  Deep learning models have gained great success in many real-world
applications. However, most existing networks are typically designed in
heuristic manners, thus lack of rigorous mathematical principles and
derivations. Several recent studies build deep structures by unrolling a
particular optimization model that involves task information. Unfortunately,
due to the dynamic nature of network parameters, their resultant deep
propagation networks do \emph{not} possess the nice convergence property as the
original optimization scheme does. This paper provides a novel proximal
unrolling framework to establish deep models by integrating experimentally
verified network architectures and rich cues of the tasks. More importantly, we
\emph{prove in theory} that 1) the propagation generated by our unrolled deep
model globally converges to a critical-point of a given variational energy, and
2) the proposed framework is still able to learn priors from training data to
generate a convergent propagation even when task information is only partially
available. Indeed, these theoretical results are the best we can ask for,
unless stronger assumptions are enforced. Extensive experiments on various
real-world applications verify the theoretical convergence and demonstrate the
effectiveness of designed deep models.
