Nonnegative Matrix Factorization With Regularizations

Matrix factorization techniques have been frequently applied in many fields. Among them, nonnegative matrix factorization (NMF) has received considerable attention for it aims to find a parts-based, linear representations of nonnegative data. Recently, many researchers propose various manifold learning algorithms to enhance learning performance by considering the local manifold smoothness assumption. However, NMF does not consider the geometrical structure of data and the local manifold smoothness does not directly ensure the representations of the data point with different labels being dissimilar. In order to find a better representation of data, we propose a novel matrix decomposition method, called nonnegative matrix factorization with Regularizations (RNMF), which incorporates three appropriate regularizations: nonnegative matrix factorization, the local manifold smoothness and a rank constraint. The representations of data learned by RNMF tend to be discriminative and sparse. By learning a Mahalanobis distance space based on labeled data, RNMF can also be extended to a semi-supervised algorithm (semi-RNMF) which has an amazing improvement on clustering performance. Our empirical study shows encouraging results of the proposed algorithm in comparison to the state-of-the-art algorithms on real-world problems.