  In this paper, we study the phase retrieval problem in the situation where
the vector to be recovered has an a priori structure that can encoded into a
regularization term. This regularizer is intended to promote solutions
conforming to some notion of simplicity or low complexity. We investigate both
noiseless recovery and stability to noise and provide a very general and
unified analysis framework that goes far beyond the sparse phase retrieval
mostly considered in the literature. In the noiseless case we provide
sufficient conditions under which exact recovery, up to global sign change, is
possible. For Gaussian measurement maps, we also provide a sample complexity
bound for exact recovery. This bound depends on the Gaussian width of the
descent cone at the soughtafter vector which is a geometric measure of the
complexity of the latter. In the noisy case, we consider both the constrained
(Mozorov) and penalized (Tikhonov) formulations. We provide sufficient
conditions for stable recovery and prove linear convergence for sufficiently
small noise. For Gaussian measurements, we again give a sample complexity bound
for linear convergence to hold with high probability. This bound scales
linearly in the intrinsic dimension of the sought-after vector but only
logarithmically in the ambient dimension.
