Machine Learning Invariants of Tensors

TL;DR

This work addresses the problem of identifying functionally independent invariants constructed from tensors under symmetry groups by introducing a data-driven algorithm that enumerates contracted tensor graphs, samples random numerical instances, and uses linear algebra to extract syzygies and generating sets. Applied to a 6d antisymmetric 3-form $H_{\mu\nu\rho}$, the method uncovers five independent invariants, demonstrated across three equivalent parameterizations: trace-based, Hodge-dual, and spinor-formulations, with explicit dictionaries connecting them. The generating set is shown to be complete, enabling a Lagrangian for $H$ to be written as $\\\\\\ ext{L}(x^{(2)},x^{(4)}_1,x^{(4)}_2,x^{(6)},x^{(8)})$ (or identical in the other variable sets), and higher-order invariants are expressible via these generators. The approach offers a practical, scalable alternative to analytical counts and decompositions, with potential to extend to other tensors and dimensions and to inform deformation analyses of tensor field theories.

Abstract

We propose a data-driven approach to identifying the functionally independent invariants that can be constructed from a tensor with a given symmetry structure. Our algorithm proceeds by first enumerating graphs, or tensor networks, that represent inequivalent contractions of a product of tensors, computing instances of these scalars using randomly generated data, and then seeking linear relations between invariants using numerical linear algebra. Such relations yield syzygies, or functional dependencies relating different invariants. We apply this approach in an extended case study of the independent invariants that can be constructed from an antisymmetric $3$-form $H_{μνρ}$ in six dimensions, finding five independent invariants. This result confirms that the most general Lagrangian for such a $3$-form, which depends on $H_{μνρ}$ but not its derivatives, is an arbitrary function of five variables, and we give explicit formulas relating other invariants to the five independent scalars in this generating set.