Journal of Fourier Analysis and Applications

, Volume 14, Issue 5, pp 877–905

Enhancing Sparsity by Reweighted 1 Minimization

Authors

  • Emmanuel J. Candès
    • Applied and Computational MathematicsCaltech
    • EngineeringColorado School of Mines
  • Stephen P. Boyd
    • Department of Electrical EngineeringStanford University
Article

DOI: 10.1007/s00041-008-9045-x

Cite this article as:
Candès, E.J., Wakin, M.B. & Boyd, S.P. J Fourier Anal Appl (2008) 14: 877. doi:10.1007/s00041-008-9045-x

Abstract

It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained 1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms 1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted 1-minimization problems where the weights used for the next iteration are computed from the value of the current solution. We present a series of experiments demonstrating the remarkable performance and broad applicability of this algorithm in the areas of sparse signal recovery, statistical estimation, error correction and image processing. Interestingly, superior gains are also achieved when our method is applied to recover signals with assumed near-sparsity in overcomplete representations—not by reweighting the 1 norm of the coefficient sequence as is common, but by reweighting the 1 norm of the transformed object. An immediate consequence is the possibility of highly efficient data acquisition protocols by improving on a technique known as Compressive Sensing.

Keywords

1-MinimizationIterative reweightingUnderdetermined systems of linear equationsCompressive sensingDantzig selectorSparsityFOCUSS

Copyright information

© Birkhäuser Boston 2008