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Non-negative matrix factorization (NMF) approximates a given matrix as aproduct of two non-negative matrices. Multiplicative algorithms deliverreliable results, but they show slow convergence for high-dimensional data andmay be stuck away from local minima. Gradient descent methods have betterbehavior, but only apply to smooth losses such as the least-squares loss. Inthis article, we propose a first-order primal-dual algorithm for non-negativedecomposition problems (where one factor is fixed) with the KL divergence,based on the Chambolle-Pock algorithm. All required computations may beobtained in closed form and we provide an efficient heuristic way to selectstep-sizes. By using alternating optimization, our algorithm readily extends toNMF and, on synthetic examples, face recognition or music source separationdatasets, it is either faster than existing algorithms, or leads to improvedlocal optima, or both.

Primal-Dual Algorithms for Non-negative Matrix Factorization with the Kullback-Leibler Divergence