Rank-constrained nonnegative matrix factorization for data representation

Abstract Graph-based regularized nonnegative matrix factorization (NMF) methods performed well in many real-world applications. However, it is still an open problem to construct an optimal graph to effectively discover the intrinsic geometric structure of data. In this paper, we propose a new data representation framework, called rank-constrained nonnegative matrix factorization (RCNMF). We impose the rank constraint on the Laplacian matrix of the learned graph, so it can ensure that the number of connected components is consistent with the number of sample categories. Instead of a fixed graph-based regularization, the proposed framework can adaptively adjust the weight of the affinity matrix in each iteration. We develop two versions of RCNMF based on the l1 and l2 norms, and introduce their optimization schemes. In addition, their convergence and the complexity analyses are also provided. Experimental results on four benchmark datasets show that our methods outperform state-of-the-art methods in clustering.