Journal
IEEE TRANSACTIONS ON SIGNAL PROCESSING
Volume 62, Issue 7, Pages 1762-1774Publisher
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TSP.2014.2304932
Keywords
Analysis model; convergence analysis; fast iterative shrinkage-thresholding algorithm; restricted isometry property; smoothing and decomposition; sparse recovery
Categories
Funding
- AFOSR [FA9550-11-1-0210]
- NSF [CCF-1014908]
- ONR [N000141310050]
- Israel Science Foundation [170/10, 253/12]
- Ollendorf Foundation
- Israel Ministry of Industry and Trade [Metro450]
- Intel Collaborative Research Institute for Computational Intelligence (ICRI-CI)
- Division of Computing and Communication Foundations
- Direct For Computer & Info Scie & Enginr [0963742] Funding Source: National Science Foundation
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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.
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