  Group convolutional neural networks (G-CNNs) have been shown to increase
parameter efficiency and model accuracy by incorporating geometric inductive
biases. In this work, we investigate the properties of representations learned
by regular G-CNNs, and show considerable parameter redundancy in group
convolution kernels. This finding motivates further weight-tying by sharing
convolution kernels over subgroups. To this end, we introduce convolution
kernels that are separable over the subgroup and channel dimensions. In order
to obtain equivariance to arbitrary affine Lie groups we provide a continuous
parameterisation of separable convolution kernels. We evaluate our approach
across several vision datasets, and show that our weight sharing leads to
improved performance and computational efficiency. In many settings, separable
G-CNNs outperform their non-separable counterpart, while only using a fraction
of their training time. In addition, thanks to the increase in computational
efficiency, we are able to implement G-CNNs equivariant to the
$\mathrm{Sim(2)}$ group; the group of dilations, rotations and translations.
$\mathrm{Sim(2)}$-equivariance further improves performance on all tasks
considered.
