Abstract
Abstract 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 𝓁 p ‐penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem. Use of such 𝓁 p ‐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.
Keywords
MathematicsOrthonormal basisQuadratic equationInverse problemNorm (philosophy)Applied mathematicsThresholdingIterative methodConstraint (computer-aided design)Basis (linear algebra)AlgorithmInverseIdeal (ethics)Nonlinear systemMathematical optimizationComputer scienceMathematical analysisImage (mathematics)Artificial intelligence
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Publication Info
- Year
- 2004
- Type
- article
- Volume
- 57
- Issue
- 11
- Pages
- 1413-1457
- Citations
- 4802
- Access
- Closed
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Cite This
Ingrid Daubechies,
Michel Defrise,
Christine De Mol
(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
Identifiers
- DOI
- 10.1002/cpa.20042