  Group equivariant neural networks have been explored in the past few years
and are interesting from theoretical and practical standpoints. They leverage
concepts from group representation theory, non-commutative harmonic analysis
and differential geometry that do not often appear in machine learning. In
practice, they have been shown to reduce sample and model complexity, notably
in challenging tasks where input transformations such as arbitrary rotations
are present. We begin this work with an exposition of group representation
theory and the machinery necessary to define and evaluate integrals and
convolutions on groups. Then, we show applications to recent SO(3) and SE(3)
equivariant networks, namely the Spherical CNNs, Clebsch-Gordan Networks, and
3D Steerable CNNs. We proceed to discuss two recent theoretical results. The
first, by Kondor and Trivedi (ICML'18), shows that a neural network is group
equivariant if and only if it has a convolutional structure. The second, by
Cohen et al. (NeurIPS'19), generalizes the first to a larger class of networks,
with feature maps as fields on homogeneous spaces.
