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.

本文提出一种新的梯度下降类型算法来解决一般的非凸和非光滑规则化的反问题，并开发了一种神经网络架构来从训练数据中自适应地学习非线性稀疏转换，该算法可以适应非凸结构的拓扑变换。数值结果表明，所提出的神经网络在各种不同的图像重建问题上的准确性和效率方面优于最先进的方法。