New Algorithms for Non-Negative Matrix Factorization in Applications to Blind Source Separation

Abstract

In this paper we develop several algorithms for non-negative matrix factorization (NMF) in applications to blind (or semi blind) source separation (BSS), when sources are generally statistically dependent under conditions that additional constraints are imposed such as nonnegativity, sparsity, smoothness, lower complexity or better predictability. We express the non-negativity constraints using a wide class of loss (cost) functions, which leads to an extended class of multiplicative algorithms with regularization. The proposed relaxed forms of the NMF algorithms have a higher convergence speed with the desired constraints. Moreover, the effects of various regularization and constraints are clearly shown. The scope of the results is vast since the discussed loss functions include quite a large number of useful cost functions such as weighted Euclidean distance, relative entropy, Kullback Leibler divergence, and generalized Hellinger, Pearson's, Neyman's distances, etc.