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Optimization algorithms for solving nonconvex inverse problem have attractedsignificant interests recently. However, existing methods require the nonconvexregularization to be smooth or simple to ensure convergence. In this paper, wepropose a novel gradient descent type algorithm, by leveraging the idea ofresidual learning and Nesterov's smoothing technique, to solve inverse problemsconsisting of general nonconvex and nonsmooth regularization with provableconvergence. Moreover, we develop a neural network architecture intimating thisalgorithm to learn the nonlinear sparsity transformation adaptively fromtraining data, which also inherits the convergence to accommodate the generalnonconvex structure of this learned transformation. Numerical resultsdemonstrate that the proposed network outperforms the state-of-the-art methodson a variety of different image reconstruction problems in terms of efficiencyand accuracy.

A Novel Learnable Gradient Descent Type Algorithm for Non-convex Non-smooth Inverse Problems