A Trust Region Method For Nonnegative Matrix Factorization

Nonnegative Matrix Factorization(NMF)is a method for data dimension reduction,decomposes a high-dimensional data matrix into the product of two smaller non-negative matrices.Because of the better interpretability,it has a broad prospect for application in many fields,such as machine learning and mass image data processing.NMF has become one of the main problems in the field of optimization.In this paper,we study the current algorithms for NMF and propose a new method for related problems.Firstly,we constructed a penalty trust region method to solve the nonnegative matrix factorization.The algorithm is based on the framework of alternating nonnegative least squares.we transform the NMF problem into unconstrained optimization problems by converting the nonnegative constraint into a penalty function in the solution of subproblems,and construct the trust region algorithm to solve the problem.The convergence of the algorithm is proved.In order to improve the computational efficiency of the algorithm,we use the stochastic singular value decomposition technique to reduce the dimension of the matrix to be processed in the iteration,and analyze the upper bound of the error generated in the process.The results of numerical experiments demonstrate that the new algorithm improves the efficiency of decomposition with high quality.Secondly,we solve the Semi Nonnegative Matrix Factorization(Semi-NMF)using the trust region method.Semi-NMF is an extended form of NMF,eliminates the nonnegative constraint on the basis matrix.We use the proposed trust region algorithm to solve the subproblems in the iterative process of SNMF.The new method is applied to the decomposition and reconstruction of face images,and the results demonstrate that the new method has better performance than multiplicative update method.