How Jellyfish Characterise Alternating Group Equivariant Neural Networks
This work addresses the problem of designing neural networks with specific symmetry properties for researchers in geometric deep learning and group theory, representing a foundational theoretical contribution.
The paper provides a complete characterization of all possible alternating group (A_n) equivariant neural networks with layers as tensor powers of ℝ^n, including a basis for learnable linear equivariant layer functions. It also extends the approach to networks equivariant to local symmetries.
We provide a full characterisation of all of the possible alternating group ($A_n$) equivariant neural networks whose layers are some tensor power of $\mathbb{R}^{n}$. In particular, we find a basis of matrices for the learnable, linear, $A_n$-equivariant layer functions between such tensor power spaces in the standard basis of $\mathbb{R}^{n}$. We also describe how our approach generalises to the construction of neural networks that are equivariant to local symmetries.