NESTA: A fast and accurate first-order method for sparse recovery. Accurate signal recovery or image reconstruction from indirect and possibly undersampled data is a topic of considerable interest; for example, the literature in the recent field of compressed sensing is already quite immense. This paper applies a smoothing technique and an accelerated first-order algorithm, both from {it Yu. Nesterov} [Math. Program. 103, No. 1 (A), 127--152 (2005; Zbl 1079.90102)], and demonstrates that this approach is ideally suited for solving large-scale compressed sensing reconstruction problems as (1) it is computationally efficient; (2) it is accurate and returns solutions with several correct digits; (3) it is flexible and amenable to many kinds of reconstruction problems; and (4) it is robust in the sense that its excellent performance across a wide range of problems does not depend on the fine tuning of several parameters. Comprehensive numerical experiments on realistic signals exhibiting a large dynamic range show that this algorithm compares favorably with recently proposed state-of-the-art methods. We also apply the algorithm to solve other problems for which there are fewer alternatives, such as total-variation minimization and convex programs seeking to minimize the $ell_1$ norm of $W_x$ under constraints, in which $W$ is not diagonal. The code is available online as a free package in the Matlab language.

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  1. Li, Qian; Bai, Yanqin; Yu, Changjun; Yuan, Ya-xiang: A new piecewise quadratic approximation approach for (L_0) norm minimization problem (2019)
  2. Renegar, James: Accelerated first-order methods for hyperbolic programming (2019)
  3. Garrigos, Guillaume; Rosasco, Lorenzo; Villa, Silvia: Iterative regularization via dual diagonal descent (2018)
  4. Li, Xingguo; Zhao, Tuo; Arora, Raman; Liu, Han; Hong, Mingyi: On faster convergence of cyclic block coordinate descent-type methods for strongly convex minimization (2018)
  5. Li, Xudong; Sun, Defeng; Toh, Kim-Chuan: A highly efficient semismooth Newton augmented Lagrangian method for solving lasso problems (2018)
  6. Mei, Jin-Jin; Huang, Ting-Zhu; Wang, Si; Zhao, Xi-Le: Second order total generalized variation for Speckle reduction in ultrasound images (2018)
  7. Shi, Xingjie; Huang, Yuan; Huang, Jian; Ma, Shuangge: A forward and backward stagewise algorithm for nonconvex loss functions with adaptive Lasso (2018)
  8. Shi, Yue Yong; Jiao, Yu Ling; Cao, Yong Xiu; Liu, Yan Yan: An alternating direction method of multipliers for MCP-penalized regression with high-dimensional data (2018)
  9. Yu, Yongchao; Peng, Jigen: The matrix splitting based proximal fixed-point algorithms for quadratically constrained (\ell_1) minimization and Dantzig selector (2018)
  10. Zhang, Richard Y.; White, Jacob K.: GMRES-accelerated ADMM for quadratic objectives (2018)
  11. Gong, Maoguo; Jiang, Xiangming; Li, Hao: Optimization methods for regularization-based ill-posed problems: a survey and a multi-objective framework (2017)
  12. Guo, Weihong; Song, Guohui; Zhang, Yue: PCM-TV-TFV: a novel two-stage framework for image reconstruction from Fourier data (2017)
  13. Karimi, Sahar; Vavasis, Stephen: IMRO: A proximal quasi-Newton method for solving (\ell_1)-regularized least squares problems (2017)
  14. Monga, Vishal; Mousavi, Hojjat Seyed; Srinivas, Umamahesh: Sparsity constrained estimation in image processing and computer vision (2017)
  15. Tran-Dinh, Quoc: Adaptive smoothing algorithms for nonsmooth composite convex minimization (2017)
  16. Wen, Bo; Chen, Xiaojun; Pong, Ting Kei: Linear convergence of proximal gradient algorithm with extrapolation for a class of nonconvex nonsmooth minimization problems (2017)
  17. Yang, Zhimin; Chai, Yi; Chen, Tao; Qu, Jianfeng: Smoothed (\ell_1)-regularization-based line search for sparse signal recovery (2017)
  18. Yun, Joo Dong; Yang, Seungjoon: ADMM in Krylov subspace and its application to total variation restoration of spatially variant blur (2017)
  19. Yu, Yaoliang; Zhang, Xinhua; Schuurmans, Dale: Generalized conditional gradient for sparse estimation (2017)
  20. Yu, Yongchao; Peng, Jigen: The Moreau envelope based efficient first-order methods for sparse recovery (2017)

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