Robust Graph Regularized Nonnegative Matrix Factorization

Nonnegative Matrix Factorization (NMF) has become a popular technique for dimensionality reduction, and been widely used in machine learning, computer vision, and data mining. Existing unsupervised NMF methods impose the intrinsic geometric constraint on the encoding matrix, which only indirectly affects the base matrix. Moreover, they ignore the global structure of the data space. To address these issues, in this paper we propose a novel unsupervised NMF learning framework, called Robust Graph regularized Nonnegative Matrix Factorization (RGNMF). RGNMF constructs a sparse graph imposed on the basis matrix to catch the global structure and preserve the discriminative information. And it models the local structure by building a k-NN graph constrained on the encoding matrix, which gains the compact representation. Consequently, RGNMF not only respects the global structure, but also depicts the local structure. In addition, it employs such a $\text{L}_{2,1}$ -norm cost function to decompose the basis matrix and encoding matrix that its robustness can be improved. Further, it imposes the $\text{L}_{2,1}$ -norm constraint on the basis matrix to choose the discriminative feature. Hence, RGNMF can gain the robust discriminative representation by combining structure learning and $\text{L}_{2,1}$ -norm constraints imposed on the basis matrix and encoding matrix. Extensive experiments on real-world problems demonstrate that RGNMF achieves better clustering results than the state-of-the-art approaches.