  The multiplicative update (MU) algorithm has been extensively used to
estimate the basis and coefficient matrices in nonnegative matrix factorization
(NMF) problems under a wide range of divergences and regularizers. However,
theoretical convergence guarantees have only been derived for a few special
divergences without regularization. In this work, we provide a conceptually
simple, self-contained, and unified proof for the convergence of the MU
algorithm applied on NMF with a wide range of divergences and regularizers. Our
main result shows the sequence of iterates (i.e., pairs of basis and
coefficient matrices) produced by the MU algorithm converges to the set of
stationary points of the non-convex NMF optimization problem. Our proof
strategy has the potential to open up new avenues for analyzing similar
problems in machine learning and signal processing.
