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

Communications on Pure and Applied Mathematics - Tập 57 Số 11 - Trang 1413-1457 - 2004
Miguel R. D. Rodrigues1, Michel Defrise2, Christine De Mol3
1Department of Mathematics, U. of Princeton
2Vrije Universiteit Brussel,
3Université libre de Bruxelles

Tóm tắt

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 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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Communications on Pure and Applied Mathematics

Tập 57 Số 11

1413-1457

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