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Model-Aware Regularization For Learning Approaches To Inverse Problems

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Document pages: 19 pages

Abstract: There are various inverse problems -- including reconstruction problemsarising in medical imaging -- where one is often aware of the forward operatorthat maps variables of interest to the observations. It is therefore natural toask whether such knowledge of the forward operator can be exploited in deeplearning approaches increasingly used to solve inverse problems.In this paper, we provide one such way via an analysis of the generalisationerror of deep learning methods applicable to inverse problems. In particular,by building on the algorithmic robustness framework, we offer a generalisationerror bound that encapsulates key ingredients associated with the learningproblem such as the complexity of the data space, the size of the training set,the Jacobian of the deep neural network and the Jacobian of the composition ofthe forward operator with the neural network. We then propose a plug-and-play regulariser that leverages the knowledge of the forward map to improve thegeneralization of the network. We likewise also propose a new method allowingus to tightly upper bound the Lipschitz constants of the relevant functionsthat is much more computational efficient than existing ones. We demonstratethe efficacy of our model-aware regularised deep learning algorithms againstother state-of-the-art approaches on inverse problems involving varioussub-sampling operators such as those used in classical compressed sensing setupand accelerated Magnetic Resonance Imaging (MRI).

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