Phase retrieval with random Gaussian sensing vectors by alternating projections

We consider a phase retrieval problem, where we want to reconstruct a $n$-dimensional vector from its phaseless scalar products with $m$ sensing vectors, independently sampled from complex normal distributions. We show that, with a suitable initalization procedure, the classical algorithm of alternating projections succeeds with high probability when $m\geq Cn$, for some $C>0$. We conjecture that this result is still true when no special initialization procedure is used, and present numerical experiments that support this conjecture.