  We consider the recovery of a (real- or complex-valued) signal from
magnitude-only measurements, known as phase retrieval. We formulate phase
retrieval as a convex optimization problem, which we call PhaseMax. Unlike
other convex methods that use semidefinite relaxation and lift the phase
retrieval problem to a higher dimension, PhaseMax is a "non-lifting" relaxation
that operates in the original signal dimension. We show that the dual problem
to PhaseMax is Basis Pursuit, which implies that phase retrieval can be
performed using algorithms initially designed for sparse signal recovery. We
develop sharp lower bounds on the success probability of PhaseMax for a broad
range of random measurement ensembles, and we analyze the impact of measurement
noise on the solution accuracy. We use numerical results to demonstrate the
accuracy of our recovery guarantees, and we showcase the efficacy and limits of
PhaseMax in practice.
