  Recovering an unknown complex signal from the magnitude of linear
combinations of the signal is referred to as phase retrieval. We present an
exact performance analysis of a recently proposed
convex-optimization-formulation for this problem, known as PhaseMax. Standard
convex-relaxation-based methods in phase retrieval resort to the idea of
"lifting" which makes them computationally inefficient, since the number of
unknowns is effectively squared. In contrast, PhaseMax is a novel convex
relaxation that does not increase the number of unknowns. Instead it relies on
an initial estimate of the true signal which must be externally provided. In
this paper, we investigate the required number of measurements for exact
recovery of the signal in the large system limit and when the linear
measurement matrix is random with iid standard normal entries. If $n$ denotes
the dimension of the unknown complex signal and $m$ the number of phaseless
measurements, then in the large system limit, $\frac{m}{n} >
\frac{4}{\cos^2(\theta)}$ measurements is necessary and sufficient to recover
the signal with high probability, where $\theta$ is the angle between the
initial estimate and the true signal. Our result indicates a sharp phase
transition in the asymptotic regime which matches the empirical result in
numerical simulations.
