Learning Equivariances and Partial Equivariances from Data

Group equivariant Convolutional Neural Networks (G-CNNs) constrain features to respect the chosen symmetries, and lead to better generalization when these symmetries appear in the data. However, if the chosen symmetries are not present, group equivariant architectures lead to overly constrained models and worse performance. Frequently, the distribution of the data can be better represented by a subset of a group than by the group as a whole, e.g., rotations in $[-90^{\circ}, 90^{\circ}]$. In such cases, a model that respects equivariance partially is better suited to represent the data. Moreover, relevant symmetries may differ for low and high-level features, e.g., edge orientations in a face, and face poses relative to the camera. As a result, the optimal level of equivariance may differ per layer. In this work, we introduce Partial G-CNNs: a family of equivariant networks able to learn partial and full equivariances from data at every layer end-to-end. Partial G-CNNs retain full equivariance whenever beneficial, e.g., for rotated MNIST, but are able to restrict it whenever it becomes harmful, e.g., for 6~/~9 or natural image classification. Partial G-CNNs perform on par with G-CNNs when full equivariance is necessary, and outperform them otherwise. Our method is applicable to discrete groups, continuous groups and combinations thereof.