Group-Algebraic Tensors: Provably-optimal Equivariant Learning and Physical Symmetry Discovery

arXiv:2605.20440v1 Announce Type: new Abstract: We introduce the star_G tensor algebra, in which any finite group G defines the multiplication rule, making equivariance an intrinsic algebraic property rather than an architectural constraint. The framework rests on three machine-verified theoretical pillars: (i)~an Eckart-Young optimality guarantee for the star_G-SVD: the first such result for symmetry-preserving tensor approximation, exact and polynomial-time; (ii)~a Kronecker factorization that composes multiple symmetries by replacing F_G with F_{G_1} otimes F_{G_2} with no architectural redesign; and (iii)a 600-line Lean4 formalization of the star_G algebra. The framework provides capabilities that equivariant neural networks (ENNs) structurally cannot: a closed-form per-irreducible-representation decomposition of every prediction, and data-driven discovery of the symmetry group that best fits a dataset. As a non-trivial empirical demonstration, decomposing QM9 molecular geometry over the chiral octahedral subgroup of SO(3) recovers the Wigner--Eckart selection rules of angular momentum from data alone, with no quantum mechanical input: scalar properties are A_1-dominated, dipole components are T_1-dominated, the isotropic polarizability is uniquely insensitive to l!=!1 as the rank-2-trace decomposition l!=!0 oplus l!=!2 requires, and the T_1/A_1 predictive-power ratio separates vector observables from scalar observables by a factor of five. On full QM9 (130{,}831 molecules), star_G-SVD with ridge regression provides closed form predictions at sim50-90imes fewer parameters than parameter-matched MLPs. Algebraic equivariance thus complements architectural equivariance not as a faster-better-cheaper alternative but as a different mathematical affordance: provably-optimal symmetry-preserving compression, per-irrep interpretability, and data-driven physical discovery.

Source: arXiv cs.LG | 2026-05-21