Novel Proximal Gradient Methods for Nonnegative Matrix Factorization with Sparsity Constraints

We consider the nonnegative matrix factorization (NMF) problem with sparsity constraints formulated as a nonconvex composite minimization problem. We introduce four novel proximal gradient based algorithms proven globally convergent to a critical point and which are applicable to sparsity constrained NMF models. Our approach builds on recent results allowing one to lift the classical global Lipschitz continuity requirement through the use of a non-Euclidean Bregman based distance. Since under the proposed framework we are not restricted by the gradient Lipschitz continuity assumption, we can consider new decomposition settings of the NMF problem. Two of the derived schemes are genuine non-Euclidean proximal methods that tackle nonstandard decompositions of the NMF problem. The two other schemes are novel extensions of the well-known state-of-the-art methods (the multiplicative and hierarchical alternating least squares), thus allowing one to significantly broaden the scope of these algorithms. Numerical experiments illustrate the performance of the proposed methods.