  We propose a general learning based framework for solving nonsmooth and
nonconvex image reconstruction problems. We model the regularization function
as the composition of the $l_{2,1}$ norm and a smooth but nonconvex feature
mapping parametrized as a deep convolutional neural network. We develop a
provably convergent descent-type algorithm to solve the nonsmooth nonconvex
minimization problem by leveraging the Nesterov's smoothing technique and the
idea of residual learning, and learn the network parameters such that the
outputs of the algorithm match the references in training data. Our method is
versatile as one can employ various modern network structures into the
regularization, and the resulting network inherits the guaranteed convergence
of the algorithm. We also show that the proposed network is parameter-efficient
and its performance compares favorably to the state-of-the-art methods in a
variety of image reconstruction problems in practice.
