A primal-dual algorithm for image reconstruction with input-convex neural network regularizers
This addresses the challenge of efficiently optimizing non-smooth, nested neural network regularizers in image reconstruction, offering a practical solution for researchers in computational imaging.
The paper tackles the optimization problem in data-driven variational image reconstruction with input-convex neural network regularizers, which suffer from slow convergence and complexity due to non-smoothness and nested structures. The result is a reformulation into a convex problem solved via a primal-dual algorithm, showing improved performance over subgradient and accelerated methods in imaging tasks.
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.