  We outline new approaches to incorporate ideas from deep learning into
wave-based least-squares imaging. The aim, and main contribution of this work,
is the combination of handcrafted constraints with deep convolutional neural
networks, as a way to harness their remarkable ease of generating natural
images. The mathematical basis underlying our method is the
expectation-maximization framework, where data are divided in batches and
coupled to additional "latent" unknowns. These unknowns are pairs of elements
from the original unknown space (but now coupled to a specific data batch) and
network inputs. In this setting, the neural network controls the similarity
between these additional parameters, acting as a "center" variable. The
resulting problem amounts to a maximum-likelihood estimation of the network
parameters when the augmented data model is marginalized over the latent
variables.
