NeNMF swMATH ID: 17586 Software Authors: N. Guan, D. Tao, Z. Luo, B. Yuan Description: NeNMF: An optimal gradient method for non-negative matrix factorization. Nonnegative matrix factorization (NMF) is a powerful matrix decomposition technique that approximates a nonnegative matrix by the product of two low-rank nonnegative matrix factors. It has been widely applied to signal processing, computer vision, and data mining. Traditional NMF solvers include the multiplicative update rule (MUR), the projected gradient method (PG), the projected nonnegative least squares (PNLS), and the active set method (AS). However, they suffer from one or some of the following three problems: slow convergence rate, numerical instability and nonconvergence. In this paper, we present a new efficient NeNMF solver to simultaneously overcome the aforementioned problems. It applies Nesterov’s optimal gradient method to alternatively optimize one factor with another fixed. In particular, at each iteration round, the matrix factor is updated by using the PG method performed on a smartly chosen search point, where the step size is determined by the Lipschitz constant. Since NeNMF does not use the time consuming line search and converges optimally at rate in optimizing each matrix factor, it is superior to MUR and PG in terms of efficiency as well as approximation accuracy. Compared to PNLS and AS that suffer from numerical instability problem in the worst case, NeNMF overcomes this deficiency. In addition, NeNMF can be used to solve -norm, -norm and manifold regularized NMF with the optimal convergence rate. Numerical experiments on both synthetic and real-world datasets show the efficiency of NeNMF for NMF and its variants comparing to representative NMF solvers. Extensive experiments on document clustering suggest the effectiveness of NeNMF. Homepage: http://ieeexplore.ieee.org/document/6166359/ Related Software: FERET; RCV1; CLUTO; SPGL1; CVX; clusfind; ITIP; BSDS; CMU PIE; Graclus; PDCO; PRMLT; ARock; TVAL3; glmnet; ICALAB; FastICA; NMF MATLAB Toolbox; GKB-FP; UTV Referenced in: 28 Publications all top 5 Referenced by 79 Authors 4 Tao, Dacheng 2 Chow, Yat Tin 2 Gillis, Nicolas 2 Huang, Yakui 2 Liu, Hongwei 1 Alquier, Pierre 1 Ang, Andersen Man Shun 1 Bao, Feng 1 Bardera, Anton 1 Ben, Chi 1 Boada, Imma 1 Bonmati, Ester 1 Chen, Wensheng 1 Cheng, Jun 1 Cheung, Vincent C. K. 1 Chung, Fu-Lai 1 Deng, Zhaohong 1 Devarajan, Karthik 1 Fang, Yixin 1 Feixas, Miquel 1 Guan, Naiyang 1 Guedj, Benjamin 1 Han, Congying 1 Ho, TuBao 1 Idier, Jérôme 1 Ito, Kazufumi 1 Jia, Chengcheng 1 Kang, Kai 1 Katayama, Jiro 1 Kuang, Yin 1 Le Thi, Hoai An 1 Leplat, Valentin 1 Li, Jicheng 1 Li, Jun 1 Li, Ting 1 Li, Wenbo 1 Li, Xuelong 1 Li, Yugao 1 Lin, Chuang 1 Liu, Jingmin 1 Liu, Tongliang 1 Liu, Xuenian 1 Liu, Yuhao 1 Liu, Ziwen 1 Luo, Zhigang 1 Maroulas, Vasileios 1 Nguyen, Duy Khuong 1 Pan, Binbin 1 Pang, Meng 1 Pang, Shu-Chao 1 Pham Dinh Tao 1 Rui, Yong 1 Sbert, Mateu 1 Schizas, Ioannis D. 1 Seki, Masato 1 Sun, Li 1 Takahashi, Norikazu 1 Takeuchi, Jun’ichi 1 Tang, Jiayi 1 Teboulle, Marc 1 Vaisbourd, Yakov 1 Vo, Xuan Thanh 1 Wan, Zhong 1 Wang, Jun 1 Wang, Junhui 1 Wang, Shitong 1 Wang, Zhiyuan 1 Wu, Tianyu 1 Xu, Dong 1 Yang, Xuejun 1 Yi, Xiaodong 1 Yi, Zhang 1 Yin, Wotao 1 Yu, Jun 1 Yu, Zhezhou 1 Yuan, Bo 1 Zhou, Sha 1 Zhou, Shuisheng 1 Zou, Jun all top 5 Referenced in 17 Serials 4 Neural Computation 3 PAA. Pattern Analysis and Applications 2 Pattern Recognition 2 Computational Statistics and Data Analysis 2 SIAM Journal on Scientific Computing 2 Mathematical Problems in Engineering 2 Data Mining and Knowledge Discovery 1 Applied Mathematics Letters 1 SIAM Journal on Matrix Analysis and Applications 1 IEEE Transactions on Signal Processing 1 Journal of Global Optimization 1 International Journal of Computer Mathematics 1 Computational Optimization and Applications 1 Mathematical Methods of Statistics 1 Journal of Applied Mathematics 1 International Journal of Wavelets, Multiresolution and Information Processing 1 SIAM Journal on Imaging Sciences all top 5 Referenced in 8 Fields 14 Computer science (68-XX) 13 Linear and multilinear algebra; matrix theory (15-XX) 13 Numerical analysis (65-XX) 10 Operations research, mathematical programming (90-XX) 8 Statistics (62-XX) 2 Information and communication theory, circuits (94-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Biology and other natural sciences (92-XX) Referencing Publications by Year