Non-negative Matrix Factorization Of The Two Algorithms

With the development of computers and information technology,matrix factorization has become a main method to deal with large-scale data for dimensionality reduction in some science fields.For example,in numerical analysis,the large-scale and complex problems can be transformed into small-scale subproblems by matrix factorization,in applied statistics,low rank approximation of original matrix for discovering the inherent structure of data can be obtained by matrix factorization. In the applications of pattern recognition,low rank approximation leads to dimensionality reduction and to lower requirement of storage and computational resource.Although the traditional matrix decompositions are powerful,they have following shortcomings:1) they can not guarantee the non-negativity of the decomposition results,yet negative elements are no practical significance;2) They express data based on the holistic rather than the part.Therefore these classic matrix factorization algorithms are limited when applied to deal with non-negative data.Fortunately,the negative matrix factorization(NMF) based on the "multiplicative" iterative rules can overcome these shortcomings.NMF is a method to obtain a representation of multivariate data using non-negative constrains:Given an initial non-negative matrix V,it is possible to find two non-negative matrices W and H to approximate the original matrix V(V≈WH),where W and H are called basis matrix and coefficient matrix respectively.Because of NMF's easy implement,faster decomposition and the physical meaning of the results,it is considered as a promising approach to reduce the dimensionality of large-scale data.According to study,NMF is a constrained optimization problem,involving the selection of target function,the derivation of rules and the convergence analysis of algorithm.Thus,two novel algorithms for NMF are proposed in this paper,by constructing the suitable objective functions.This paper is structured as follows:The first part is introduction.In this section,the history,background and actual applications of NMF algorithm are described.And,Lee-Sung-NMF algorithm including the selection of objective function,the derivation of iteration rules and the analysis of convergence is presented.In the second part,first,the Bergman distance function and its special quality suitable to NMF is introduced.Then the objective function on the basis of Bergman distance function is built,and the iterative rules are constructed,thus a NMF algorithm based on Bergman distance function(BNMF) is presented,and its convergence is analyzed.Finally,the BNMF is applied in ORL facial images leaming to show the validity of the algorithm.The experiment results indicate that,under some mild condition,the BNMF algorithm is feasible and effective. In the last part,NMF is considered as the linear mixture model with additive noise,Based on this,the objective function suitable to NMF is proposed from the perspective of Statistics,the iterative rules are deduced,thus a NMF algorithm based on exponential distribution(ENMF) is presented and its convergence is analyzed.Finally,we applied it to UMIST facial images learning. The experimental results show that,the ENMF algorithm is feasible and effective.