Probabilistic sparse non-negative matrix factorization

In this paper, we propose a probabilistic sparse non-negative matrix factorization model that extends a recently proposed variational Bayesian non-negative matrix factorization model to explicitly account for sparsity. We assess the influence of imposing sparsity within a probabilistic framework on either the loading matrix, score matrix, or both and further contrast the influence of imposing an exponential or truncated normal distribution as prior. The probabilistic methods are compared to conventional maximum likelihood based NMF and sparse NMF on three image datasets; (1) A (synthetic) swimmer dataset, (2) The CBCL face dataset, and (3) The MNIST handwritten digits dataset. We find that the probabilistic sparse NMF is able to automatically learn the level of sparsity and find that the existing probabilistic NMF as well as the proposed probabilistic sparse NMF prunes inactive components and thereby automatically learns a suitable number of components. We further find that accounting for sparsity can provide more part based representations but for the probabilistic modeling the choice of priors and how sparsity is imposed can have a strong influence on the extracted representations.