A dual split Bregman method for fast ℓ1 minimization

In this paper we propose a new algorithm for fast l1 minimization as frequently arising in compressed sensing. Our method is based on a split Bregman algorithm applied to the dual of the problem of minimizing ∥u∥1 + 1 2α ∥u∥2 such that u solves the under determined linear system Au = f , which was recently investigated in the context of linearized Bregman methods. Furthermore, we provide a convergence analysis for split Bregman methods in general and show with our compressed sensing example that a split Bregman approach to the primal energy can lead to a different type of convergence than split Bregman applied to the dual, thus making the analysis of different ways to minimize the same energy interesting for a wide variety of optimization problems.

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