Data-Consistent Learning of Inverse Problems
Markus Haltmeier, Gyeongha Hwang
TL;DR
The paper addresses ill-posed linear inverse problems by integrating data-consistent learning with classical regularization. It introduces null-space networks that constrain learned corrections to the forward operator's null space, ensuring data fidelity while enabling a two-step reconstruction $g_{\theta,\delta} = f_\theta \circ \mathcal{B}_\delta$ to be a convergent regularization method. The authors prove convergence and derive rates under standard regularization theory, leveraging a regularizing filter $g_\alpha$ and appropriate parameter choices, and illustrate the approach with a numerical example showing robustness to distribution shifts. This framework bridges traditional inverse problem theory and modern data-driven methods, offering reliable reconstructions that benefit from learned priors without sacrificing measurement-consistency.
Abstract
Inverse problems are inherently ill-posed, suffering from non-uniqueness and instability. Classical regularization methods provide mathematically well-founded solutions, ensuring stability and convergence, but often at the cost of reduced flexibility or visual quality. Learned reconstruction methods, such as convolutional neural networks, can produce visually compelling results, yet they typically lack rigorous theoretical guarantees. DC (DC) networks address this gap by enforcing the measurement model within the network architecture. In particular, null-space networks combined with a classical regularization method as an initial reconstruction define a convergent regularization method. This approach preserves the theoretical reliability of classical schemes while leveraging the expressive power of data-driven learning, yielding reconstructions that are both accurate and visually appealing.