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The multiplicative update (MU) algorithm has been extensively used toestimate 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 specialdivergences without regularization. In this work, we provide a conceptuallysimple, self-contained, and unified proof for the convergence of the MUalgorithm applied on NMF with a wide range of divergences and regularizers. Ourmain result shows the sequence of iterates (i.e., pairs of basis andcoefficient matrices) produced by the MU algorithm converges to the set ofstationary points of the non-convex NMF optimization problem. Our proofstrategy has the potential to open up new avenues for analyzing similarproblems in machine learning and signal processing.

A Unified Convergence Analysis of the Multiplicative Update Algorithm for Regularized Nonnegative Matrix Factorization