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. Garrigos, Guillaume; Rosasco, Lorenzo; Villa, Silvia: Iterative regularization via dual diagonal descent (2018)
  2. Mei, Jin-Jin; Huang, Ting-Zhu; Wang, Si; Zhao, Xi-Le: Second order total generalized variation for Speckle reduction in ultrasound images (2018)
  3. Yu, Yongchao; Peng, Jigen: The matrix splitting based proximal fixed-point algorithms for quadratically constrained $\ell_1$ minimization and Dantzig selector (2018)
  4. Karimi, Sahar; Vavasis, Stephen: IMRO: A proximal quasi-Newton method for solving $\ell_1$-regularized least squares problems (2017)
  5. Tran-Dinh, Quoc: Adaptive smoothing algorithms for nonsmooth composite convex minimization (2017)
  6. Wen, Bo; Chen, Xiaojun; Pong, Ting Kei: Linear convergence of proximal gradient algorithm with extrapolation for a class of nonconvex nonsmooth minimization problems (2017)
  7. Yang, Zhimin; Chai, Yi; Chen, Tao; Qu, Jianfeng: Smoothed $\ell_1$-regularization-based line search for sparse signal recovery (2017)
  8. Yun, Joo Dong; Yang, Seungjoon: ADMM in Krylov subspace and its application to total variation restoration of spatially variant blur (2017)
  9. Yu, Yongchao; Peng, Jigen: The Moreau envelope based efficient first-order methods for sparse recovery (2017)
  10. Yu, Yongchao; Peng, Jigen; Han, Xuanli; Cui, Angang: A primal Douglas-Rachford splitting method for the constrained minimization problem in compressive sensing (2017)
  11. De Asmundis, Roberta; di Serafino, Daniela; Landi, Germana: On the regularizing behavior of the SDA and SDC gradient methods in the solution of linear ill-posed problems (2016)
  12. Fountoulakis, Kimon; Gondzio, Jacek: Performance of first- and second-order methods for $\ell_1$-regularized least squares problems (2016)
  13. Fountoulakis, Kimon; Gondzio, Jacek: A second-order method for strongly convex $\ell _1$-regularization problems (2016)
  14. Giryes, Raja: Sampling in the analysis transform domain (2016)
  15. Hager, William W.; Yashtini, Maryam; Zhang, Hongchao: An $\mathcal O(1/k)$ convergence rate for the variable stepsize Bregman operator splitting algorithm (2016)
  16. Li, Jueyou; Chen, Guo; Dong, Zhaoyang; Wu, Zhiyou: A fast dual proximal-gradient method for separable convex optimization with linear coupled constraints (2016)
  17. Pereyra, Marcelo: Proximal Markov chain Monte Carlo algorithms (2016)
  18. Shefi, Ron; Teboulle, Marc: A dual method for minimizing a nonsmooth objective over one smooth inequality constraint (2016)
  19. Dassios, Ioannis; Fountoulakis, Kimon; Gondzio, Jacek: A preconditioner for a primal-dual Newton conjugate gradient method for compressed sensing problems (2015)
  20. He, Niao; Juditsky, Anatoli; Nemirovski, Arkadi: Mirror Prox algorithm for multi-term composite minimization and semi-separable problems (2015)

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