Group Equivariant Convolutional Networks

TL;DR

The paper addresses the limitation of standard CNNs in exploiting only translation symmetry by introducing group-equivariant CNNs ($G$-CNNs) that use $G$-convolutions to share weights across a discrete symmetry group $G$. It develops a formal mathematical framework for functions on groups, detailing the groups $p4$ and $p4m$, and proves that G-convolutions, pooling, and nonlinearities preserve equivariance to $G$. The authors present efficient implementation strategies, including filter transformations and augmented filter banks, enabling practical use with negligible overhead. Empirical results on Rotated MNIST and CIFAR-10 demonstrate that G-CNNs attain state-of-the-art performance, often with fewer parameters than comparable planar architectures. The work highlights the value of incorporating structured symmetry into deep networks and outlines avenues for extending to larger or continuous groups and 3D domains.

Abstract

We introduce Group equivariant Convolutional Neural Networks (G-CNNs), a natural generalization of convolutional neural networks that reduces sample complexity by exploiting symmetries. G-CNNs use G-convolutions, a new type of layer that enjoys a substantially higher degree of weight sharing than regular convolution layers. G-convolutions increase the expressive capacity of the network without increasing the number of parameters. Group convolution layers are easy to use and can be implemented with negligible computational overhead for discrete groups generated by translations, reflections and rotations. G-CNNs achieve state of the art results on CIFAR10 and rotated MNIST.

Figures (2)

• Figure 1: A p4 feature map and its rotation by $r$.
• Figure 2: A p4m feature map and its rotation by $r$.