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New Conditions on Stable Recovery of Weighted Sparse Signals via Weighted \(l_1\) Minimization

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Abstract

A problem of recovering weighted sparse signals via weighted \(l_1\) minimization has recently drawn considerable attention with application to function interpolation. The weighted robust null space property (NSP) of order s and the weighted restricted isometry property (RIP) with the weighted 3s-RIP constant \(\delta _{\mathbf {w},3s}\) have been proposed and proved to be sufficient conditions for guaranteeing stable recovery of weighted s-sparse signals. In this paper, we propose two new sufficient conditions, i.e., the weighted \(l_q\)-robust NSP of order s and the weighted RIP with \(\delta _{\mathbf {w},2s}\). Different from the existing results, the weighted \(l_q\)-robust NSP of order s is more general and weaker than the weighted robust NSP of order s, and the weighted RIP is characterized by \(\delta _{\mathbf {w},2s}\) instead of \(\delta _{\mathbf {w},3s}\). Accordingly, the reconstruction error estimations based on the newly proposed recovery conditions are also derived, respectively. Moreover, we demonstrate that the weighted RIP with small \(\delta _{\mathbf {w},2s}\) implies the weighted \(l_1\)-robust NSP of order s.

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Acknowledgements

The authors thank the referees very much for carefully reading the paper and for elaborate and valuable suggestions.

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Authors and Affiliations

  1. Department of Mathematics, School of Sciences, Nanchang University, Nanchang, 330031, Jiangxi, China

    Haiye Huo

  2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, China

    Wenchang Sun

  3. Department of Electrical and Computer Engineering, University of Delaware, Newark, DE, 19716, USA

    Li Xiao

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  1. Haiye Huo
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  2. Wenchang Sun
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  3. Li Xiao
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Corresponding author

Correspondence to Wenchang Sun.

Additional information

This work was partially supported by the National Natural Science Foundation of China (11371200, 11525104 and 11531013).

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Huo, H., Sun, W. & Xiao, L. New Conditions on Stable Recovery of Weighted Sparse Signals via Weighted \(l_1\) Minimization. Circuits Syst Signal Process 37, 2866–2883 (2018). https://doi.org/10.1007/s00034-017-0691-6

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  • DOI: https://doi.org/10.1007/s00034-017-0691-6

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