Research On Nonnegative Matrix Factorization Algorithm

Matrix factorization is an important multidimensional or N-way array analysis method. There are a lot of multidimensional data with nonnegative constraints in science and industry. Nonnegative matrix factorization (NMF) attracts many researchers'attentions. Under nonnegative constraints, we can obtain the part-based representation of the data matrix. The nonnegative base matrix and the representative matrix have their physical meanings in practical applications.In this thesis, we focus on the fast and high efficiency algorithm of NMF. Currently, the most used algorithm of NMF is LS algorithm proposed by Daniel D. Lee and H. Sebastian Seung. The LS algorithm contains two update rules. The research and practice show that the convergence rate of LS algorithm is low. Firstly, we briefly introduce the problem of NMF, the fundamental knowledge of mathematics which will be used in the study, the developments and the research states of NMF. Secondly, we proposed the interior-point gradient BB algorithm. The subproblem of NMF is a box-constrained or bound constrained programming. Currently, the box-constrained programming is widely studied, there are some valuable results. We can use the methods of solving the box-constrained programming to solve the subproblem of NMF. In Chapter 2, by using some powerful, state-of-the-art, techniques, we propose one decent direction and estimate the step size of nonmonotone line research. The convergence analysis show that the consequence generated by the proposed algorithm can convergent to a stationary point. Theoretical analysis and numerical experiments show that our proposed algorithm not only has high rate of the convergence, but also has good performance of numerical experiments and practical problem. Thirdly, we propose Rank-2 HALS algorithm and modified Rank-2 HALS algorithm. The multiplication of two matrices can be written as the sum of some rank-1 matrices. By analyzing the advantage and disadvantage of HALS algorithm, the solution properties of two special quadratic programming, we obtain our proposed algorithm. The convergence analysis show that the consequence generated by the proposed algorithms can convergent to a stationary point. In Blind Source Separation problem, the Modified Rank-two HALS algorithm has a high performance. Finally, we give a brief introduction of the tensor d ecomposition and the problem we need to solve.