An interior point trust region method for nonnegative matrix factorization

Recently, Nonnegative Matrix Factorization (NMF) is a developed method for dimension reduction, feature extraction and data mining, etc. In this paper, we propose an interior point trust region (IPTR) method, which can find a better solution in global region for NMF with general cost functions. First, to control the growth in the size of the solution with noise and regularize the solution in iterations, two auxiliary constraints are added into NMF. Then we introduce the logarithmic barrier function to eliminate the nonnegative regularization, and obtain an equivalent quadratic trust region problem by some mathematical calculation. According to the necessary and sufficient conditions of the trust region problem, we obtain a solution of the original problem by solving a parameterized linear system. We apply this method into NMF with different cost functions, including α-divergenceα-divergence, β-divergenceβ-divergence, KL-divergence, dual KL(DKL)-divergence, where different cost functions are imposed on different types of data. Numerical experiments demonstrate the high performance of the proposed method.