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Verifiable conditions of ℓ 1-recovery for sparse signals with sign restrictions

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Abstract

We propose necessary and sufficient conditions for a sensing matrix to be “s-semigood” – to allow for exact ℓ 1-recovery of sparse signals with at most s nonzero entries under sign restrictions on part of the entries. We express error bounds for imperfect ℓ 1-recovery in terms of the characteristics underlying these conditions. These characteristics, although difficult to evaluate, lead to verifiable sufficient conditions for exact sparse ℓ 1-recovery and thus efficiently computable upper bounds on those s for which a given sensing matrix is s-semigood. We examine the properties of proposed verifiable sufficient conditions, describe their limits of performance and provide numerical examples comparing them with other verifiable conditions from the literature.

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Authors and Affiliations

  1. LJK, Université J. Fourier, B.P. 53, 38041, Grenoble Cedex 9, France

    Anatoli Juditsky

  2. Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Fatma Kılınç Karzan & Arkadi Nemirovski

Authors
  1. Anatoli Juditsky
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  2. Fatma Kılınç Karzan
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  3. Arkadi Nemirovski
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Correspondence to Anatoli Juditsky.

Additional information

Research of the second and the third authors was supported by the Office of Naval Research grant # N000140811104.

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Juditsky, A., Kılınç Karzan, F. & Nemirovski, A. Verifiable conditions of ℓ 1-recovery for sparse signals with sign restrictions. Math. Program. 127, 89–122 (2011). https://doi.org/10.1007/s10107-010-0418-y

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  • DOI: https://doi.org/10.1007/s10107-010-0418-y

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