  We consider a phase retrieval problem, where we want to reconstruct a
$n$-dimensional vector from its phaseless scalar products with $m$ sensing
vectors. We assume the sensing vectors to be independently sampled from complex
normal distributions. We propose to solve this problem with the classical
non-convex method of alternating projections. We show that, when $m\geq Cn$ for
$C$ large enough, alternating projections succeed with high probability,
provided that they are carefully initialized. We also show that there is a
regime in which the stagnation points of the alternating projections method
disappear, and the initialization procedure becomes useless. However, in this
regime, $m$ has to be of the order of $n^2$. Finally, we conjecture from our
numerical experiments that, in the regime $m=O(n)$, there are stagnation
points, but the size of their attraction basin is small if $m/n$ is large
enough, so alternating projections can succeed with probability close to $1$
even with no special initialization.
