Brauer's Group Equivariant Neural Networks
This work addresses a gap in the theoretical understanding of equivariant neural networks for specific symmetry groups, which is incremental as it extends existing frameworks to new groups.
The paper tackles the problem of characterizing all possible group equivariant neural networks for three symmetry groups (O(n), SO(n), Sp(n)) missing in machine learning literature, and provides a spanning set of matrices for learnable linear equivariant layers in tensor power spaces.
We provide a full characterisation of all of the possible group equivariant neural networks whose layers are some tensor power of $\mathbb{R}^{n}$ for three symmetry groups that are missing from the machine learning literature: $O(n)$, the orthogonal group; $SO(n)$, the special orthogonal group; and $Sp(n)$, the symplectic group. In particular, we find a spanning set of matrices for the learnable, linear, equivariant layer functions between such tensor power spaces in the standard basis of $\mathbb{R}^{n}$ when the group is $O(n)$ or $SO(n)$, and in the symplectic basis of $\mathbb{R}^{n}$ when the group is $Sp(n)$.