Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems

Many problems in signal processing and statistical inference involve finding sparse solutions to under-determined, or ill-conditioned, linear systems of equations. A standard approach consists in minimizing an objective function which includes a quadratic (squared ) error term combined with a sparse...

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Bibliographic Details
Published in:IEEE journal of selected topics in signal processing 2007-12, Vol.1 (4), p.586-597
Main Authors: Figueiredo, M.A.T., Nowak, R.D., Wright, S.J.
Format: Article
Language:English
Subjects:
Algorithms
Compressed sensing
convex optimization
Deconvolution
Degradation
Equations
gradient projection
Inverse problems
Linear systems
Operations research
Quadratic programming
Signal processing
Signal processing algorithms
sparse reconstruction
sparseness
Studies
Testing
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Summary:Many problems in signal processing and statistical inference involve finding sparse solutions to under-determined, or ill-conditioned, linear systems of equations. A standard approach consists in minimizing an objective function which includes a quadratic (squared ) error term combined with a sparseness-inducing regularization term. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), wavelet-based deconvolution, and compressed sensing are a few well-known examples of this approach. This paper proposes gradient projection (GP) algorithms for the bound-constrained quadratic programming (BCQP) formulation of these problems. We test variants of this approach that select the line search parameters in different ways, including techniques based on the Barzilai-Borwein method. Computational experiments show that these GP approaches perform well in a wide range of applications, often being significantly faster (in terms of computation time) than competing methods. Although the performance of GP methods tends to degrade as the regularization term is de-emphasized, we show how they can be embedded in a continuation scheme to recover their efficient practical performance.
ISSN:1932-4553
1941-0484
DOI:10.1109/JSTSP.2007.910281