Enabling equivariance for arbitrary Lie groups

Although provably robust to translational perturbations, convolutional neural networks (CNNs) are known to suffer from extreme performance degradation when presented at test time with more general geometric transformations of inputs. Recently, this limitation has motivated a shift in focus from CNNs to Capsule Networks (CapsNets). However, CapsNets suffer from admitting relatively few theoretical guarantees of invariance. We introduce a rigourous mathematical framework to permit invariance to any Lie group of warps, exclusively using convolutions (over Lie groups), without the need for capsules. Previous work on group convolutions has been hampered by strong assumptions about the group, which precludes the application of such techniques to common warps in computer vision such as affine and homographic. Our framework enables the implementation of group convolutions over \emph{any} finite-dimensional Lie group. We empirically validate our approach on the benchmark affine-invariant classification task, where we achieve $\sim$30\% improvement in accuracy against conventional CNNs while outperforming the state-of-the-art CapsNet. As further illustration of the generality of our framework, we train a homography-convolutional model which achieves superior robustness on a homography-perturbed dataset, where CapsNet results degrade.