A General Theory of Equivariant CNNs on Homogeneous Spaces
Abstract
We present a general theory of Group equivariant Convolutional Neural Networks (G-CNNs) 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 G-CNNs 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 one-to-one with convolutions using equivariant kernels, and characterize the space of such kernels.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2018
- DOI:
- 10.48550/arXiv.1811.02017
- arXiv:
- arXiv:1811.02017
- Bibcode:
- 2018arXiv181102017C
- Keywords:
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- Computer Science - Machine Learning;
- Computer Science - Artificial Intelligence;
- Computer Science - Computational Geometry;
- Computer Science - Computer Vision and Pattern Recognition;
- Statistics - Machine Learning
- E-Print:
- Advances in Neural Information Processing Systems 32 (NeurIPS 2019) 9142-9153