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

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- by Laurent Demanet and Xiangxiong Zhang PDF
- Math. Comp.
**85**(2016), 209-238 Request permission

## 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.## References

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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
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp.
**85**(2016), 209-238 - MSC (2010): Primary 49M29, 65K10, 90C25
- DOI: https://doi.org/10.1090/mcom/2965
- MathSciNet review: 3404448