Non-negative Matrix Factorization Algorithm Based On Method Of Least Square And Its Applications

In scientific research,matrix factorization in different decomposition meaning and different decomposition ways are often used to analyse and process massive data. After ma trix decomposition,the low rank matrix approximation of primitive matrix can be obtained and the dimension of data matrix can be greatly reduced,thus saving storage spaces and computing resourses. As a new kind of matrix decomposition method ,non-negative matrix factorization has caused extensive concern of scientific community once it was put forward,for it has special decomposition condition,that is"the matrices are all be non-negative matrices". Under this conditon,one non-negaitve matrix can be decomposed approximatively two non-negative matrices multiplication by looking for low rank matrices step by step and it reflects the potential linear relation of the data. Compared with classical method of matrix factorization,non-negative matrix has more advantages,such as simple algorithm,interpretable decomposition results and small storage spaces.Based on the original theories of non-negative matrix factorization,an algorithm more superior than the original ones was proposed,that is,the non-negative matrix factorization algorithm based on method of least square,and the algorithm was realized through program.The main research works in this thesis are as follows:First,the existing theories of non–negative matrix factorization and its applications both in domestic and foreign were summarized and concluded.Second,the non-negative matrix factorization algorithm based on method of least squ -are was proposed. The approximation degree of non-negative matrix factorization was measured by Euclidean distance and the matrix factorization problem has been transformed into least square optimization problem. And it is easily to verify that the computing complexity of this algorithm was reduced to a certain extent. Two improved algorithms was proposed,and the speed and precision of the two algorithms can be improved tosome degree.Finally,some examples were given at the end of this section to verify the effectiveness of these algorithms.Third,Combining matrix full rank decomposition theory,this paper proposed the non-negative matrix full rank factorization algorithm was proposed based on method of least square. And the related properties were analysed. Finally,the advantages and disadvantages of the algorithm were enumerated,and also some examples were showed to explain the related conclusion.Fourth, the algorithm program was realized by combining the related software theories of VC6.0 and Lingo.This program realized circular computations by calling VC,thus it can raise the computation speed in great degree.The most important is that the program is is also applicable to image processing,and it can dirrectly call image files,this is very convenient and efficient.Fifth,the proposed algorithm,that is, the non-negative matrix factorization algorithm based on method of least square ,can be applied in image processing filed,including process three-dimensional figure and small resolution image.In this paper,a new non-negative matrix factorization algorithm based on method of least square was proposed,and the algorithm was applied in image processing,and also writ ed the program in order to realize rapid circulating operation. And it can provide useful references for the further research on the theories and applications of non-negative matrix factorization.