Fast computation of permutation equivariant layers with the partition algebra

Linear neural network layers that are either equivariant or invariant to permutations of their inputs form core building blocks of modern deep learning architectures. Examples include the layers of DeepSets, as well as linear layers occurring in attention blocks of transformers and some graph neural networks. The space of permutation equivariant linear layers can be identified as the invariant subspace of a certain symmetric group representation, and recent work parameterized this space by exhibiting a basis whose vectors are sums over orbits of standard basis elements with respect to the symmetric group action. A parameterization opens up the possibility of learning the weights of permutation equivariant linear layers via gradient descent. The space of permutation equivariant linear layers is a generalization of the partition algebra, an object first discovered in statistical physics with deep connections to the representation theory of the symmetric group, and the basis described above generalizes the so-called orbit basis of the partition algebra. We exhibit an alternative basis, generalizing the diagram basis of the partition algebra, with computational benefits stemming from the fact that the tensors making up the basis are low rank in the sense that they naturally factorize into Kronecker products. Just as multiplication by a rank one matrix is far less expensive than multiplication by an arbitrary matrix, multiplication with these low rank tensors is far less expensive than multiplication with elements of the orbit basis. Finally, we describe an algorithm implementing multiplication with these basis elements.