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 ` 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 `-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 Peyré, 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.