  Recently, deep unfolding methods that guide the design of deep neural
networks (DNNs) through iterative algorithms have received increasing attention
in the field of inverse problems. Unlike general end-to-end DNNs, unfolding
methods have better interpretability and performance. However, to our
knowledge, their accuracy and stability in solving inverse problems cannot be
fully guaranteed. To bridge this gap, we modified the training procedure and
proved that the unfolding method is an iterative regularization method. More
precisely, we jointly learn a convex penalty function adversarially by an
input-convex neural network (ICNN) to characterize the distance to a real data
manifold and train a DNN unfolded from the proximal gradient descent algorithm
with this learned penalty. Suppose the real data manifold intersects the
inverse problem solutions with only the unique real solution. We prove that the
unfolded DNN will converge to it stably. Furthermore, we demonstrate with an
example of MRI reconstruction that the proposed method outperforms conventional
unfolding methods and traditional regularization methods in terms of
reconstruction quality, stability and convergence speed.
