Research On Convex Nonnegative Matrix Factorization For Data Representation

With the rapid development of computer technology,the way and ability of people to obtain data have been greatly improved.At the same time,massive data and dimensionality disasters have become a major challenge.How to design a reasonable data dimensionality reduction algorithm so as to obtain effective low dimensional data representation is an urgent problem to be solved in data analysis.As an effective feature extraction technique and low dimensional data representation method,non-negative matrix factorization is widely used in feature extraction,pattern recognition,computer vision,image engineering and other fields.Unlike traditional principal component analysis(PCA)and linear discriminant analysis(LDA),which are based on global features,NMF only allows the additive combination of original data.By combining non-negative constraints,the partial representation of original data can be obtained.Such local features are more consistent with the concept that "parts constitute whole" in people’s thinking.Based on the existing non-negative matrix factorization techniques,this paper proposes two improved non-negative matrix factorization algorithms by incorporating sparse constraint and total variation with convex NMF.Specifically,including the following two parts:First,we propose graph-regularized sparse convex non-negative matrix factorization(GSCNMF).This model combines sparse constraint and graph regularization into the convex NMF,which can be as sparse as possible while processing the inherent geometric structure of the image data,so as to better obtain data representation in the low-dimensional space.The traditional Lagrange multiplier is used to solve the the objective function,and the convergence under the iterative update rule is proved.Second,We propose total variation constrained graph-regularized convex non-negative matrix factorization(TV-GCNMF).In this model,the feature details of the data are preserved by a diffusion coefficient based on the gradient information.The graph regularization and convex constraints reveal the intrinsic geometry and structure information of the features;thereby,obtaining sparse and parts-based representations.Furthermore,we give the multiplicative update rules and prove convergence of the proposed algorithm.The results of clustering experiments on multiple datasets,under various noise conditions,show the effectiveness and robustness of the proposed method compared to state-of-the-art clustering methods and other related work.