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We consider a recently proposed convex formulation, known as the PhaseMaxmethod, for solving the phase retrieval problem. Using the replica method fromstatistical mechanics, we analyze the performance of PhaseMax in thehigh-dimensional limit. Our analysis predicts the \emph{exact} asymptoticperformance of PhaseMax. In particular, we show that a sharp phase transitionphenomenon takes place, with a simple analytical formula characterizing thephase transition boundary. This result shows that the oversampling ratiorequired by existing performance bounds in the literature can be significantlyreduced. Numerical results confirm the validity of our replica analysis,showing that the theoretical predictions are in excellent agreement with theactual performance of the algorithm, even for moderate signal dimensions.

Fundamental Limits of PhaseMax for Phase Retrieval: A Replica Analysis