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Title: Equivariant neural networks for inverse problems Authors:  Ferdia Sherry - University of Cambridge (United Kingdom) [presenting]
Abstract: In recent years, the use of convolutional layers to encode an inductive bias (translational equivariance) in neural networks has proven to be a very fruitful idea. The successes of this approach have motivated a line of research into incorporating other symmetries into deep learning methods, in the form of group equivariant convolutional neural networks. Much of this work has been focused on the roto-translational symmetry of Euclidean spaces, but other examples are the scaling symmetry of Euclidean spaces and the rotational symmetry of the sphere. We demonstrate that group equivariant convolutional operations can naturally be incorporated into learned reconstruction methods for inverse problems that are motivated by the variational regularisation approach. Indeed, if the regularisation functional is invariant under a group symmetry, the corresponding proximal operator will satisfy an equivariance property with respect to the same group symmetry. As a result of this observation, we design learned iterative methods in which proximal operators are modelled as group equivariant neural networks. We use roto-translationally equivariant operations in the proposed methodology and apply it to the problems of low-dose CT reconstruction and subsampled MRI reconstruction. The proposed methodology is demonstrated to improve the reconstruction quality of a learned reconstruction method with a little extra computational cost at training time but without any extra cost at test time.