Nonlinear non-negative component analysis algorithms

In this paper, general solutions for nonlinear non-negative component analysis for data representation and recognition are proposed. Motivated by a combination of the non-negative matrix factorization (NMF) algorithm and kernel theory, which has lead to a recently proposed NMF algorithm in a polynomial feature space, we propose a general framework where one can build a nonlinear non-negative component analysis method using kernels, the so-called projected gradient kernel non-negative matrix factorization (PGKNMF). In the proposed approach, arbitrary positive definite kernels can be adopted while at the same time it is ensured that the limit point of the procedure is a stationary point of the optimization problem. Moreover, we propose fixed point algorithms for the special case of Gaussian radial basis function (RBF) kernels. We demonstrate the power of the proposed methods in face and facial expression recognition applications.