This paper provides theoretical guarantees for denoising performance of greedy-like methods. Those include Compressive Sampling Matching Pursuit (CoSaMP), Subspace Pursuit (SP), and Iterative Hard Thresholding (IHT). Our results show that the denoising obtained with these algorithms is a constant and a log-factor away from the oracle's performance, if the signal's representation is sufficiently sparse. Turning to practice, we show how to convert these algorithms to work without knowing the target cardinality, and instead constrain the solution to an error-budget. Denoising tests on synthetic data and image patches show the potential in this stagewise technique as a replacement of the classical OMP.
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