  This article introduces new multiplicative updates for nonnegative matrix
factorization with the $\beta$-divergence and sparse regularization of one of
the two factors (say, the activation matrix). It is well known that the norm of
the other factor (the dictionary matrix) needs to be controlled in order to
avoid an ill-posed formulation. Standard practice consists in constraining the
columns of the dictionary to have unit norm, which leads to a nontrivial
optimization problem. Our approach leverages a reparametrization of the
original problem into the optimization of an equivalent scale-invariant
objective function. From there, we derive block-descent
majorization-minimization algorithms that result in simple multiplicative
updates for either $\ell_{1}$-regularization or the more "aggressive"
log-regularization. In contrast with other state-of-the-art methods, our
algorithms are universal in the sense that they can be applied to any
$\beta$-divergence (i.e., any value of $\beta$) and that they come with
convergence guarantees. We report numerical comparisons with existing heuristic
and Lagrangian methods using various datasets: face images, an audio
spectrogram, hyperspectral data, and song play counts. We show that our methods
obtain solutions of similar quality at convergence (similar objective values)
but with significantly reduced CPU times.
