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.

本文针对深度学习的无监督学习，将群不变和群等变表示学习扩展到了该领域。我们提出了一种基于编码器 - 解码器框架的通用学习策略，其中潜在表示被分为不变项和等变群作用项。在利用预测适当的群作用来对齐输入和输出姿势以解决重建任务时，网络可以学习将数据编码和解码为群不变表示。我们导出依变编码器的必要条件，并针对旋转，平移和置换明确描述了我们的构造。我们在不同网络架构下使用不同数据类型进行各种实验，测试了我们方法的有效性和鲁棒性。