Mathematics of Computation

Author(s): Jian-Feng Cai; Stanley Osher; Zuowei Shen.
Journal: Math. Comp. 78 (2009), 1515-1536.
MSC (2000): Primary 65K05, 65F22; Secondary 65T99
Posted: October 22, 2008
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Abstract: Finding a solution of a linear equation

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

, while

and

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

as shown by numerical simulations. Hence, it can be used as another choice of an efficient tool in compressed sensing.

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Retrieve articles in Mathematics of Computation with MSC (2000): 65K05, 65F22, 65T99

Jian-Feng Cai
Affiliation: Temasek Laboratories, National University of Singapore, 2 Science Drive 2, Singapore 117543
Email: tslcaij@nus.edu.sg

Stanley Osher
Affiliation: Department of Mathematics, UCLA, 520 Portola Plaza, Los Angeles, California 90095
Email: sjo@math.ucla.edu

Zuowei Shen
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Email: matzuows@nus.edu.sg

DOI: 10.1090/S0025-5718-08-02189-3
PII: S 0025-5718(08)02189-3
Received by editor(s): February 27, 2008
Received by editor(s) in revised form: June 25, 2008
Posted: October 22, 2008
Additional Notes: The first author's research was supported by the Wavelets and Information Processing Programme under a grant from DSTA, Singapore.
The second author's research was partially supported by ONR grant N000140710810, and by the Department of Defense, USA
The third author's research was supported in part by Grant R-146-000-113-112 at the National University of Singapore.
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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