Using and Abusing Equivariance

In this paper we show how Group Equivariant Convolutional Neural Networks use subsampling to learn to break equivariance to the rotation and reflection symmetries. We focus on the 2D rotations and reflections and investigate the impact of the broken equivariance on network performance. We show that a change in the input dimension of a network as small as a single pixel can be enough for commonly used architectures to become approximately equivariant, rather than exactly. We investigate the impact of networks not being exactly equivariant and find that approximately equivariant networks generalise significantly worse to unseen symmetries compared to their exactly equivariant counterparts. However, when the symmetries in the training data are not identical to the symmetries of the network, we find that approximately equivariant networks can relax their equivariance constraints, matching or outperforming exactly equivariant networks on common benchmarks.