Nonnegative Matrix Factorization Algorithm With Sparseness Constraints And Its Applications

In recent years,the matrix decomposition methods has developed rapidly in the fields of high dimensional data such as face feature extraction and information reconstruction.It maps the high-dimensional data to the low dimensional subspace,and finds the local representation of the original data while realizing dimensionality reduction.Nonnegative matrix factorization is a partial matrix decomposition method.It searches for two low rank nonnegative matrices to approximate the original data matrix.Because of the non-negative constraints in NMF,it can find the local features in the decomposition results without negative value.The parts-based and pure additive representation makes it superior to other feature extraction methods.NMF can discover the intrinsic correlation of the original data and reveal the essential characteristics of the data.Since the algorithm was proposed by Lee and Seung,it has been widely used in the fields of computer vision,text clustering and signal processing.Although NMF can be relatively sparse or localized representation of raw data,the sparsity of the decomposition results is only a by-product of non-negative constraints,and is not the target of the NMF.The degree of sparsity cannot be guaranteed,so how to improve the ability of sparse representation of NMF has become a research hotspot.In this paper,the matrix decomposition methods with sparsity constraint has been studied.Strong sparse constraints were incorporated into NMF.The common method is to add L1 norm constraints on base matrix or coefficient matrix.Compared with L1 norm,L0 norm is a more direct way to measure sparsity which directly reflects the number of nonzero elements in a matrix.A strong constraint matrix decomposition method was proposed by applying L0 norm constraint on basis matrix or coefficient matrix.The non-convexity of the L0 norm renders a problem NP-hard which requires a detailed combination search.Thus,this paper presents an approximate solution.The main work and innovation of this paper are as follows:(1)In order to solve the problem that the sparse representation ability of normal subspace non negative matrix decomposition(NMFOS)method is weak,the L1 norm constraint was introduced into the NMFOS model,called NMFOS-SC.By adding regularized sparse constraints to NMFOS,the algorithm makes the result of matrix decomposition more sparse which improves the quality of decomposition.The iterative updating rule was given in this paper.Through text clustering and face feature extraction experiments,it has proved that this algorithm has good clustering and sparse expression ability.(2)The sparse constrained NMF proposed by Hoyer was to add regularization terms of L1 norm constraint in objective function of NMF,combining NMF decomposition and sparse representation,which built a linear model that balance the reconstruction error.In this paper,the L0 norm constraint was added to the model of NMF,because the L0 norm can measure the sparsity of the matrix more intuitively,and it directly reflects the number of non-zero elements of the matrix.When the constraintwas applied on the coefficient matrix H,due to the non-convexity of the L0 norm,that rendered a NP-hard problem in the sparse coding stage.To solve this problem,this paper adopted the idea of reverse matching to improve NNLS,and proposed rs NNLS.When the L0 norm constraint was on the base matrix W,the base vectors were projected to the closest non negative vectors in the space.In each process of iteration,the minimum item in the vector was deleted until achieving the expected sparsity.Through the experiments on face recognition and face feature extraction,it has proved that the algorithm not only has high recognition rate,but also can represent face image more locally.