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

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 regula...

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Bibliographic Details
Published in:Communications on pure and applied mathematics Vol. 57; no. 11; pp. 1413 - 1457
Main Authors: Daubechies, I., Defrise, M., De Mol, C.
Format: Journal Article
Language:English
Published: Hoboken Wiley Subscription Services, Inc., A Wiley Company 01.11.2004
Wiley
John Wiley and Sons, Limited
Subjects:
Algebra
Algorithmics. Computability. Computer arithmetics
Algorithms
Applied sciences
Computer science; control theory; systems
Exact sciences and technology
Linear and multilinear algebra, matrix theory
Linear equations
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Sciences and techniques of general use
Theoretical computing
Linear problem
Thresholding algorithm
Applied mathematics
Ideal
Iterative method
Mathematical expansion
Penalty method
Inverse problem
ISSN:0010-3640, 1097-0312
Online Access:Get full text
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Summary: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.
Bibliography:ArticleID:CPA20042
NSF Grants - No. DMS-0070689; No. DMS-0219233
Action de Recherche Concertée - No. Nb 02/07-281
IAP-network in Statistics P5/24
ark:/67375/WNG-Z83KNMKD-7
FWO (Fund for Scientific Research - Flanders), Belgium - No. G.0174.03
AFOSR Grant - No. F49620-01-1-0099
istex:39034D4C676B5611F8DC991B07FD147B65460AD7
SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 14
ISSN:0010-3640
1097-0312
DOI:10.1002/cpa.20042