Exploiting Redundancy: Separable Group Convolutional Networks on Lie Groups

TL;DR

This paper introduces separable group convolutional networks on Lie groups, reducing parameter redundancy and improving efficiency and performance in vision tasks by leveraging subgroup sharing and continuous parameterisation.

Contribution

It proposes a novel separable convolution kernel design for G-CNNs, enabling equivariance to affine Lie groups and enhancing efficiency and accuracy.

Findings

Separable G-CNNs outperform non-separable versions in accuracy.

Parameter sharing reduces training time significantly.

Implementation of $ ext{Sim(2)}$-equivariant G-CNNs improves results across tasks.

Abstract

Group convolutional neural networks (G-CNNs) have been shown to increase parameter efficiency and model accuracy by incorporating geometric inductive biases. In this work, we investigate the properties of representations learned by regular G-CNNs, and show considerable parameter redundancy in group convolution kernels. This finding motivates further weight-tying by sharing convolution kernels over subgroups. To this end, we introduce convolution kernels that are separable over the subgroup and channel dimensions. In order to obtain equivariance to arbitrary affine Lie groups we provide a continuous parameterisation of separable convolution kernels. We evaluate our approach across several vision datasets, and show that our weight sharing leads to improved performance and computational efficiency. In many settings, separable G-CNNs outperform their non-separable counterpart, while only using a fraction of their training time. In addition, thanks to the increase in computational efficiency, we are able to implement G-CNNs equivariant to the $\mathrm{Sim(2)}$ group; the group of dilations, rotations and translations. $\mathrm{Sim(2)}$-equivariance further improves performance on all tasks considered.