Linearized Bregman iterations for compressed sensing

Abstract

Finding a solution of a linear equation Au = f with various minimization properties arises from many applications. One such application is compressed sensing, where an efficient and robust-to-noise algorithm to find a minimal ℓ1norm solution is needed. This means that the algorithm should be tailored for large scale and completely dense matrices A, while Au and ATu can be computed by fast transforms and the solution we seek is sparse. Recently, a simple and fast algorithm based on linearized Bregman iteration was proposed in [28,32] for this purpose. This paper is to analyze the convergence of linearized Bregman iterations and the minimization properties of their limit. Based on our analysis here, we derive also a new algorithm that is proven to be convergent with a rate. Furthermore, the new algorithm is simple and fast in approximating a minimal ℓ1norm solution of Au = f as shown by numerical simulations. Hence, it can be used as another choice of an efficient tool in compressed sensing. © 2008 American Mathematical Society.