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

An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,10.1002/cpa.20042,Communications on Pure and Applied Mathem

An iterative thresholding algorithm for linear inverse problems with a sparsity constraint   (Citations: 593)
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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.
Journal: Communications on Pure and Applied Mathematics - COMMUN PURE APPL MATH , vol. 57, no. 11, pp. 1413-1457, 2004
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