  Optimization algorithms for solving nonconvex inverse problem have attracted
significant interests recently. However, existing methods require the nonconvex
regularization to be smooth or simple to ensure convergence. In this paper, we
propose a novel gradient descent type algorithm, by leveraging the idea of
residual learning and Nesterov's smoothing technique, to solve inverse problems
consisting of general nonconvex and nonsmooth regularization with provable
convergence. Moreover, we develop a neural network architecture intimating this
algorithm to learn the nonlinear sparsity transformation adaptively from
training data, which also inherits the convergence to accommodate the general
nonconvex structure of this learned transformation. Numerical results
demonstrate that the proposed network outperforms the state-of-the-art methods
on a variety of different image reconstruction problems in terms of efficiency
and accuracy.
