Pareto front of bi-objective kernel-based nonnegative matrix factorization

The nonnegative matrix factorization (NMF) is a powerful data analysis and dimensionality reduction technique. So far, the NMF has been limited to a single-objective problem in either its linear or nonlinear kernel-based formulation. This paper presents a novel bi-objective NMF model based on kernel machines, where the decomposition is performed simultaneously in both input and feature spaces. The problem is solved employing the sum-weighted approach. Without loss of generality, we study the case of the Gaussian kernel, where the multiplicative update rules are derived and the Pareto front is approximated. The performance of the proposed method is demonstrated for unmixing hyperspectral images.