Multidimensional inverse problems in imaging and identification using low-complexity models, optimal mass transport, and machine learning

Abstract: This thesis, which mainly consists of six appended papers, primarily considers a number of inverse problems in imaging and system identification.In particular, the first two papers generalize results for the rational covariance extension problem from one to higher dimensions. The rational covariance extension problem stems from system identification and can be formulated as a trigonometric moment problem, but with a complexity constraint on the sought measure. The papers investigate a solution method based on varia tional regularization and convex optimization. We prove the existence and uniqueness of a solution to the variational problem, both when enforcing exact moment matching and when considering two different versions of approximate moment matching. A number of related questions are also considered, such as well-posedness, and the theory is illustrated with a number of examples.The third paper considers the maximum delay margin problem in robust control: To find the largest time delay in a feedback loop for a linear dynamical system so that there still exists a single controller that stabilizes the system for all delays smaller than or equal to this time delay. A sufficient condition for robust stabilization is recast as an analytic interpolation problem, which leads to an algorithm for computing a lower bound on the maximum delay margin. The algorithm is based on bisection, where positive semi-definiteness of a Pick matrix is used as selection criteria.Paper four investigate the use of optimal transport as a regularizing functional to incorporate prior information in variational formulations for image reconstruction. This is done by observing that the so-called Sinkhorn iterations, which are used to solve large scale optimal transport problems, can be seen as coordinate ascent in a dual optimization problem. Using this, we extend the idea of Sinkhorn iterations and derive a iterative algorithm for computing the proximal operator. This allows us to solve large-scale convex optimization problems that include an optimal transport term.In paper five, optimal transport is used as a loss function in machine learning for inverse problems in imaging. This is motivated by noise in the training data which has a geometrical characteristic. We derive theoretical results that indicate that optimal transport is better at compensating for this type of noise, compared to the standard 2-norm, and the effect is demonstrated in a numerical experiment.The sixth paper considers using machine learning techniques for solving large-scale convex optimization problems. We first parametrizes a family of algorithms, from which a new optimization algorithm is derived. Then we apply machine learning techniques to learn optimal parameters for given families of optimization problems, while imposing a fixed number of iterations in the scheme. By constraining the parameters appropriately, this gives learned optimization algorithms with provable convergence.