Adaptive method for nonsmooth nonnegative matrix factorization

Nonnegative matrix factorization (NMF) is an emerging tool for meaningful low-rank matrix representation. In NMF, explicit constraints are usually required, such that NMF generates desired products (or factorizations), especially when the products have significant sparseness features. It is known that the ability of NMF in learning sparse representation can be improved by embedding a smoothness factor between the products. Motivated by this result, we propose an adaptive nonsmooth NMF (Ans-NMF) method in this paper. In our method, the embedded factor is obtained by using a data-related approach, so it matches well with the underlying products, implying a superior faithfulness of the representations. Besides, due to the usage of an adaptive selection scheme to this factor, the sparseness of the products can be separately constrained, leading to wider applicability and interpretability. Furthermore, since the adaptive selection scheme is processed through solving a series of typical linear programming problems, it can be easily implemented. Simulations using computer-generated data and real-world data show the advantages of the proposed Ans-NMF method over the state-of-the-art methods.