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. Wu, Caiying; Wang, Jing; Alcantara, Jan Harold; Chen, Jein-Shan: Smoothing strategy along with conjugate gradient algorithm for signal reconstruction (2021)
  2. Dai, Xiangguang; Su, Xiaojie; Zhang, Wei; Xue, Fangzheng; Li, Huaqing: Robust Manhattan non-negative matrix factorization for image recovery and representation (2020)
  3. Ding, Liang; Han, Weimin: A projected gradient method for (\alpha\ell_1-\beta\ell_2) sparsity regularization (2020)
  4. Kanno, Yoshihiro: An accelerated Uzawa method for application to frictionless contact problem (2020)
  5. Ren, Sheng; Kang, Emily L.; Lu, Jason L.: MCEN: a method of simultaneous variable selection and clustering for high-dimensional multinomial regression (2020)
  6. Shen, Chungen; Xue, Wenjuan; Zhang, Lei-Hong; Wang, Baiyun: An active-set proximal-Newton algorithm for (\ell_1) regularized optimization problems with box constraints (2020)
  7. Tu, Kai; Zhang, Haibin; Gao, Huan; Feng, Junkai: A hybrid Bregman alternating direction method of multipliers for the linearly constrained difference-of-convex problems (2020)
  8. Zheng, Peng; Aravkin, Aleksandr: Relax-and-split method for nonconvex inverse problems (2020)
  9. Biau, G.; Cadre, B.; Rouvière, L.: Accelerated gradient boosting (2019)
  10. Chen, Yunmei; Lan, Guanghui; Ouyang, Yuyuan; Zhang, Wei: Fast bundle-level methods for unconstrained and ball-constrained convex optimization (2019)
  11. Fan, Ya-Ru; Buccini, Alessandro; Donatelli, Marco; Huang, Ting-Zhu: A non-convex regularization approach for compressive sensing (2019)
  12. Feng, Lei; Sun, Huaijiang; Zhu, Jun: Robust image compressive sensing based on half-quadratic function and weighted Schatten-(p) norm (2019)
  13. Hien, Le Thi Khanh; Nguyen, Cuong V.; Xu, Huan; Lu, Canyi; Feng, Jiashi: Accelerated randomized mirror descent algorithms for composite non-strongly convex optimization (2019)
  14. Koep, Niklas; Behboodi, Arash; Mathar, Rudolf: An introduction to compressed sensing (2019)
  15. Li, Qian; Bai, Yanqin; Yu, Changjun; Yuan, Ya-xiang: A new piecewise quadratic approximation approach for (L_0) norm minimization problem (2019)
  16. Liu, Zexian; Liu, Hongwei; Wang, Xiping: Accelerated augmented Lagrangian method for total variation minimization (2019)
  17. Renegar, James: Accelerated first-order methods for hyperbolic programming (2019)
  18. Shen, Yuan; Ji, Lei: Partial convolution for total variation deblurring and denoising by new linearized alternating direction method of multipliers with extension step (2019)
  19. Wu, Caiying; Zhan, Jiaming; Lu, Yue; Chen, Jein-Shan: Signal reconstruction by conjugate gradient algorithm based on smoothing (l_1)-norm (2019)
  20. Yang, Tianbao; Zhang, Lijun; Jin, Rong; Zhu, Shenghuo; Zhou, Zhi-Hua: A simple homotopy proximal mapping algorithm for compressive sensing (2019)

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