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. Karimi, Sahar; Vavasis, Stephen: IMRO: A proximal quasi-Newton method for solving $\ell_1$-regularized least squares problems (2017)
  2. Tran-Dinh, Quoc: Adaptive smoothing algorithms for nonsmooth composite convex minimization (2017)
  3. Wen, Bo; Chen, Xiaojun; Pong, Ting Kei: Linear convergence of proximal gradient algorithm with extrapolation for a class of nonconvex nonsmooth minimization problems (2017)
  4. Yun, Joo Dong; Yang, Seungjoon: ADMM in Krylov subspace and its application to total variation restoration of spatially variant blur (2017)
  5. 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)
  6. Fountoulakis, Kimon; Gondzio, Jacek: A second-order method for strongly convex $\ell _1$-regularization problems (2016)
  7. Fountoulakis, Kimon; Gondzio, Jacek: Performance of first- and second-order methods for $\ell_1$-regularized least squares problems (2016)
  8. Giryes, Raja: Sampling in the analysis transform domain (2016)
  9. Hager, William W.; Yashtini, Maryam; Zhang, Hongchao: An $\mathcal O(1/k)$ convergence rate for the variable stepsize Bregman operator splitting algorithm (2016)
  10. Li, Jueyou; Chen, Guo; Dong, Zhaoyang; Wu, Zhiyou: A fast dual proximal-gradient method for separable convex optimization with linear coupled constraints (2016)
  11. Pereyra, Marcelo: Proximal Markov chain Monte Carlo algorithms (2016)
  12. Shefi, Ron; Teboulle, Marc: A dual method for minimizing a nonsmooth objective over one smooth inequality constraint (2016)
  13. Dassios, Ioannis; Fountoulakis, Kimon; Gondzio, Jacek: A preconditioner for a primal-dual Newton conjugate gradient method for compressed sensing problems (2015)
  14. He, Niao; Juditsky, Anatoli; Nemirovski, Arkadi: Mirror Prox algorithm for multi-term composite minimization and semi-separable problems (2015)
  15. Jakeman, J.D.; Eldred, M.S.; Sargsyan, K.: Enhancing $\ell_1$-minimization estimates of polynomial chaos expansions using basis selection (2015)
  16. Landi, G.: A modified Newton projection method for $\ell _1$-regularized least squares image deblurring (2015)
  17. Lan, Guanghui: Bundle-level type methods uniformly optimal for smooth and nonsmooth convex optimization (2015)
  18. Li, Xinxin; Yuan, Xiaoming: A proximal strictly contractive Peaceman-Rachford splitting method for convex programming with applications to imaging (2015)
  19. Li, Yusheng; Xie, Xinchang; Yang, Zhouwang: Alternating direction method of multipliers for solving dictionary learning models (2015)
  20. Ouyang, Yuyuan; Chen, Yunmei; Lan, Guanghui; Pasiliao, Eduardo jun.: An accelerated linearized alternating direction method of multipliers (2015)

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