Learned regularisation for inverse problems

求解逆问题，诸如在成像应用中出现的那些逆问题（例如，计算机断层摄影）由于其固有的不适定性和高维性而成为一项具有挑战性的任务。传统上，人们使用基于模型的方法来解决它们，其中涉及固定正则化器的变分正则化模型是最流行的。最近，已经考虑了用于确定正则化算子和正则化参数的更多数据驱动的方法。本研究的目标是开发新的逆问题求解器：这些新的策略应该获得机器学习方法通常所能看到的良好的经验结果，同时保持与传统的基于模型的方法相关联的理论基础。一种尝试将联合收割机机器学习和基于模型的方法相结合的方法是所谓的双层学习问题，其中，寻求使训练集的损失函数（传统上是重构误差）最小化的参数，该参数受到求解逆问题的重构的影响，并且依赖于这些参数。由于固有的嵌套性质，解决这些问题的计算成本很高。本研究将探讨将双层问题转化为单层鞍点问题，并在寻求大量参数时推导出理论结果和有效的求解方法。例如，当采用神经网络来反演测试数据时，会出现双层学习问题，并且应该计算最佳网络权重。最近的工作提出了独立于所考虑的逆问题的解决方案来训练这样的神经网络，从而降低了否则遇到的计算成本。在这个博士项目中研究的单级重构可以为这样一个网络的训练提供计算可行性，而不需要将其与所考虑的逆问题的解决方案解耦。将得出关于使用解耦方法训练的网络与使用双层框架训练的网络相比如何的理论结果。

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.