcomplete representation is unavoidable.