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Recovery algorithms for vector valued data with joint sparsity constraints (2006)
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BibTeX
@MISC{Fornasier06recoveryalgorithms,
author = {Massimo Fornasier and Holger Rauhut},
title = {Recovery algorithms for vector valued data with joint sparsity constraints},
year = {2006}
}
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Abstract
Vector valued data appearing in concrete applications often possess sparse expansions with respect to a preassigned frame for each vector component individually. Additionally, different components may also exhibit common sparsity patterns. Recently, there were introduced sparsity measures that take into account such joint sparsity patterns, promoting coupling of non-vanishing components. These measures are typically constructed as weighted ℓ1 norms of componentwise ℓq norms of frame coefficients. We show how to compute solutions of linear inverse problems with such joint sparsity regularization constraints by fast thresholded Landweber algorithms. Next we discuss the adaptive choice of suitable weights appearing in the definition of sparsity measures. The weights are interpreted as indicators of the sparsity pattern and are iteratively up-dated after each new application of the thresholded Landweber algorithm. The resulting two-step algorithm is interpreted as a double-minimization scheme for a suitable target functional. We show its ℓ2-norm convergence. An implementable version of the algorithm is also formulated, and its norm convergence is proven. Numerical experiments in color image restoration are presented.
Keyphrases
recovery algorithm joint sparsity constraint two-step algorithm suitable target 2-norm convergence vector component different component concrete application linear inverse problem frame coefficient adaptive choice color image restoration non-vanishing component norm convergence landweber algorithm common sparsity pattern thresholded landweber algorithm double-minimization scheme joint sparsity regularization constraint implementable version suitable weight numerical experiment new application componentwise norm sparsity measure posse sparse expansion account joint sparsity pattern sparsity pattern
