An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
Abstract
We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary pre-assigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted l^p-penalties on the coefficients of such expansions, with 1 < or = p < or =2, still regularizes the problem. If p < 2, regularized solutions of such l^p-penalized problems will have sparser expansions, with respect to the basis under consideration. To compute the corresponding regularized solutions we propose 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. We also review some potential applications of this method.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- July 2003
- DOI:
- 10.48550/arXiv.math/0307152
- arXiv:
- arXiv:math/0307152
- Bibcode:
- 2003math......7152D
- Keywords:
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- Mathematics - Functional Analysis;
- Mathematics - Numerical Analysis
- E-Print:
- 30 pages, 3 figures