A Unified Convergence Analysis of the Multiplicative Update Algorithm for Regularized NMF with General Divergences

The multiplicative update (MU) algorithm has been used extensively to estimate the basis and coefficient matrices in nonnegative matrix factorization (NMF) problems under a wide range of divergences and regularizations. However, theoretical convergence guarantees have only been derived for a few special divergences and without regularizers. 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 with $\ell_1$ and Tikhonov regularizations. Our 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 NMF (optimization) problem. Our proof strategy has the potential to open up new avenues for analyzing similar problems.