  We propose a robust and efficient approach to the problem of compressive
phase retrieval in which the goal is to reconstruct a sparse vector from the
magnitude of a number of its linear measurements. The proposed framework relies
on constrained sensing vectors and a two-stage reconstruction method that
consists of two standard convex programs that are solved sequentially.
  In recent years, various methods are proposed for compressive phase
retrieval, but they have suboptimal sample complexity or lack robustness
guarantees. The main obstacle has been that there is no straightforward convex
relaxations for the type of structure in the target. Given a set of
underdetermined measurements, there is a standard framework for recovering a
sparse matrix, and a standard framework for recovering a low-rank matrix.
However, a general, efficient method for recovering a jointly sparse and
low-rank matrix has remained elusive.
  Deviating from the models with generic measurements, in this paper we show
that if the sensing vectors are chosen at random from an incoherent subspace,
then the low-rank and sparse structures of the target signal can be effectively
decoupled. We show that a recovery algorithm that consists of a low-rank
recovery stage followed by a sparse recovery stage will produce an accurate
estimate of the target when the number of measurements is
$\mathsf{O}(k\,\log\frac{d}{k})$, where $k$ and $d$ denote the sparsity level
and the dimension of the input signal. We also evaluate the algorithm through
numerical simulation.
