Analysis of Optimal Thresholding Algorithms for Compressed Sensing
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The optimal thresholding is a new technique that has recently been developed for compressed sensing and signal approximation. The optimal k-thresholding (OT) and optimal k-thresholding pursuit (OTP) provide basic algorithmic frameworks that are fundamental to the development of practical and efficient optimal-thresholding-type algorithms for compressed sensing. In this paper, we first prove an improved bound for the guaranteed performance of these basic frameworks in terms of the kth order restricted isometry property of the sensing matrices. More importantly, we analyze the newly developed algorithms called relaxed optimal thresholding (ROTω) and relaxed optimal thresholding pursuit (ROTPω) which are derived from the tightest convex relaxation of the OT and OTP. Numerical results in <cit.> have demonstrated that such approaches are truly efficient to recover a wide range of signals and can remarkably outperform the existing hard-thresholding-type methods as well as the classic ℓ_1-minimization in numerous situations. However, the guaranteed performance/convergence of these algorithms with ω≥ 2 has not yet established. The main purpose of this paper is to establish the first guaranteed performance results for the ROTω and ROTPω.
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