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

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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- General Mathematics
- 4904 Pure mathematics
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- 0102 Applied Mathematics
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*Communications on Pure and Applied Mathematics*,

*57*(11), 1413–1457. https://doi.org/10.1002/cpa.20042

*Communications on Pure and Applied Mathematics*57, no. 11 (November 1, 2004): 1413–57. https://doi.org/10.1002/cpa.20042.

*Communications on Pure and Applied Mathematics*, vol. 57, no. 11, Nov. 2004, pp. 1413–57.

*Scopus*, doi:10.1002/cpa.20042.

## Published In

## DOI

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

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