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. 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≥ 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 n2. 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.