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Bounded Gradient Projection Methods for Sparse Signal Recovery

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Abstract

The $\ell_2$-$\ell_1$ sparse signal minimization problem can be solved efficiently by gradient projection. In many applications, the signal to be estimated is known to lie in some range of values. With these additional constraints on the estimate, the resulting constrained minimization problem is more challenging to solve. In previous work, we proposed a gradient projection approach for solving this type of minimization problem with nonnegativity constraints. In this paper, we generalize this approach to solve any bound-constrained $\ell_2$-$\ell_1$ minimization problem. Our method is based on solving the Lagrangian dual problem, and we show that by constraining the solution to known a priori bounds within the optimization method, we can obtain a more accurate estimate than simply thresholding the solution from the unconstrained minimization problem. Numerical results are presented to demonstrate the effectiveness of this approach.

Citation

J. Hernandez, Z. T. Harmany, D. O. Thompson and R. F. Marcia, “Bounded gradient projection methods for sparse signal recovery”, in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2011, 949–952.

BibTeX

@inproceedings{hernandez-icassp2011-boundedgradientprojection,
  doi = {10.1109/ICASSP.2011.5946562},
  title = {Bounded gradient projection methods for sparse signal recovery},
  author = {Hernandez, James and Harmany, Zachary T. and Thompson, Daniel O. and Marcia, Roummel F.},
  booktitle = {IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)},
  month = {may},
  year = {2011},
  pages = {949–952},
  abstract = {The $\ell_2$-$\ell_1$ sparse signal minimization problem can be solved efficiently by gradient projection. In many applications, the signal to be estimated is known to lie in some range of values. With these additional constraints on the estimate, the resulting constrained minimization problem is more challenging to solve. In previous work, we proposed a gradient projection approach for solving this type of minimization problem with nonnegativity constraints. In this paper, we generalize this approach to solve any bound-constrained $\ell_2$-$\ell_1$ minimization problem. Our method is based on solving the Lagrangian dual problem, and we show that by constraining the solution to known a priori bounds within the optimization method, we can obtain a more accurate estimate than simply thresholding the solution from the unconstrained minimization problem. Numerical results are presented to demonstrate the effectiveness of this approach.}
}