A Precise Analysis of PhaseMax in Phase Retrieval

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.