Nonconvex Nonseparable Sparse Nonnegative Matrix Factorization for Hyperspectral Unmixing

Hyperspectral unmixing is an important step to learn the material categories and corresponding distributions in a scene. Over the past decade, nonnegative matrix factorization (NMF) has been utilized for this task, thanks to its good physical interpretation. The solution space of NMF is very huge due to its nonconvex objective function for both variables simultaneously. Many convex and nonconvex sparse regularizations are embedded into NMF to limit the number of trivial solutions. Unfortunately, they either produce biased sparse solutions or unbiased sparse solutions with the sacrifice of the convex objective function of NMF with respect to individual variable. In this article, we enhance NMF by introducing a generalized minimax concave (GMC) sparse regularization. The GMC regularization is nonconvex and nonseparable, enabling promotion of unbiased and sparser results while simultaneously preserving the convexity of NMF for each variable separately. Therefore, GMC-NMF better avoids being trapped into local minimals, and thereby produce physically meaningful and accurate results. Extensive experimental results on synthetic data and real-world data verify its utility when compared with several state-of-the-art approaches.