Generative models for inverse problems

Inverse problems are ubiquitous in scientific and engineering applications, where the goal is to recover an unknown signal from indirect and often corrupted observations. These problems are typically ill posed, requiring the incorporation of prior information to ensure stable and meaningful solutions. In recent years, deep generative models have emerged as powerful tools for encoding complex data distributions, offering a promising alternative to classical regularization techniques. This thesis investigates the use of generative models for solving inverse problems, with a particular focus on medical imaging applications and theoretical advancements in generative sensing. The work is structured into two main parts. In the first part, we address the problem of motion artifact correction in brain Magnetic Resonance Imaging (MRI). We evaluate two primary applications of diffusion models: deploying them for direct motion correction, and utilizing them for data augmentation to generate synthetic pairs for supervised training. Through extensive numerical experiments on real MRI datasets, we quantitatively and qualitatively outline the strengths and weaknesses of these methods, while highlighting important considerations regarding model reliability and clinical applicability. The second part focuses on a theoretical extension of generative approaches to inverse problems. We introduce an infinite dimensional formulation of generative compressed sensing, where signals are assumed to lie in the range of a generative operator rather than being sparse in some predefined basis. Within this framework, we establish recovery guarantees and analyze the role of the sampling strategy, showing how adaptive measurements informed by the generative model can significantly improve reconstruction performance. Finally we perform numerical experiments, including applications to function space, to support the theoretical findings. Overall, this thesis demonstrates that generative models provide a flexible and effective framework for tackling inverse problems, bridging the gap between practical applications and theoretical understanding, and opening new directions for data driven reconstruction methods.