A dual split Bregman method for fast $\ell ^1$ minimization

Abstract

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. ©2013 American Mathematical Society.