  The big empirical success of group equivariant networks has led in recent
years to the sprouting of a great variety of equivariant network architectures.
A particular focus has thereby been on rotation and reflection equivariant CNNs
for planar images. Here we give a general description of $E(2)$-equivariant
convolutions in the framework of Steerable CNNs. The theory of Steerable CNNs
thereby yields constraints on the convolution kernels which depend on group
representations describing the transformation laws of feature spaces. We show
that these constraints for arbitrary group representations can be reduced to
constraints under irreducible representations. A general solution of the kernel
space constraint is given for arbitrary representations of the Euclidean group
$E(2)$ and its subgroups. We implement a wide range of previously proposed and
entirely new equivariant network architectures and extensively compare their
performances. $E(2)$-steerable convolutions are further shown to yield
remarkable gains on CIFAR-10, CIFAR-100 and STL-10 when used as a drop-in
replacement for non-equivariant convolutions.
