Enabling equivariance for arbitrary Lie groups

TL;DR

This paper introduces a rigorous mathematical framework for implementing group convolutions over any finite-dimensional Lie group, enabling invariance to complex geometric transformations in neural networks, and demonstrates significant empirical improvements over CNNs and CapsNets.

Contribution

The authors develop a general framework for Lie group convolutions that does not rely on capsules, allowing invariance to a wide range of geometric transformations in neural networks.

Findings

30% accuracy improvement on affine-invariant classification

Superior robustness on homography-perturbed datasets

Effective invariance to complex geometric transformations

Abstract

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.