  We propose a flexible convex relaxation for the phase retrieval problem that
operates in the natural domain of the signal. Therefore, we avoid the
prohibitive computational cost associated with "lifting" and semidefinite
programming (SDP) in methods such as PhaseLift and compete with recently
developed non-convex techniques for phase retrieval. We relax the quadratic
equations for phaseless measurements to inequality constraints each of which
representing a symmetric "slab". Through a simple convex program, our proposed
estimator finds an extreme point of the intersection of these slabs that is
best aligned with a given anchor vector. We characterize geometric conditions
that certify success of the proposed estimator. Furthermore, using classic
results in statistical learning theory, we show that for random measurements
the geometric certificates hold with high probability at an optimal sample
complexity. Phase transition of our estimator is evaluated through simulations.
Our numerical experiments also suggest that the proposed method can solve phase
retrieval problems with coded diffraction measurements as well.
