Group word operations can be learned by small two-layer networks because the associated word tensor has low rank, decomposable through the fusion algebra of the group's self-conjugate representations.
Tensor Rank and Complexity
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abstract
These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will be expressed in terms of tensor (border) rank, (border) symmetric rank and the asymptotic rank of certain tensors. We introduce the multilinear rank of a tensor as well, deal with the concept of tensor equivalence and study prehomogeneous vector spaces with the castling transform. Moreover, we treat Apolarity Theory and use it to determine the symmetric rank (Waring rank) of some symmetric tensors.
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Learning words in groups: fusion algebras, tensor ranks and grokking
Group word operations can be learned by small two-layer networks because the associated word tensor has low rank, decomposable through the fusion algebra of the group's self-conjugate representations.