Group Equivariant Subsampling

TL;DR

This paper introduces group equivariant subsampling and upsampling layers to build CNNs that maintain exact translation and group equivariance, improving representation learning and data efficiency.

Contribution

It proposes novel group equivariant subsampling/upsampling layers and demonstrates their effectiveness in learning low-dimensional equivariant representations.

Findings

Representations are equivariant to translations and rotations.

GAEs generalize well to unseen positions and orientations.

Improved data efficiency and object decomposition in multi-object datasets.

Abstract

Subsampling is used in convolutional neural networks (CNNs) in the form of pooling or strided convolutions, to reduce the spatial dimensions of feature maps and to allow the receptive fields to grow exponentially with depth. However, it is known that such subsampling operations are not translation equivariant, unlike convolutions that are translation equivariant. Here, we first introduce translation equivariant subsampling/upsampling layers that can be used to construct exact translation equivariant CNNs. We then generalise these layers beyond translations to general groups, thus proposing group equivariant subsampling/upsampling. We use these layers to construct group equivariant autoencoders (GAEs) that allow us to learn low-dimensional equivariant representations. We empirically verify on images that the representations are indeed equivariant to input translations and rotations, and thus generalise well to unseen positions and orientations. We further use GAEs in models that learn object-centric representations on multi-object datasets, and show improved data efficiency and decomposition compared to non-equivariant baselines.