Separation of nonnegative alpha-stable sources

We address the problem of decomposing nonnegative data into meaningful structured components, which is referred to as latent variable analysis or source separation, depending on the community. It is an active research topic in many areas, such as music and image signal processing, applied physics and text mining. In this paper, we introduce the Positive $\alpha$-stable (P$\alpha$S) 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 L\'evy distribution, a particular case of P$\alpha$S, we propose a mixture model called L\'evy Nonnegative Matrix Factorization (L\'evy 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, which leads to an iterative procedure under multiplicative update rules. We also derive an estimator of the sources given the parameters, which extends the validity of the generalized Wiener filtering to the P$\alpha$S case. Experiments on synthetic data show that L\'evy NMF compares favorably with state-of-the art techniques in terms of robustness to impulsive noise. The analysis of two types of realistic signals is also considered: musical spectrograms and fluorescence spectra of chemical species. The results highlight the potential of the L\'evy NMF model for decomposing nonnegative data.