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.

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  1. Leplat, Valentin; Gillis, Nicolas; Idier, Jérôme: Multiplicative updates for NMF with (\beta)-divergences under disjoint equality constraints (2021)
  2. Li, Jicheng; Li, Wenbo; Liu, Xuenian: An efficient nonmonotone projected Barzilai-Borwein method for nonnegative matrix factorization with extrapolation (2021)
  3. Li, Ting; Tang, Jiayi; Wan, Zhong: An alternating nonmonotone projected Barzilai-Borwein algorithm of nonnegative factorization of big matrices (2021)
  4. Teboulle, Marc; Vaisbourd, Yakov: Novel proximal gradient methods for nonnegative matrix factorization with sparsity constraints (2020)
  5. Ang, Andersen Man Shun; Gillis, Nicolas: Accelerating nonnegative matrix factorization algorithms using extrapolation (2019)
  6. Chen, Wen-Sheng; Liu, Jingmin; Pan, Binbin; Li, Yugao: Block kernel nonnegative matrix factorization for face recognition (2019)
  7. Sun, Li; Han, Congying; Liu, Ziwen: Active set type algorithms for nonnegative matrix factorization in hyperspectral unmixing (2019)
  8. Kang, Kai; Maroulas, Vasileios; Schizas, Ioannis; Bao, Feng: Improved distributed particle filters for tracking in a wireless sensor network (2018)
  9. Takahashi, Norikazu; Katayama, Jiro; Seki, Masato; Takeuchi, Jun’ichi: A unified global convergence analysis of multiplicative update rules for nonnegative matrix factorization (2018)
  10. Alquier, Pierre; Guedj, Benjamin: An oracle inequality for quasi-Bayesian nonnegative matrix factorization (2017)
  11. Chow, Yat Tin; Wu, Tianyu; Yin, Wotao: Cyclic coordinate-update algorithms for fixed-point problems: analysis and applications (2017)
  12. Nguyen, Duy Khuong; Ho, Tu Bao: Accelerated parallel and distributed algorithm using limited internal memory for nonnegative matrix factorization (2017)
  13. Ben, Chi; Wang, Zhiyuan; Yang, Xuejun; Yi, Xiaodong: Non-negative matrix semi-tensor factorization for image feature extraction and clustering (2016)
  14. Chow, Yat Tin; Ito, Kazufumi; Zou, Jun: Analysis on a nonnegative matrix factorization and its applications (2016)
  15. Devarajan, Karthik; Cheung, Vincent C. K.: A quasi-likelihood approach to nonnegative matrix factorization (2016)
  16. Le Thi, Hoai An; Vo, Xuan Thanh; Dinh, Tao Pham: Efficient nonnegative matrix factorization by DC programming and DCA (2016)
  17. Liu, Tongliang; Tao, Dacheng; Xu, Dong: Dimensionality-dependent generalization bounds for (k)-dimensional coding schemes (2016)
  18. Bonmati, Ester; Bardera, Anton; Boada, Imma; Feixas, Miquel; Sbert, Mateu: Hierarchical clustering based on the information bottleneck method using a control process (2015)
  19. Huang, Yakui; Liu, Hongwei; Zhou, Sha: An efficient monotone projected Barzilai-Borwein method for nonnegative matrix factorization (2015)
  20. Huang, Yakui; Liu, Hongwei; Zhou, Shuisheng: Quadratic regularization projected Barzilai-Borwein method for nonnegative matrix factorization (2015)

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