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Dictionary-Sparse Recovery via Thresholding-Based Algorithms

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Abstract

It is shown that the iterative hard thresholding and hard thresholding pursuit algorithms provide the same theoretical guarantees as \(\ell _1\)-minimization for the recovery from imperfect compressive measurements of signals that have almost sparse analysis expansions in a fixed dictionary. Unlike other signal space algorithms targeting the recovery of signals with sparse synthesis expansions, the ability to compute (near) best approximations by synthesis-sparse signals is not necessary. The results are first established for tight frame dictionaries, before being extended to arbitrary dictionaries modulo an adjustment of the measurement process.

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Notes

  1. Meaning that a bound on \(\Vert \mathbf{D}^* \mathbf{f}- \mathbf{D}^* \mathbf{f}^\sharp \Vert _p\) was established—for \(p=2\) and for tight frames, it reduces to the bound (1) on \(\Vert \mathbf{f}- \mathbf{f}^\sharp \Vert _2\).

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Acknowledgments

Simon Foucart is partially supported by NSF Grant number DMS-1120622.

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  1. University of Georgia, Athens, GA, USA

    Simon Foucart

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  1. Simon Foucart
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Correspondence to Simon Foucart.

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Communicated by Roman Vershynin.

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Foucart, S. Dictionary-Sparse Recovery via Thresholding-Based Algorithms. J Fourier Anal Appl 22, 6–19 (2016). https://doi.org/10.1007/s00041-015-9411-4

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  • DOI: https://doi.org/10.1007/s00041-015-9411-4

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