We study the stable recovery of complex k-sparse signals from as few phaseless measurements as possible. The main result is to show that one can employ ${\ell }_1$ minimization to stably recover complex k-sparse signals from $m\ge O(k\log (n/k))$ complex Gaussian random quadratic measurements with high probability. To do that, we establish that Gaussian random measurements satisfy the restricted isometry property over rank-2 and sparse matrices with high probability. This paper presents the first theoretical estimation of the measurement number for stably recovering complex sparse signals from complex Gaussian quadratic measurements.

中文翻译：

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从少量无相位测量中恢复复杂的稀疏信号

我们从尽可能少的无相位测量中研究了复杂k稀疏信号的稳定恢复。主要结果是表明一个人可以雇用${\ell }_{\mathrm{1个}}$最小化以稳定地从中恢复复杂的k稀疏信号$米\ge Ø（ķ\mathrm{日志}（ñ/ķ））$高概率的复杂高斯随机二次测量。为此，我们确定高斯随机度量满足秩2和稀疏矩阵的受限等距特性，并且概率很高。本文提出了用于从稳定的高斯二次测量中稳定恢复出复杂稀疏信号的测量数量的第一个理论估计。