Projected gradient method for kernel discriminant nonnegative matrix factorization and the applications

Nonnegative matrix factorization (NMF) is a technique for analyzing the data structure when nonnegative constraints are imposed. However, NMF aims at minimizing the objective function from the viewpoint of data reconstruction and thus it may produce undesirable performances in classification tasks. In this paper, we develop a novel NMF algorithm (called KDNMF) by optimizing the objective function in a feature space under nonnegative constraints and discriminant constraints. The KDNMF method exploits the geometrical structure of data points and seeks the tradeoff between data reconstruction errors and the geometrical structure of data. The projected gradient method is used to solve KDNMF since directly using the multiplicative update algorithm to update nonnegative matrices is impractical for Gaussian kernels. Experiments on facial expression images and face images are conducted to show the effectiveness of the proposed method.