A General Theory of Equivariant CNNs on Homogeneous Spaces
Abstract
We present a general theory of Group equivariant Convolutional Neural Networks (GCNNs) on homogeneous spaces such as Euclidean space and the sphere. Feature maps in these networks represent fields on a homogeneous base space, and layers are equivariant maps between spaces of fields. The theory enables a systematic classification of all existing GCNNs in terms of their symmetry group, base space, and field type. We also consider a fundamental question: what is the most general kind of equivariant linear map between feature spaces (fields) of given types? Following Mackey, we show that such maps correspond onetoone with convolutions using equivariant kernels, and characterize the space of such kernels.
 Publication:

arXiv eprints
 Pub Date:
 November 2018
 DOI:
 10.48550/arXiv.1811.02017
 arXiv:
 arXiv:1811.02017
 Bibcode:
 2018arXiv181102017C
 Keywords:

 Computer Science  Machine Learning;
 Computer Science  Artificial Intelligence;
 Computer Science  Computational Geometry;
 Computer Science  Computer Vision and Pattern Recognition;
 Statistics  Machine Learning
 EPrint:
 Advances in Neural Information Processing Systems 32 (NeurIPS 2019) 91429153