Research On Non-Negative Matrix Factorization Algorithm Based On The Semi-Tensor Product

With the rapid development of modern acquisition technology and the increasing demand for the fineness of the objective world,the dimension of massive,diverse,and high-speed big data has also increased.Therefore,when mining high-dimensional big data features or potential structural relationships,it is necessary to consider both the positive/negative characteristics and the sparsity characteristics of the data to improve the accuracy and timeliness,so as to mine the relevance of data faster and more accurately.These characteristics of the current big data pose great challenges to data processing methods in terms of time complexity,spatial complexity,and algorithm accuracy.These problems with large-scale data are transformed into corresponding high-dimensional matrices.The low-rank approximation of non-negative matrix factorization achieves the dimensionality reduction function of the original non-negative matrix.The two non-negative factor matrices obtained by the factorization have goo;d interpretability and clear physical meaning,and can save storage space,facilitate parallelism and reduce calculation time.The traditional non-negative matrix factorization algorithm factorizes the input non-negative matrix into low-order non-negative basis matrix(basis image)and coefficient matrix.However,these two matrices should satisfy the dimensional matching condition in the matrix multiplication,that is,the number of columns of the basis matrix and the number of rows of the coefficient matrix are equal.In this paper,the two factor matrices obtained by the factorization are not limited by the dimension matching conditions in matrix multiplication.The generalized non-negative matrix factorization algorithm based on semi-tensor product and its optimization algorithm are studied.The main research results and innovations are described as follows:(1)Aiming at the dimension matching condition restriction of traditional non-negative matrix factorization process,based on the definition of matrix semi-tensor product and its basic properties,the objective function and the loss calculation model of the generalized non-negative matrix factorization based on the semi-tensor product(including the generalized non-negative matrix factorization based on the left semi-tensor product and the generalized non-negative matrix factorization based on the right semi-tensor product)are proposed.The multiplicative iterative algorithm of generalized updating basis matrix and coefficient matrix is studied and proposed,which not only improves the convergence speed of the algorithm,but also reduces the complexity of the algorithm.The proof of convergence and computational complexity of the generalized non-negative matrix factorization are given,the validity and time complexity of generalized non-negative matrix factorization are verified,the problem of dimension matching constraint between factor matrices in traditional non-negative matrix factorization is solved.Based on the specific advantages of the basis matrix and the coefficient matrix of the proposed generalized non-negative matrix factorization based on left semi?tensor product,a high-capacity digital watermarking algorithm based on left semi-tensor product generalized non-negative matrix factorization is proposed.We test and analyze the digital watermark under different attacks,test and analyze the digital watermark under different tampering,and compare it with the existing typical algorithm.The algorithm can embed high capacity(same resolution with high gray level)to the original image.The watermark image has high robustness at the same time.(2)Aiming at the difficulties faced by static data processing method in dynamic data set processing,a dynamic incremental generalized non-negative matrix factorization algorithm is proposed,which mainly includes dynamic incremental generalized non-negative matrix factorization algorithm based on left semi-tensor product and right semi-tensor.Aiming at the problem of low recognition rate of information in generalized non-negative matrix factorization,a sparse generalized non-negative matrix factorization algorithm based on left semi-tensor product is proposed.In addition,a face recognition calculation process of generalized non-negative matrix factorization based on left semi-tensor product and sparse generalized non-negative matrix factorization based on left semi-tensor is given.The JAFFE database and ORL database are used as data sets to compare the performance of face training and recognition based on dynamic incremental generalized non-negative matrix algorithm and dynamic incremental non-negative matrix algorithm.The proposed dynamic incremental generalized non-negative matrix factorization algorithm effectively saves the factor matrix storage space and effectively reduces the recognition time.To compare the performance of face training and recognition based on non-negative matrix factorization,sparse non-negative matrix factorization,generalized non-negative matrix factorization based on left semi-tensor and sparse generalized non-negative matrix factorization based on left semi-tensor,the proposed sparse generalized non-negative matrix factorization algorithm based on left semi-tensor effectively saves the storage space of basis matrix,reduces face recognition time and improves face recognition accuracy.(3)Aiming at the problem of the attributed social network community mining,based on the semi-tensor product,the joint generalized non-negative matrix factorization algorithm and the Laplace joint generalized non-negative matrix factorization algorithm are proposed,and corresponding algorithm analysis is given,and the mining attributed weights with excavation of the attributed social network community is realized.To compare the performance of the Laplace generalized non-negative matrix factorization and the Laplace joint generalized non-negative matrix factorization on the two indexes,the density of the vertex link of the community and the entropy of the vertex attribute of the community,the proposed algorithm has certain advantages.