Smoothing and decomposition for analysis sparse recovery

Zhao Tan, Yonina C. Eldar, Amir Beck, Arye Nehorai

Research output: Contribution to journalArticlepeer-review

77 Scopus citations


We consider algorithms and recovery guarantees for the analysis sparse model in which the signal is sparse with respect to a highly coherent frame. We consider the use of a monotone version of the fast iterative shrinkage-thresholding algorithm (MFISTA) to solve the analysis sparse recovery problem. Since the proximal operator in MFISTA does not have a closed-form solution for the analysis model, it cannot be applied directly. Instead, we examine two alternatives based on smoothing and decomposition transformations that relax the original sparse recovery problem, and then implement MFISTA on the relaxed formulation. We refer to these two methods as smoothing-based and decomposition-based MFISTA. We analyze the convergence of both algorithms and establish that smoothing-based MFISTA converges more rapidly when applied to general nonsmooth optimization problems. We then derive a performance bound on the reconstruction error using these techniques. The bound proves that our methods can recover a signal sparse in a redundant tight frame when the measurement matrix satisfies a properly adapted restricted isometry property. Numerical examples demonstrate the performance of our methods and show that smoothing-based MFISTA converges faster than the decomposition-based alternative in real applications, such as MRI image reconstruction.

Original languageEnglish
Article number6733349
Pages (from-to)1762-1774
Number of pages13
JournalIEEE Transactions on Signal Processing
Issue number7
StatePublished - 1 Apr 2014
Externally publishedYes


FundersFunder number
National Science FoundationCCF-1014908
Office of Naval ResearchN000141310050
National Science Foundation
Directorate for Computer and Information Science and Engineering1014908


    • Analysis model
    • convergence analysis
    • fast iterative shrinkage-thresholding algorithm
    • restricted isometry property
    • smoothing and decomposition
    • sparse recovery


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