The Study And Application Of Nonnegative Matrix Factorization Algorithms

Nonnegative Matrix factorization (NMF) is a new analysis and processing of large-scale data method proposed in recent years. Unlike the general matrix factorization, NMF is a matrix factorization method under the condition of all elements are non-negative, and this non-negative condition is consistent with the requirement of a lot information in reality. NMF which can be seen as a non-negative matrix factorization into the product of a base matrix and its projection matrix in this base matrix, has good interpretability and practicality and has been widely applied to various fields, such as image processing, face recognition, pattern recognition, text analysis and so on.On the basis of analysis a variety of existing algorithms, we found that the main problem of the current algorithms is convergent slow or utility extension poor. In order to improve the convergence speed and ensure the practicality of algorithms, a lot of research we have done. There are two main models for NMF, one of which is based on Euclidean distance, and the other is based on K-L divergence. First of all, we transform the Euclidean distance model into a sequence of least squares problems, from the non-negativity constrained least squares’KKT conditions we have a linear complementarity problem(LCP), and then a new algorithm for NMF based on LCP is proposed. The new algorithm has loose initial conditions requirement and fast convergence. But we find that the based on rank one decomposition fast NMF algorithm has faster convergence than all other algorithms, however, its practicality is limited by its positive constraints. We improve the algorithm and present a practical fast NMF algorithm which is practical and also has fast convergence. For the second model, we introduce ordered subsets acceleration method to the multiplicative iterative algorithm, and we have the NMF algorithm of ordered subsets acceleration, which greatly improves the convergence rate of the original algorithm. Meanwhile, we extended it to the Euclidean model of multiplicative iterative method, and the convergence rate is also greatly improved.In addition, there are also a lot of other problems to be solved for non-negative matrix factorization problem, such as how to get the global optimal solution, how to choose the decom-position rank, and what is the decomposition results criteria. In this paper, we have do some attempt study for the problem of how to determine decomposition rank, and present a sequence rank one decomposition method. We did not completely resolve the problem, but a direction of a feasibility study is proposed.