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


Authors: Laurent Demanet and Xiangxiong Zhang
Journal: Math. Comp. 85 (2016), 209-238
MSC (2010): Primary 49M29, 65K10, 90C25
DOI: https://doi.org/10.1090/mcom/2965
Published electronically: May 15, 2015
MathSciNet review: 3404448
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Abstract: We provide a simple analysis of the Douglas-Rachford splitting algorithm in the context of $ \ell ^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 $ \ell ^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.


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Additional Information

Laurent Demanet
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: laurent@math.mit.edu

Xiangxiong Zhang
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Address at time of publication: Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, Indiana 47907
Email: zhan1966@purdue.edu

DOI: https://doi.org/10.1090/mcom/2965
Keywords: Basis pursuit, Douglas-Rachford, generalized Douglas-Rachford, Peaceman-Rachford, relaxation parameter, asymptotic linear convergence rate
Received by editor(s): January 24, 2013
Received by editor(s) in revised form: April 27, 2013, June 11, 2013, and May 5, 2014
Published electronically: May 15, 2015
Article copyright: © Copyright 2015 American Mathematical Society