Learning Difference-of-Convex Regularizers for Inverse Problems: A Flexible Framework with Theoretical Guarantees
This work addresses the problem of balancing empirical performance with theoretical guarantees in learned regularizers for inverse problems, which is important for scientific and engineering applications, though it appears incremental by extending from weak convexity to DC functions.
The paper tackles the challenge of learning effective regularizers for ill-posed inverse problems by proposing a framework based on difference-of-convex (DC) functions, which achieves strong performance in computed tomography reconstruction tasks and consistently outperforms other weakly supervised learned regularizers.
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}.