The recovery of complex sparse signals from few phaseless measurements

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 $m\geq O(k\log (n/k))$ 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.