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Sparse Reconstruction by Separable Approximation (2007)
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BibTeX
@MISC{Wright07sparsereconstruction,
author = {Stephen J. Wright and Robert D. Nowak and Mário A. T. Figueiredo},
title = {Sparse Reconstruction by Separable Approximation },
year = {2007}
}
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Abstract
Finding sparse approximate solutions to large underdetermined linear systems of equations is a common problem in signal/image processing and statistics. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), wavelet-based deconvolution and reconstruction, and compressed sensing (CS) are a few well-known areas in which problems of this type appear. One standard approach is to minimize an objective function that includes a quadratic (ℓ2) error term added to a sparsity-inducing (usually ℓ1) regularizer. We present an algorithmic framework for the more general problem of minimizing the sum of a smooth convex function and a nonsmooth, possibly nonconvex, sparsity-inducing function. We propose iterative methods in which each step is an optimization subproblem involving a separable quadratic term (diagonal Hessian) plus the original sparsity-inducing term. Our approach is suitable for cases in which this subproblem can be solved much more rapidly than the original problem. In addition to solving the standard ℓ2 − ℓ1 case, our approach handles other problems, e.g., ℓp regularizers with p � = 1, or group-separable (GS) regularizers. Experiments with CS problems show that our approach provides state-of-the-art speed for the standard ℓ2 − ℓ1 problem, and is also efficient on problems with GS regularizers. Index Terms — sparse approximation, compressed sensing, optimization, reconstruction.
Keyphrases
sparse reconstruction separable approximation diagonal hessian problem formulation state-of-the-art speed original sparsity-inducing term objective function type appear wavelet-based deconvolution sparse approximate solution optimization subproblem general problem large underdetermined linear system original problem absolute shrinkage standard approach c problem signal image processing common problem smooth convex function g regularizers separable quadratic term basis pursuit well-known area selection operator index term sparse approximation error term algorithmic framework sparsity-inducing function iterative method
