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2016

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

4 years 3 months ago
Eventual linear convergence of the Douglas-Rachford iteration for basis pursuit
We provide a simple analysis of the Douglas-Rachford splitting algorithm in the context of 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 2 -regularization and over-relaxation including the dual split Bregman method [24] are also treated. We make no attempt at characterizing the transient regime preceding the onset of linear convergence. Acknowledgments: The authors are grateful to Jalal Fadili, Stanley Osher, Gabriel Peyr´e, Ming Yan, Yi Yang and Wotao Yin for discussions on modern methods of optimization that were very instructive to us. The authors are supported by the National Science Foundation and the Alfred P. Sloan Foundation.
Laurent Demanet, Xiangxiong Zhang
Added 08 Apr 2016
Updated 08 Apr 2016
Type Journal
Year 2016
Where MOC
Authors Laurent Demanet, Xiangxiong Zhang
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