Group-Algebraic Tensors for Provably-Optimal Equivariant Learning

A novel mathematical framework known as the ⋆_G tensor algebra presents group-equivariant learning as an inherent algebraic feature instead of a limitation of architecture. This framework rests on three verified theoretical foundations: an Eckart-Young optimality assurance for the ⋆_G-SVD, marking the first instance of symmetry-preserving tensor approximation; a Kronecker factorization that integrates various symmetries without needing to alter the architecture; and a 600-line formalization of the ⋆_G algebra in Lean 4. It offers functionalities unattainable by conventional equivariant neural networks, such as a closed-form decomposition for each prediction per irreducible representation and the ability to identify the symmetry group that aligns best with a dataset. An empirical example showcases the decomposition of QM9 molecular data.