Abstract
The aim of this paper is to study the stability of the \(\ell _1\) minimization for the compressive phase retrieval and to extend the instance-optimality in compressed sensing to the real phase retrieval setting. We first show that \(m={\mathcal {O}}(k\log (N/k))\) measurements are enough to guarantee the \(\ell _1\) minimization to recover k-sparse signals stably provided the measurement matrix A satisfies the strong RIP property. We second investigate the phaseless instance-optimality presenting a null space property of the measurement matrix A under which there exists a decoder \(\Delta \) so that the phaseless instance-optimality holds. We use the result to study the phaseless instance-optimality for the \(\ell _1\) norm. This builds a parallel for compressive phase retrieval with the classical compressive sensing.
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Acknowledgments
Yang Wang was supported in part by the AFOSR grant FA9550-12-1-0455 and NSF grant IIS-1302285. Zhiqiang Xu was supported by NSFC grant (11171336, 11422113, 11021101, 11331012) and by National Basic Research Program of China (973 Program 2015CB856000).
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Communicated by Peter G. Casazza.
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Gao, B., Wang, Y. & Xu, Z. Stable Signal Recovery from Phaseless Measurements. J Fourier Anal Appl 22, 787–808 (2016). https://doi.org/10.1007/s00041-015-9434-x
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DOI: https://doi.org/10.1007/s00041-015-9434-x