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Analysis ℓ 1-recovery with Frames and Gaussian Measurements

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Abstract

This paper provides novel results for the recovery of signals from undersampled measurements based on analysis ℓ 1-minimization, when the analysis operator is given by a frame. We both provide so-called uniform and nonuniform recovery guarantees for cosparse (analysis-sparse) signals using Gaussian random measurement matrices. The nonuniform result relies on a recovery condition via tangent cones and the uniform recovery guarantee is based on an analysis version of the null space property. Examining these conditions for Gaussian random matrices leads to precise bounds on the number of measurements required for successful recovery. In the special case of standard sparsity, our result improves a bound due to Rudelson and Vershynin concerning the exact reconstruction of sparse signals from Gaussian measurements with respect to the constant and extends it to stability under passing to approximately sparse signals and to robustness under noise on the measurements.

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Acknowledgements

M. Kabanava and H. Rauhut acknowledge support by the Hausdorff Center for Mathematics, University of Bonn, and by the European Research Council through the grant StG 258926.

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  1. Lehrstuhl C für Mathematik (Analysis), RWTH Aachen University, Templergraben 55, 52062, Aachen, Germany

    Maryia Kabanava & Holger Rauhut

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  1. Maryia Kabanava
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  2. Holger Rauhut
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Correspondence to Maryia Kabanava.

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Kabanava, M., Rauhut, H. Analysis ℓ 1-recovery with Frames and Gaussian Measurements. Acta Appl Math 140, 173–195 (2015). https://doi.org/10.1007/s10440-014-9984-y

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  • DOI: https://doi.org/10.1007/s10440-014-9984-y

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