Nonnegative Matrix And Tensor Factorization And Their Applications

Low rank approximation of matrix is a method to approximate the matrix. With this method, people can discover the latent information of the large scale data matrix. Nonnegative matrix factorization is one of the low rank approximation methods. It means that the entries of the matrix are nonnegative and the factors of decomposition are also nonnegative.Because in the real applications, most of the data are nonnegative, nonnegative factorization can express the intrinsic quality of the data, as well as the nonnegative matrix factorization model is in accordance with the law that people understand the world:the whole is made up from the parts. It was widely studied and used in some fields after the nonnegative matrix factorization method was proposed, and also it has been successfully applied in many fields. When the indexes of the array are more than two, the table can be expressed by a tensor. Tensor can be regarded as an extension of vector and matrix in high dimensional space. The research of tensor algebra and its applications is a research topic. Although many research results have been done about the research of matrix and tensor factorization and their applications, there still are some research topic needs to be done. In this thesis, we mainly have done some researches on nonnegative matrix factorization and its applications, as well as tensor factorization and its applications. The following problems are discussed about nonnegative matrix and tensor factorization and their applications: 1. A model and accordingly algorithm for nonnegative matrix factorization are given based on the quadratic programming. By using the interior point penalty function, we convert the nonnegative matrix factorization into a quadratic programming without any constraints and then some system equations. In the process of solving the quadratic programming, we introduce a method to reduce the dimension of the equation systems which can save the costs of the computation. The algorithm has better convergence quality compared with other NMF algorithms. Numerical experiments are presented in this thesis.2. Fragmental image recognition is a research branch in computer vision. Based on NMF algorithm, we presented a strategy and a method for fragmental image recognition. The strategy mainly deals with the fragmental image which the missing data are known. Experiments showed that our algorithm and strategy is effective.3. Image data missing often happens in real problems. How to extract the feature of the image by nonnegative matrix factorization when the data of the training set missed? This problem was discussed in this thesis. First we classify the missing data and then extract the local feature using NMF based algorithms. Numerical experiments are given for the model and the algorithms presented in this thesis.4. Online face recognition is a research branch in machine learning. The data scale is large in face recognition, when the training set was changed, the computation of extract local features for training sets needed to be done again to extract the new local features in the recognition, which will cost more computation and need more saving space to save the data. We presented a method to deal with this problem in the thesis; the method can deal with the case which is incremental or decremental, experiments showed that our method is effective, it can save the computation costs and saving space.5. When dealing with the problem of nonnegative matrix factorization, the nonnegative constraints make the algorithms run slowly. And also in the real applications, we only need the nonnegativity of the base matrix. What will happen if we give partly nonnegative constraints to the factors? Can the base matrix express the local features again? These questions have been discussed in the thesis.6. As an extension of vector and matrix, tensor can be regarded as a high dimensional matrix. Tensor model can be used to deal with the problem whose indices more than two. Based on tensor factorization model, we give web community discovering model and accordingly algorithms. The model considers three factors:authority, hub, text. This model is given to avoid the drift of the topic and make the search result more accurate. Experiments showed that the model is effective.At the end of the thesis, the situation of the NMF and tensor factorization research is given, also the futural works of the research are proposed.