Convergent Projective Non-negative Matrix Factorization with Kullback–Leibler Divergence

•The problem of convergence for Projective Non-negative Matrix Factorization is solved.•A new iterative formula for basis matrix is derived strictly.•A proof of algorithm convergence is provided.•The orthogonality and the sparseness of the basis matrix are better.•There is higher recognition accuracy in face recognition.

In order to solve the problem of algorithm convergence in Projective Non-negative Matrix Factorization (P-NMF), a method, called Convergent Projective Non-negative Matrix Factorization with Kullback–Leibler Divergence (CP-NMF-DIV) is proposed. In CP-NMF-DIV, an objective function of Kullback–Leibler Divergence is considered. The Taylor series expansion and the Newton iteration formula of solving root are used. An iterative algorithm for basis matrix is derived, and a proof of algorithm convergence is provided. Experimental results show that the convergence speed of the algorithm is higher; relative to Non-negative Matrix Factorization (NMF), the orthogonality and the sparseness of the basis matrix are better, however the reconstructed results of data show that the basis matrix is still approximately orthogonal; in face recognition, there is higher recognition accuracy and it is stable in most cases which the ranks of the basis matrices are set with different values. The method for CP-NMF-DIV is effective.