Phase Retrieval by Alternating Minimization With Random Initialization

We consider the phase retrieval problem, where the goal is to reconstruct an n-dimensional complex vector from its phaseless scalar products with m sensing vectors, independently sampled from complex normal distributions. We show that, if m ≥ Mn^3/2 log^7/2 n for some M > 0, then the classical algorithm of alternating minimization with random initialization succeeds with high probability as n, m → ∞. This is a step toward proving the conjecture in, which conjectures that the algorithm succeeds when m = O(n). The analysis depends on an approach that enables the decoupling of the dependency between the algorithmic iterates and the sensing vectors.