Provably Convergent Proximal Gradient Methods with Tailored Input Convex Neural Networks Regularization for Inverse Problems

Abstract

This study presents a novel Deep Proximal Gradient Descent framework for ill-posed problems by employing a tailored second-order differentiable Input-Convex Neural Networks (ICNNs) as a learned regularizer. A key contribution is the design of convex residual mapping, which preserves the convexity of the regularized objective, thereby enhancing the interpretability of the deep network without sacrificing its expressive power. Based on this framework, we develop two types of algorithms. For linear problems, the ICNN-based regularizer is embedded into the standard proximal gradient structure. For nonlinear problems, we introduce an innovative formulation that employs the learned residual to guide gradient descent, while using the traditional data misfit as a proximal regularizer to avoid network-dominated spurious solutions. Building on this iterative scheme, we establish groundbreaking convergence results for both algorithms, complete with rigorous proofs. Extensive numerical experiments, particularly on real low-dose Computed Tomography data, validate the superior imaging quality and high computational efficiency of our algorithms.