  Subsampling is used in convolutional neural networks (CNNs) in the form of
pooling or strided convolutions, to reduce the spatial dimensions of feature
maps and to allow the receptive fields to grow exponentially with depth.
However, it is known that such subsampling operations are not translation
equivariant, unlike convolutions that are translation equivariant. Here, we
first introduce translation equivariant subsampling/upsampling layers that can
be used to construct exact translation equivariant CNNs. We then generalise
these layers beyond translations to general groups, thus proposing group
equivariant subsampling/upsampling. We use these layers to construct group
equivariant autoencoders (GAEs) that allow us to learn low-dimensional
equivariant representations. We empirically verify on images that the
representations are indeed equivariant to input translations and rotations, and
thus generalise well to unseen positions and orientations. We further use GAEs
in models that learn object-centric representations on multi-object datasets,
and show improved data efficiency and decomposition compared to non-equivariant
baselines.
