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We consider a phase retrieval problem, where the goal is to reconstruct a$n$-dimensional complex vector from its phaseless scalar products with $m$sensing vectors, independently sampled from complex normal distributions. Weshow that, with a random initialization, the classical algorithm of alternatingminimization succeeds with high probability as $n,m\rightarrow\infty$ when${m}/{\log^3m}\geq Mn^{3/2}\log^{1/2}n$ for some $M>0$. This is a step towardproving the conjecture in \cite{Waldspurger2016}, which conjectures that thealgorithm succeeds when $m=O(n)$. The analysis depends on an approach thatenables the decoupling of the dependency between the algorithmic iterates andthe sensing vectors.

Phase Retrieval by Alternating Minimization with Random Initialization