  We introduce a deep learning (DL) based network and an associated exact
recovery theory for imaging from intensity-only measurements. The network
architecture uses a recurrent structure that unrolls the Wirtinger Flow (WF)
algorithm with a deep decoding prior that enables performing the algorithm
updates in a lower dimensional encoded image space. We use a separate deep
network (DN), referred to as the encoding network, for transforming the
spectral initialization used in the WF algorithm to an appropriate initial
value for the encoded domain. The unrolling scheme models a fixed number of
iterations of the underlying optimization algorithm into a recurrent neural
network (RNN). Furthermore, it facilitates simultaneous learning of the
parameters of the decoding and encoding networks and the RNN. We establish a
sufficient condition to guarantee exact recovery under deterministic forward
models. Additionally, we demonstrate the relation between the Lipschitz
constants of the trained decoding prior and encoding networks to the
convergence rate of the WF algorithm. We show the practical applicability of
our method in synthetic aperture imaging using high fidelity simulation data
from the PCSWAT software. Our numerical study shows that the decoding prior and
the encoding network facilitate improvements in sample complexity.
