Linearized Bregman iterations for compressed sensing
- 1 September 2009
- journal article
- research article
- Published by American Mathematical Society (AMS) in Mathematics of Computation
- Vol. 78 (267), 1515-1536
- https://doi.org/10.1090/s0025-5718-08-02189-3
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 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 norm solution of as shown by numerical simulations. Hence, it can be used as another choice of an efficient tool in compressed sensing.This publication has 25 references indexed in Scilit:
- A framelet-based image inpainting algorithmApplied and Computational Harmonic Analysis, 2008
- Compressive samplingPublished by European Mathematical Society - EMS - Publishing House GmbH ,2007
- Sparsity and incoherence in compressive samplingInverse Problems, 2007
- Iteratively solving linear inverse problems under general convex constraintsInverse Problems & Imaging, 2007
- Regularization and Variable Selection Via the Elastic NetJournal of the Royal Statistical Society Series B: Statistical Methodology, 2005
- An iterative thresholding algorithm for linear inverse problems with a sparsity constraintCommunications on Pure and Applied Mathematics, 2004
- On sparse representation in pairs of basesIEEE Transactions on Information Theory, 2003
- A generalized uncertainty principle and sparse representation in pairs of basesIEEE Transactions on Information Theory, 2002
- De-noising by soft-thresholdingIEEE Transactions on Information Theory, 1995
- Robust Regression: Asymptotics, Conjectures and Monte CarloThe Annals of Statistics, 1973