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

2026-05-19

AI Summary

This work addresses key limitations of conventional equivariant neural networks, which struggle with optimal symmetry-preserving compression, interpretable decomposition into irreducible representations, and data-driven discovery of symmetries due to architectural constraints. The authors propose a $\star_G$ tensor algebra grounded in finite group $G$, internalizing equivariance as an intrinsic algebraic structure, and introduce the $\star_G$-SVD—a polynomial-time algorithm that yields exact symmetry-preserving approximations with Eckart–Young optimality guarantees. This framework seamlessly combines multiple symmetries without network redesign, enables prediction decomposition by irreducible representations, and automatically identifies the optimal symmetry group from data. On the QM9 dataset, it achieves closed-form predictions with 50–90× fewer parameters, reproduces Wigner–Eckart selection rules for angular momentum using only geometric inputs, and demonstrates a fivefold higher predictive accuracy for vector (T₁) versus scalar (A₁) observables.

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 Lean~4 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 $\sim50-90\times$ 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.

Problem

• equivariant learning
• symmetry discovery
• tensor approximation
• group algebra
• physical symmetry

Innovation

• group-equivariant learning
• tensor algebra
• symmetry discovery
• optimal SVD
• irreducible representation