  Equivariant neural networks, whose hidden features transform according to
representations of a group G acting on the data, exhibit training efficiency
and an improved generalisation performance. In this work, we extend group
invariant and equivariant representation learning to the field of unsupervised
deep learning. We propose a general learning strategy based on an
encoder-decoder framework in which the latent representation is separated in an
invariant term and an equivariant group action component. The key idea is that
the network learns to encode and decode data to and from a group-invariant
representation by additionally learning to predict the appropriate group action
to align input and output pose to solve the reconstruction task. We derive the
necessary conditions on the equivariant encoder, and we present a construction
valid for any G, both discrete and continuous. We describe explicitly our
construction for rotations, translations and permutations. We test the validity
and the robustness of our approach in a variety of experiments with diverse
data types employing different network architectures.
