Connecting Permutation Equivariant Neural Networks and Partition Diagrams
This work provides a theoretical foundation for designing permutation equivariant neural networks, which is incremental as it builds on existing duality concepts to offer a new computational approach.
The paper tackles the problem of constructing permutation equivariant neural networks by showing that all weight matrices in these networks can be derived from Schur-Weyl duality between the symmetric group and the partition algebra, resulting in a diagrammatic method for calculating these matrices.
Permutation equivariant neural networks are often constructed using tensor powers of $\mathbb{R}^{n}$ as their layer spaces. We show that all of the weight matrices that appear in these neural networks can be obtained from Schur-Weyl duality between the symmetric group and the partition algebra. In particular, we adapt Schur-Weyl duality to derive a simple, diagrammatic method for calculating the weight matrices themselves.