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 any finite-dimensional Lie group. We empirically
validate our approach on the benchmark affine-invariant classification task,
where we achieve 30% improvement in accuracy against conventional CNNs while
outperforming most CapsNets. 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.

本文介绍了一种使用李群上的卷积（group convolutions over Lie groups）来实现任何形变的不变性的严谨数学框架，经实验证明在具有仿射不变性的分类任务中，我们的方法比传统 CNN 提高了 30％的准确性，同时优于大多数 CapsNets。