`2,1 Norm and Hessian Regularized Non-Negative Matrix Factorization with Discriminability for Data Representation

Matrix factorization based methods have widely been used in data representation. Among them, Non-negative Matrix Factorization (NMF) is a promising technique owing to its psychological and physiological interpretation of spontaneously occurring data. On one hand, although traditional Laplacian regularization can enhance the performance of NMF, it still suffers from the problem of its weak extrapolating ability. On the other hand, standard NMF disregards the discriminative information hidden in the data and cannot guarantee the sparsity of the factor matrices. In this paper, a novel algorithm called `2,1 norm and Hessian Regularized Non-negative Matrix Factorization with Discriminability (`2,1HNMFD), is developed to overcome the aforementioned problems. In `2,1HNMFD, Hessian regularization is introduced in the framework of NMF to capture the intrinsic manifold structure of the data. `2,1 norm constraints and approximation orthogonal constraints are added to assure the group sparsity of encoding matrix and characterize the discriminative information of the data simultaneously. To solve the objective function, an efficient optimization scheme is developed to settle it. Our experimental results on five benchmark data sets have demonstrated that `2,1HNMFD can learn better data representation and provide better clustering results.