Convergent Regularization

Convergent regularization aims to develop mathematically sound methods for solving ill-posed inverse problems, particularly in image reconstruction, by incorporating learned regularizers. Current research focuses on establishing convergence guarantees for various algorithms, including those based on invertible residual networks (iResNets), primal-dual methods with weakly convex regularizers, and plug-and-play (PnP) schemes with linear denoisers. This work is crucial for ensuring the reliability and robustness of data-driven approaches in sensitive applications like medical imaging, where theoretical understanding is paramount for trustworthy results. The development of provably convergent methods bridges the gap between the empirical success of deep learning and the need for rigorous mathematical justification.