Learning Distributionally Robust Solutions for Inverse Problems using the Wasserstein Distance

This thesis proposes a novel data-driven framework that integrates Wasserstein robustness into inverse problem modeling. It aims to bridge the research fields of inverse problems and Wasserstein robustness, providing insights into the relationship between regularization and robustness, which are critical for developing stable and reliable solutions in various applications. We introduce a general framework to find distributionally robust solutions and prove a new strong duality result that can be modified to be applied in many fields. For an academic impact case, the dual representation of an inverse problem robust to Gaussian noise in the measurement space is further explored and reduced to a convex, finite-dimensional problem, making it computationally tractable. The framework is validated through numerical simulations, demonstrating that it can learn solutions for inverse problems that are robust to perturbations in the Wasserstein distance. This work expands the theoretical foundations of both distributionally robust optimization and inverse problems and is applicable to various other types of problems. The developed framework holds promise for practical applications in diverse fields, including more complex inverse problems and higher-dimensional applications.