FPC_AS (fixed-point continuation and active set) is a MATLAB solver for the l1-regularized least squares problem: A fast algorithm for sparse reconstruction based on shrinkage, subspace optimization, and continuation. We propose a fast algorithm for solving the ℓ 1 -regularized minimization problem min x∈ℝ n μ∥x∥ 1 +∥Ax-b∥ 2 2 for recovering sparse solutions to an undetermined system of linear equations Ax=b. The algorithm is divided into two stages that are performed repeatedly. In the first stage a first-order iterative “shrinkage” method yields an estimate of the subset of components of x likely to be nonzero in an optimal solution. Restricting the decision variables x to this subset and fixing their signs at their current values reduces the ℓ 1 -norm ∥x∥ 1 to a linear function of x. The resulting subspace problem, which involves the minimization of a smaller and smooth quadratic function, is solved in the second phase. Our code FPC_AS embeds this basic two-stage algorithm in a continuation (homotopy) approach by assigning a decreasing sequence of values to μ. This code exhibits state-of-the-art performance in terms of both its speed and its ability to recover sparse signals

References in zbMATH (referenced in 68 articles , 1 standard article )

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  1. Bian, Fengmiao; Zhang, Xiaoqun: A three-operator splitting algorithm for nonconvex sparsity regularization (2021)
  2. Cheng, Wanyou; Dai, Yu-Hong: An active set Newton-CG method for (\ell_1) optimization (2021)
  3. Huang, Yong; Beck, James L.; Li, Hui; Ren, Yulong: Sequential sparse Bayesian learning with applications to system identification for damage assessment and recursive reconstruction of image sequences (2021)
  4. Jiang, Shan; Fang, Shu-Cherng; Jin, Qingwei: Sparse solutions by a quadratically constrained (\ellq) ((0 < q< 1)) minimization model (2021)
  5. Tendero, Yohann; Ciril, Igor; Darbon, Jérôme; Serna, Susana: An algorithm solving compressive sensing problem based on maximal monotone operators (2021)
  6. Wang, Yifei; Jia, Zeyu; Wen, Zaiwen: Search direction correction with normalized gradient makes first-order methods faster (2021)
  7. Xiao, Nachuan; Liu, Xin; Yuan, Ya-xiang: Exact penalty function for (\ell_2,1) norm minimization over the Stiefel manifold (2021)
  8. Yu, Tengteng; Liu, Xin-Wei; Dai, Yu-Hong; Sun, Jie: Stochastic variance reduced gradient methods using a trust-region-like scheme (2021)
  9. Zhu, Wenxing; Huang, Zilin; Chen, Jianli; Peng, Zheng: Iteratively weighted thresholding homotopy method for the sparse solution of underdetermined linear equations (2021)
  10. Cheng, Wanyou; Chen, Zixin; Hu, Qingjie: An active set Barzilar-Borwein algorithm for (l_0) regularized optimization (2020)
  11. Mousavi, Ahmad; Rezaee, Mehdi; Ayanzadeh, Ramin: A survey on compressive sensing: classical results and recent advancements (2020)
  12. 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)
  13. Wang, Guoqiang; Wei, Xinyuan; Yu, Bo; Xu, Lijun: An efficient proximal block coordinate homotopy method for large-scale sparse least squares problems (2020)
  14. Zhang, Chao; Chen, Xiaojun: A smoothing active set method for linearly constrained non-Lipschitz nonconvex optimization (2020)
  15. Azmi, Behzad; Kunisch, Karl: A hybrid finite-dimensional RHC for stabilization of time-varying parabolic equations (2019)
  16. Becker, Stephen; Fadili, Jalal; Ochs, Peter: On quasi-Newton forward-backward splitting: proximal calculus and convergence (2019)
  17. Cheng, Wanyou; Hu, Qingjie; Li, Donghui: A fast conjugate gradient algorithm with active set prediction for (\ell_1) optimization (2019)
  18. Esmaeili, Hamid; Shabani, Shima; Kimiaei, Morteza: A new generalized shrinkage conjugate gradient method for sparse recovery (2019)
  19. Lin, Meixia; Liu, Yong-Jin; Sun, Defeng; Toh, Kim-Chuan: Efficient sparse semismooth Newton methods for the clustered Lasso problem (2019)
  20. Nutini, Julie; Schmidt, Mark; Hare, Warren: “Active-set complexity” of proximal gradient: how long does it take to find the sparsity pattern? (2019)

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