Biobjective Nonnegative Matrix Factorization: Linear Versus Kernel-Based Models

Nonnegative matrix factorization (NMF) is a powerful class of feature extraction techniques that has been successfully applied in many fields, particularly in signal and image processing. Current NMF techniques have been limited to a single-objective optimization problem, in either its linear or nonlinear kernel-based formulation. In this paper, we propose to revisit the NMF as a multiobjective problem, particularly a biobjective one, where the objective functions defined in both input and feature spaces are taken into account. By taking the advantage of the sum-weighted method from the literature of multiobjective optimization, the proposed biobjective NMF determines a set of nondominated, Pareto optimal, solutions. Moreover, the corresponding Pareto front is approximated and studied. Experimental results on unmixing synthetic and real hyperspectral images confirm the efficiency of the proposed biobjective NMF compared with the state-of-the-art methods.