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 initialization procedure, the classical algorithm of alternating projections (Gerchberg-Saxton) succeeds with high probability when m ≥ 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.