Learned regularisation for inverse problems

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Solving inverse problems, such as those that arise in imaging applications (e.g., computed tomography), is a challenging task due to their inherent ill-posedness and high dimensionality. Traditionally, one solves them using model-based methods, with variational regularisation models involving a fixed regulariser being the most popular. Recently, more data-driven approaches for determining regularisation operators and regularisation parameters have been considered. The goal of this research is to develop new solvers for inverse problems: these new strategies should attain the good empirical results commonly witnessed from machine learning methods, while maintaining theoretical underpinnings associated to the traditional model-based methods.One such approach that attempts to combine machine learning and model-based methods is the so-called bi-level learning problem, wherein one seeks parameters that minimise a loss function (classically a reconstruction error) of a training set, subject to the reconstructions solving an inverse problem - and being dependent on such parameters. Solving these problems is computationally expensive due to the inherent nested nature. This research will explore reformulations of the bi-level problem into a single level, saddle-point point problem, and derive theoretical results and efficient solvers for the reformulated problem when a large number of parameters are sought. Bi-level learning problems arise, for instance, when a neural network is employed to invert test data, and optimal network weights should be computed. Recent work has proposed training such a neural network independently to the solution of the considered inverse problem, decreasing the computational cost otherwise encountered. The single-level reformulation investigated within this PhD project may provide computational feasibility to the training of such a network without the need to decouple it from the solution of the considered inverse problems. Theoretical results regarding how a network trained using the decoupling approach compares to a network trained using the bi-level framework will be derived.