General $E(2)$-Equivariant Steerable CNNs

TL;DR

This paper provides a comprehensive theoretical framework for $E(2)$-equivariant steerable CNNs, deriving kernel constraints for arbitrary group representations, and demonstrates their superior performance on standard image classification benchmarks.

Contribution

It introduces a general solution for $E(2)$-equivariant kernels, unifies existing architectures, and shows improved results on multiple datasets.

Findings

Remarkable accuracy gains on CIFAR-10, CIFAR-100, STL-10

Unified framework for $E(2)$-equivariant kernels

Implementation of both existing and novel equivariant architectures

Abstract

The big empirical success of group equivariant networks has led in recent years to the sprouting of a great variety of equivariant network architectures. A particular focus has thereby been on rotation and reflection equivariant CNNs for planar images. Here we give a general description of $E(2)$-equivariant convolutions in the framework of Steerable CNNs. The theory of Steerable CNNs thereby yields constraints on the convolution kernels which depend on group representations describing the transformation laws of feature spaces. We show that these constraints for arbitrary group representations can be reduced to constraints under irreducible representations. A general solution of the kernel space constraint is given for arbitrary representations of the Euclidean group $E(2)$ and its subgroups. We implement a wide range of previously proposed and entirely new equivariant network architectures and extensively compare their performances. $E(2)$-steerable convolutions are further shown to yield remarkable gains on CIFAR-10, CIFAR-100 and STL-10 when used as a drop-in replacement for non-equivariant convolutions.