Learning Difference-of-Convex Regularizers for Inverse Problems: A Flexible Framework with Theoretical Guarantees

TL;DR

This paper introduces a flexible framework for inverse problems using difference-of-convex regularizers, combining empirical performance with strong theoretical guarantees and enabling the use of specialized optimization algorithms.

Contribution

It extends neural network-based regularizers to the broader class of DC functions, providing improved empirical results and convergence guarantees for inverse problem solutions.

Findings

Outperforms existing regularizers in CT reconstruction tasks.

Enables use of DCA and PSM algorithms for better optimization.

Provides theoretical conditions for expressing optimal regularizers as DC functions.

Abstract

Learning effective regularization is crucial for solving ill-posed inverse problems, which arise in a wide range of scientific and engineering applications. While data-driven methods that parameterize regularizers using deep neural networks have demonstrated strong empirical performance, they often result in highly nonconvex formulations that lack theoretical guarantees. Recent work has shown that incorporating structured nonconvexity into neural network-based regularizers, such as weak convexity, can strike a balance between empirical performance and theoretical tractability. In this paper, we demonstrate that a broader class of nonconvex functions, difference-of-convex (DC) functions, can yield improved empirical performance while retaining strong convergence guarantees. The DC structure enables the use of well-established optimization algorithms, such as the Difference-of-Convex Algorithm (DCA) and a Proximal Subgradient Method (PSM), which extend beyond standard gradient descent. Furthermore, we provide theoretical insights into the conditions under which optimal regularizers can be expressed as DC functions. Extensive experiments on computed tomography (CT) reconstruction tasks show that our approach achieves strong performance across sparse and limited-view settings, consistently outperforming other weakly supervised learned regularizers. Our code is available at \url{https://github.com/YasminZhang/ADCR}.