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An iterative thresholding algorithm for linear inverse problems with a sparsity constraint

Publication ,  Journal Article
Daubechies, I; Defrise, M; De Mol, C
Published in: Communications on Pure and Applied Mathematics
November 1, 2004

We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary preassigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted of ℓP - penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem. Use of such ℓP- penalized problems with p < 2 is often advocated when one expects the underlying ideal noiseless solution to have a sparse expansion with respect to the basis under consideration. To compute the corresponding regularized solutions, we analyze an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. © 2004 Wiley Periodicals, Inc.

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Published In

Communications on Pure and Applied Mathematics

DOI

ISSN

0010-3640

Publication Date

November 1, 2004

Volume

57

Issue

11

Start / End Page

1413 / 1457

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 4901 Applied mathematics
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics
 

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Daubechies, I., Defrise, M., & De Mol, C. (2004). An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Communications on Pure and Applied Mathematics, 57(11), 1413–1457. https://doi.org/10.1002/cpa.20042
Daubechies, I., M. Defrise, and C. De Mol. “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint.” Communications on Pure and Applied Mathematics 57, no. 11 (November 1, 2004): 1413–57. https://doi.org/10.1002/cpa.20042.
Daubechies I, Defrise M, De Mol C. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Communications on Pure and Applied Mathematics. 2004 Nov 1;57(11):1413–57.
Daubechies, I., et al. “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint.” Communications on Pure and Applied Mathematics, vol. 57, no. 11, Nov. 2004, pp. 1413–57. Scopus, doi:10.1002/cpa.20042.
Daubechies I, Defrise M, De Mol C. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Communications on Pure and Applied Mathematics. 2004 Nov 1;57(11):1413–1457.
Journal cover image

Published In

Communications on Pure and Applied Mathematics

DOI

ISSN

0010-3640

Publication Date

November 1, 2004

Volume

57

Issue

11

Start / End Page

1413 / 1457

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 4901 Applied mathematics
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics