  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 roto-translational symmetry of $\mathbf R^d$, but other examples are
the scaling symmetry of $\mathbf R^d$ and rotational symmetry of the sphere. In
this work, 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 the proximal operators are modelled as group
equivariant convolutional neural networks. We use roto-translationally
equivariant operations in the proposed methodology and apply it to the problems
of low-dose computerised tomography reconstruction and subsampled magnetic
resonance imaging 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.
