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Eventual linear convergence of the Douglas-Rachford iteration for basis pursuit

Author(s)
Demanet, Laurent; Zhang, Xiangxiong
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Abstract
We provide a simple analysis of the Douglas-Rachford splitting algorithm in the context of ℓ[superscript 1] minimization with linear constraints, and quantify the asymptotic linear convergence rate in terms of principal angles between relevant vector spaces. In the compressed sensing setting, we show how to bound this rate in terms of the restricted isometry constant. More general iterative schemes obtained by ℓ[superscript 2]-regularization and over-relaxation including the dual split Bregman method are also treated, which answers the question of how to choose the relaxation and soft-thresholding parameters to accelerate the asymptotic convergence rate. We make no attempt at characterizing the transient regime preceding the onset of linear convergence.
Date issued
2015-05
URI
http://hdl.handle.net/1721.1/110312
Department
Massachusetts Institute of Technology. Department of Mathematics
Journal
Mathematics of Computation
Citation
Demanet, Laurent, and Xiangxiong Zhang. “Eventual Linear Convergence of the Douglas-Rachford Iteration for Basis Pursuit.” Math. Comp. 85, no. 297 (May 15, 2015): 209–238. © 2015 American Mathematical Society
Version: Final published version
ISSN
0025-5718
1088-6842

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