Premium
Iterated hard‐thresholding for linear inverse problems with sparsity constraints
Author(s) -
Lorenz Dirk,
Bredies Kristian
Publication year - 2007
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200700906
Subject(s) - tikhonov regularization , iterated function , regularization (linguistics) , thresholding , mathematics , frank–wolfe algorithm , convergence (economics) , inverse , minification , rate of convergence , algorithm , inverse problem , mathematical optimization , computer science , artificial intelligence , mathematical analysis , image (mathematics) , key (lock) , convex set , geometry , computer security , convex optimization , regular polygon , economics , economic growth
We describe an iterative algorithm for the minimization of Tikhonov type functionals which involve sparsity constraints in form of ℓ p ‐penalties which have been proposed recently for the regularization of ill‐posed problems. In contrast to the well‐known algorithm considered by Daubechies, Defrise and De Mol, it uses hard instead of soft thresholding. This hard thresholding algorithm is based on the generalized conditional gradient method. General results on the convergence of the generalized conditional gradient method enable us to prove strong convergence of the iterates. Furthermore we are able to establish convergence rates of O ( n –1/2 ) and O ( λ n ) for p = 1 and 1 < p ≤ 2 respectively. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)