  This article proposes new multiplicative updates for nonnegative matrix
factorization (NMF) with the $\beta$-divergence objective function. Our new
updates are derived from a joint majorization-minimization (MM) scheme, in
which an auxiliary function (a tight upper bound of the objective function) is
built for the two factors jointly and minimized at each iteration. This is in
contrast with the classic approach in which a majorizer is derived for each
factor separately. Like that classic approach, our joint MM algorithm also
results in multiplicative updates that are simple to implement. They however
yield a significant drop of computation time (for equally good solutions), in
particular for some $\beta$-divergences of important applicative interest, such
as the squared Euclidean distance and the Kullback-Leibler or Itakura-Saito
divergences. We report experimental results using diverse datasets: face
images, an audio spectrogram, hyperspectral data and song play counts.
Depending on the value of $\beta$ and on the dataset, our joint MM approach can
yield CPU time reductions from about $13\%$ to $78\%$ in comparison to the
classic alternating scheme.
