  This paper addresses the estimation of the latent dimensionality in
nonnegative matrix factorization (NMF) with the \beta-divergence. The
\beta-divergence is a family of cost functions that includes the squared
Euclidean distance, Kullback-Leibler and Itakura-Saito divergences as special
cases. Learning the model order is important as it is necessary to strike the
right balance between data fidelity and overfitting. We propose a Bayesian
model based on automatic relevance determination in which the columns of the
dictionary matrix and the rows of the activation matrix are tied together
through a common scale parameter in their prior. A family of
majorization-minimization algorithms is proposed for maximum a posteriori (MAP)
estimation. A subset of scale parameters is driven to a small lower bound in
the course of inference, with the effect of pruning the corresponding spurious
components. We demonstrate the efficacy and robustness of our algorithms by
performing extensive experiments on synthetic data, the swimmer dataset, a
music decomposition example and a stock price prediction task.
