Convergence of the linearized Bregman iteration for ℓ₁-norm minimization

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

doi:10.1090/S0025-5718-09-02242-X

Convergence of the linearized Bregman iteration for $\ell _1$-norm minimization
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by Jian-Feng Cai, Stanley Osher and Zuowei Shen;

Math. Comp. 78 (2009), 2127-2136

DOI: https://doi.org/10.1090/S0025-5718-09-02242-X

Published electronically: March 6, 2009

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Abstract:

One of the key steps in compressed sensing is to solve the basis pursuit problem $\min _{u\in \mathbb {R}^n}\{\|u\|_1:Au=f\}$. Bregman iteration was very successfully used to solve this problem in [40]. Also, a simple and fast iterative algorithm based on linearized Bregman iteration was proposed in [40], which is described in detail with numerical simulations in [35]. A convergence analysis of the smoothed version of this algorithm was given in [11]. The purpose of this paper is to prove that the linearized Bregman iteration proposed in [40] for the basis pursuit problem indeed converges.

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