Complete set of rotation invariants for higher-order tensors/polynomials

Establish a complete set of rotation invariants for order r ≥ 3 homogeneous polynomial tensors (or higher-order central moment tensors) in R^d such that agreement of these invariants for two tensors implies the existence of an orthogonal rotation mapping one to the other, and clarify the connection of this completeness problem to graph isomorphism.

Background

The paper proposes extending PCA-based rotation-invariant descriptors from covariance matrices (order-2 tensors) to higher-order central moments and polynomial-times-Gaussian representations. For r = 1 and r = 2, complete rotation invariants are known (vector norm and matrix trace-power invariants).

For higher-order tensors, the authors develop diagrammatic, graph-based constructions of many rotation invariants but note that agreement of these invariants is only necessary, not sufficient, to guarantee that two shapes differ only by rotation. They point out that identifying a complete, sufficient set of invariants for r ≥ 3 appears difficult and is linked to the graph isomorphism problem.

References

However, while agreement of such invariants is necessary conditions for rotation invariance, to be certain that $[p]\sim[q]$ we would also need sufficient condition: a complete set of invariants, which agreement allows to conclude differing only by rotation. While for $r=1,2$ it is known, for higher orders it seems a difficult open problem, which resolution should also solve graph isomorphism problem.

— Higher order PCA-like rotation-invariant features for detailed shape descriptors modulo rotation  (2601.03326 - Duda, 6 Jan 2026) in Section "Rotation invariants for tensors/polynomials", Subsection "General rotation invariants"

Raising the body order or increasing the tensor degree resolves progressively more symmetric cases, yet counterexamples are known already at low levels, and no finite level has been shown to achieve completeness across all configurations.

— Why Multi-Layer Message Passing Works: Completeness Theory for Graph Neural Network Interatomic Potentials  (2609.00528 - Ming et al., 1 Sep 2026) in Section Discussion and perspective

Consequently, it remains unclear how applicable this approach would be to tensors of even moderately higher ranks.

— Cartesian isotropic tensors revisited  (2609.04576 - Bouzas, 4 Sep 2026) in Introduction