Deep unfolding as iterative regularization for imaging inverse problems

TL;DR

This paper introduces a novel deep unfolding approach as an iterative regularization method for inverse problems, enhancing stability and accuracy by jointly learning a convex penalty function and proving convergence guarantees.

Contribution

The authors propose a new training procedure for deep unfolding methods, demonstrating their stability and convergence as iterative regularization techniques with learned penalties.

Findings

Outperforms traditional methods in MRI reconstruction quality

Provides stability and convergence guarantees for unfolding methods

Jointly learns a convex penalty function using adversarial training

Abstract

Recently, deep unfolding methods that guide the design of deep neural networks (DNNs) through iterative algorithms have received increasing attention in the field of inverse problems. Unlike general end-to-end DNNs, unfolding methods have better interpretability and performance. However, to our knowledge, their accuracy and stability in solving inverse problems cannot be fully guaranteed. To bridge this gap, we modified the training procedure and proved that the unfolding method is an iterative regularization method. More precisely, we jointly learn a convex penalty function adversarially by an input-convex neural network (ICNN) to characterize the distance to a real data manifold and train a DNN unfolded from the proximal gradient descent algorithm with this learned penalty. Suppose the real data manifold intersects the inverse problem solutions with only the unique real solution. We prove that the unfolded DNN will converge to it stably. Furthermore, we demonstrate with an example of MRI reconstruction that the proposed method outperforms conventional unfolding methods and traditional regularization methods in terms of reconstruction quality, stability and convergence speed.