title: Efficient Nonnegative Matrix Factorization by DC Programming and DCA.
abstract: In this letter, we consider the nonnegative matrix factorization (NMF) problem and several NMF variants. Two approaches based on DC (difference of convex functions) programming and DCA (DC algorithm) are developed. The first approach follows the alternating framework that requires solving, at each iteration, two nonnegativity-constrained least squares subproblems for which DCA-based schemes are investigated. The convergence property of the proposed algorithm is carefully studied. We show that with suitable DC decompositions, our algorithm generates most of the standard methods for the NMF problem. The second approach directly applies DCA on the whole NMF problem. Two algorithms-one computing all variables and one deploying a variable selection strategy-are proposed. The proposed methods are then adapted to solve various NMF variants, including the nonnegative factorization, the smooth regularization NMF, the sparse regularization NMF, the multilayer NMF, the convex/convex-hull NMF, and the symmetric NMF. We also show that our algorithms include several existing methods for these NMF variants as special versions. The efficiency of the proposed approaches is empirically demonstrated on both real-world and synthetic data sets. It turns out that our algorithms compete favorably with five state-of-the-art alternating nonnegative least squares algorithms.
authors.affiliation: Department for Management of Science and Technology Development and Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam, and Laboratory of Theoretical and Applied Computer Science EA 3097, University of Lorraine, Ile du Saulcy, 57045 Metz, France lethihoaian@tdt.edu.vn.
authors.forename: Hoai An
authors.initials: HA
authors.lastname: Le Thi
authors.affiliation: Laboratory of Theoretical and Applied Computer Science EA 3097, University of Lorraine, Ile du Saulcy, 57045 Metz, France xuan-thanh.vo@univ-lorraine.fr.
authors.forename: Xuan Thanh
authors.initials: XT
authors.lastname: Vo
authors.affiliation: Laboratory of Mathematics, INSA-Rouen, University of Normandie, 76801 Saint-Etienne-du-Rouvray Cedex, France pham@insa-rouen.fr.
authors.forename: Tao Pham
authors.initials: TP
authors.lastname: Dinh