A primal-dual algorithm for image reconstruction with input-convex neural network regularizers

TL;DR

This paper introduces a primal-dual algorithm for image reconstruction that efficiently handles input-convex neural network regularizers by reformulating the problem into a convex optimization task, improving convergence and training.

Contribution

It proposes a novel reformulation of ICNN-based regularization problems, enabling the use of primal-dual methods for faster and more effective image reconstruction.

Findings

Outperforms subgradient and accelerated methods in experiments

Facilitates training of the neural network regularizer

Efficiently solves non-smooth optimization problems

Abstract

We address the optimization problem in a data-driven variational reconstruction framework, where the regularizer is parameterized by an input-convex neural network (ICNN). While gradient-based methods are commonly used to solve such problems, they struggle to effectively handle non-smooth problems which often leads to slow convergence. Moreover, the nested structure of the neural network complicates the application of standard non-smooth optimization techniques, such as proximal algorithms. To overcome these challenges, we reformulate the problem and eliminate the network's nested structure. By relating this reformulation to epigraphical projections of the activation functions, we transform the problem into a convex optimization problem that can be efficiently solved using a primal-dual algorithm. We also prove that this reformulation is equivalent to the original variational problem. Through experiments on several imaging tasks, we show that the proposed approach not only outperforms subgradient methods and even accelerated methods in the smooth setting, but also facilitates the training of the regularizer itself.