SPGL1: A solver for large-scale sparse reconstruction: Probing the Pareto frontier for basis pursuit solutions. The basis pursuit problem seeks a minimum one-norm solution of an underdetermined least-squares problem. Basis Pursuit DeNoise (BPDN) fits the least-squares problem only approximately, and a single parameter determines a curve that traces the optimal trade-off between the least-squares fit and the one-norm of the solution. We prove that this curve is convex and continuously differentiable over all points of interest, and show that it gives an explicit relationship to two other optimization problems closely related to BPDN. We describe a root-finding algorithm for finding arbitrary points on this curve; the algorithm is suitable for problems that are large scale and for those that are in the complex domain. At each iteration, a spectral gradient-projection method approximately minimizes a least-squares problem with an explicit one-norm constraint. Only matrix-vector operations are required. The primal-dual solution of this problem gives function and derivative information needed for the root-finding method. Numerical experiments on a comprehensive set of test problems demonstrate that the method scales well to large problems.

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  11. Thesing, L.; Hansen, A. C.: Non-uniform recovery guarantees for binary measurements and infinite-dimensional compressed sensing (2021)
  12. Zeng, Liaoyuan; Yu, Peiran; Pong, Ting Kei: Analysis and algorithms for some compressed sensing models based on L1/L2 minimization (2021)
  13. Abubakar, Auwal Bala; Kumam, Poom; Mohammad, Hassan: A note on the spectral gradient projection method for nonlinear monotone equations with applications (2020)
  14. Daskalakis, Emmanouil; Herrmann, Felix J.; Kuske, Rachel: Accelerating sparse recovery by reducing chatter (2020)
  15. Ding, Liang; Han, Weimin: A projected gradient method for (\alpha\ell_1-\beta\ell_2) sparsity regularization (2020)
  16. Estrin, Ron; Friedlander, Michael P.: A perturbation view of level-set methods for convex optimization (2020)
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  18. Jiang, Shan; Fang, Shu-Cherng; Nie, Tiantian; Xing, Wenxun: A gradient descent based algorithm for (\ell_p) minimization (2020)
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  20. Le Gia, Quoc Thong; Sloan, Ian H.; Womersley, Robert S.; Wang, Yu Guang: Isotropic sparse regularization for spherical harmonic representations of random fields on the sphere (2020)

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