Categorification of Group Equivariant Neural Networks
This work addresses a theoretical bottleneck in deep learning for researchers, but it is incremental as it builds on existing group equivariant networks.
The paper tackles the problem of understanding and computing linear layers in group equivariant neural networks for specific groups by applying category theory, resulting in new insights and an algorithm for faster computation.
We present a novel application of category theory for deep learning. We show how category theory can be used to understand and work with the linear layer functions of group equivariant neural networks whose layers are some tensor power space of $\mathbb{R}^{n}$ for the groups $S_n$, $O(n)$, $Sp(n)$, and $SO(n)$. By using category theoretic constructions, we build a richer structure that is not seen in the original formulation of these neural networks, leading to new insights. In particular, we outline the development of an algorithm for quickly computing the result of a vector that is passed through an equivariant, linear layer for each group in question. The success of our approach suggests that category theory could be beneficial for other areas of deep learning.