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. Ren, Sheng; Kang, Emily L.; Lu, Jason L.: MCEN: a method of simultaneous variable selection and clustering for high-dimensional multinomial regression (2020)
  2. 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)
  3. Biau, G.; Cadre, B.; Rouvière, L.: Accelerated gradient boosting (2019)
  4. Chen, Yunmei; Lan, Guanghui; Ouyang, Yuyuan; Zhang, Wei: Fast bundle-level methods for unconstrained and ball-constrained convex optimization (2019)
  5. Fan, Ya-Ru; Buccini, Alessandro; Donatelli, Marco; Huang, Ting-Zhu: A non-convex regularization approach for compressive sensing (2019)
  6. 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)
  7. Koep, Niklas; Behboodi, Arash; Mathar, Rudolf: An introduction to compressed sensing (2019)
  8. Li, Qian; Bai, Yanqin; Yu, Changjun; Yuan, Ya-xiang: A new piecewise quadratic approximation approach for (L_0) norm minimization problem (2019)
  9. Liu, Zexian; Liu, Hongwei; Wang, Xiping: Accelerated augmented Lagrangian method for total variation minimization (2019)
  10. Renegar, James: Accelerated first-order methods for hyperbolic programming (2019)
  11. Shen, Yuan; Ji, Lei: Partial convolution for total variation deblurring and denoising by new linearized alternating direction method of multipliers with extension step (2019)
  12. Wu, Caiying; Zhan, Jiaming; Lu, Yue; Chen, Jein-Shan: Signal reconstruction by conjugate gradient algorithm based on smoothing (l_1)-norm (2019)
  13. Yang, Tianbao; Zhang, Lijun; Jin, Rong; Zhu, Shenghuo; Zhou, Zhi-Hua: A simple homotopy proximal mapping algorithm for compressive sensing (2019)
  14. Bolte, Jérôme; Sabach, Shoham; Teboulle, Marc: Nonconvex Lagrangian-based optimization: monitoring schemes and global convergence (2018)
  15. Garrigos, Guillaume; Rosasco, Lorenzo; Villa, Silvia: Iterative regularization via dual diagonal descent (2018)
  16. Irannejad, Maziar; Mahdavi-Nasab, Homayoun: Block matching video compression based on sparse representation and dictionary learning (2018)
  17. 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)
  18. Li, Xudong; Sun, Defeng; Toh, Kim-Chuan: A highly efficient semismooth Newton augmented Lagrangian method for solving lasso problems (2018)
  19. Mei, Jin-Jin; Huang, Ting-Zhu; Wang, Si; Zhao, Xi-Le: Second order total generalized variation for Speckle reduction in ultrasound images (2018)
  20. Park, Dohyung; Kyrillidis, Anastasios; Caramanis, Constantine; Sanghavi, Sujay: Finding low-rank solutions via nonconvex matrix factorization, efficiently and provably (2018)

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