Geometric Deep Learning and Equivariant Neural Networks
This work addresses the challenge of incorporating geometric priors into deep learning for researchers and practitioners in fields like computer vision and physics, though it is largely incremental as it builds on existing equivariant network frameworks.
The paper tackles the problem of developing neural networks that respect geometric symmetries by introducing gauge equivariant convolutional networks on arbitrary manifolds and group equivariant networks for homogeneous spaces, demonstrating their application in tasks like semantic segmentation and object detection.
We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds $\mathcal{M}$ using principal bundles with structure group $K$ and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces $\mathcal{M}=G/K$, which are instead equivariant with respect to the global symmetry $G$ on $\mathcal{M}$. Group equivariant layers can be interpreted as intertwiners between induced representations of $G$, and we show their relation to gauge equivariant convolutional layers. We analyze several applications of this formalism, including semantic segmentation and object detection networks. We also discuss the case of spherical networks in great detail, corresponding to the case $\mathcal{M}=S^2=\mathrm{SO}(3)/\mathrm{SO}(2)$. Here we emphasize the use of Fourier analysis involving Wigner matrices, spherical harmonics and Clebsch-Gordan coefficients for $G=\mathrm{SO}(3)$, illustrating the power of representation theory for deep learning.