Lévy NMF for robust nonnegative source separation

Source separation, which consists in decomposing data into meaningful structured components, is an active research topic in many fields including music signal processing. In this paper, we introduce the Positive α-stable (PαS) distributions to model the latent sources, which are a subclass of the stable distributions family. They notably permit us to model random variables that are both nonnegative and impulsive. Considering the Levy distribution, the only PαS distribution whose density is tractable, we propose a mixture model called Lévy Nonnegative Matrix Factorization (Lévy NMF). This model accounts for low-rank structures in nonnegative data that possibly has high variability or is corrupted by very adverse noise. The model parameters are estimated in a maximum-likelihood sense. We also derive an estimator of the sources, which extends the validity of the Wiener filtering to the PαS case. Experiments on synthetic data and realistic music signals show that Lévy NMF compares favorably with state-of-the art techniques in terms of robustness to impulsive noise and highlight its potential for decomposing nonnegative data.