Fast algorithms for l&amp;lt;inf&amp;gt;p&amp;lt;/inf&amp;gt;deconvolution

Yarlagadda, R.; Bednar, J. Bee; Watt, T.

doi:10.1109/tassp.1985.1164508

Cited by 87 publications

(27 citation statements)

References 11 publications

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“…where ρ(·) is a penalty function such as the ℓ 1 norm [39,42]. This minimization can be accomplished by solving a sequence of weighted least-squares problems where the weights {w i } depend on the previous residual w i = ρ ′ (r i )/r i .…”

Section: Historical Progressionmentioning

confidence: 99%

Enhancing Sparsity by Reweighted ℓ 1 Minimization

Candès

Wakin

Boyd

2008

J Fourier Anal Appl

4,309

3,421

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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 ℓ 1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms ℓ 1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted ℓ 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 ℓ 1 norm of the coefficient sequence as is common, but by reweighting the ℓ 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 compressed sensing.

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Section: Historical Progressionmentioning

confidence: 99%

Enhancing Sparsity by Reweighted ℓ 1 Minimization

Candès

Wakin

Boyd

2008

J Fourier Anal Appl

4,309

3,421

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“…We obtained (8) by applying a fixed point approach to the NSC (3) but the equivalent algorithm (5) can also be obtained by applying an iterative reweighted least squares (IRLS) approach to (2), see [20]- [24]. Other iterative algorithms [27]- [30] have also been proposed for similar criteria having a different penalization term. They are usually applied to either strictly convex or convex differentiable criterion.…”

Section: Iterative Algorithmmentioning

confidence: 99%

“…A standard trick consists in replacing the small components in or by , for instance, or by intervening similarely on in (9). Introducing Huber's function in place of the penalization in (2), as proposed in [30], can also be considered. Here, however, no such modifications were implemented.…”

Section: B a Time Varying Casementioning

confidence: 99%

Convergence of a Sparse Representations Algorithm Applicable to Real or Complex Data

Fuchs

2007

IEEE J. Sel. Top. Signal Process.

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Abstract-Sparse representations has become an important topic in recent years. It consists in representing, say, a signal (vector) as a linear combination of as few as possible components (vectors) from a redundant basis (of the vector space). This is usually performed, either iteratively (adding a component at a time), or globally (selecting simultaneously all the needed components). We consider a specific algorithm, that we obtain as a fixed point algorithm, but that can also be seen as an iteratively reweighted least-squares algorithm. We analyze it thoroughly and show that it converges to the global optimum. We detail the proof in the real case and indicate how to extend it to the complex case. We illustrate the result with some easily reproducible toy simulations, that further illustrate the potential tracking properties of the proposed algorithm.Index Terms-Convergence of numerical methods, fixed-point algorithms, iterative methods, minimization methods, spectral analysis.

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“…Note that our convergence analysis can be extended to the inhomogeneous case in a straightforward manner. This allows one to address signal processing applications such as autoregressive modeling [18] or digital filter synthesis [19].…”

Section: Introductionmentioning

confidence: 99%

On global and local convergence of half-quadratic algorithms

Allain¹,

Idier²,

Goussard

2006

IEEE Trans. on Image Process.

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This paper provides original results on the global and local convergence properties of half-quadratic (HQ) algorithms resulting from the Geman and Yang (GY) and Geman and Reynolds (GR) primal-dual constructions. First, we show that the convergence domain of the GY algorithm can be extended with the benefit of an improved convergence rate. Second, we provide a precise comparison of the convergence rates for both algorithms. This analysis shows that the GR form does not benefit from a better convergence rate in general. Moreover, the GY iterates often take advantage of a low cost implementation. In this case, the GY form is usually faster than the GR form from the CPU time viewpoint.

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Fast algorithms for l<inf>p</inf>deconvolution

Cited by 87 publications

References 11 publications

Enhancing Sparsity by Reweighted ℓ 1 Minimization

Enhancing Sparsity by Reweighted ℓ 1 Minimization

Convergence of a Sparse Representations Algorithm Applicable to Real or Complex Data

On global and local convergence of half-quadratic algorithms

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