Coseparable Nonnegative Matrix Factorization
Junjun Pan, Michael K. Ng · SIAM Journal on Matrix Analysis and Applications · 2023
Abstract. Nonnegative matrix factorization (NMF) is a popular model in the field of pattern recognition. The aim is to find a low rank approximation for nonnegative matrix [Formula: see text] by a product of two nonnegative matrices [Formula: see text] and [Formula: see text]. In general, NMF is NP-hard to solve while it can be solved efficiently under a separability assumption, which requires that the columns of the factor matrix are some columns of the input matrix [Formula: see text]. In this paper, we generalize the separability assumption based on 3-factor NMF ([Formula: see text]), and require that [Formula: see text] is a submatrix of the input matrix [Formula: see text]. We refer to this NMF as a Coseparable NMF (CoS-NMF). In the paper, we discuss and study mathematical properties of CoS-NMF, and present its relationships with other matrix factorizations such as generalized separable NMF, tri-symNMF, biorthogonal trifactorization and CUR decomposition. An optimization method for CoS-NMF is proposed, and an alternating fast gradient method is employed to determine the rows and the columns of [Formula: see text] for the submatrix [Formula: see text]. Numerical experiments on synthetic data sets, document data sets, and facial data sets are conducted to verify the effectiveness of the proposed CoS-NMF model. By comparison with state-of-the-art methods, the CoS-NMF model performs very well in a coclustering task by finding useful features, and keeps a good approximation to the input data matrix as well.