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.
Cite
@article{arxiv.math/0307152,
title = {An iterative thresholding algorithm for linear inverse problems with a sparsity constraint},
author = {Ingrid Daubechies and Michel Defrise and Christine De Mol},
journal= {arXiv preprint arXiv:math/0307152},
year = {2025}
}
Comments
30 pages, 3 figures; this is version 2 - changes with respect to v1: small correction in proof (but not statement of) lemma 3.15; description of Besov spaces in intro and app A clarified (and corrected); smaller pointsize (making 30 instead of 38 pages)