Linearized Bregman iterations for compressed sensing

Cai, Jian-Feng; Osher, Stanley; Shen, Zuowei

doi:10.1090/S0025-5718-08-02189-3

Linearized Bregman iterations for compressed sensing
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by Jian-Feng Cai, Stanley Osher and Zuowei Shen;

Math. Comp. 78 (2009), 1515-1536

DOI: https://doi.org/10.1090/S0025-5718-08-02189-3

Published electronically: October 22, 2008

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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 $\ell _1$ norm solution is needed. This means that the algorithm should be tailored for large scale and completely dense matrices $A$, while $Au$ and $A^Tu$ 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 $\ell _1$ norm solution of $Au=f$ as shown by numerical simulations. Hence, it can be used as another choice of an efficient tool in compressed sensing.

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