  Source separation, which consists in decomposing data into meaningful
structured components, 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, the only P$\alpha$S
distribution whose density is tractable, 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. 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.
