Testing or selecting approximate symmetry at the crossover scale

Construct a data-driven test or model selector for approximate invariance that compares the symmetry defect \(A_G(C)\) with the anti-invariant estimation risk and has power at the crossover scale \(A_n^\star\).

Background

For a finite-group projection, the paper derives an exact risk decomposition: projection removes anti-invariant estimation error but incurs the squared symmetry misspecification cost AG(C)A_G(C). The projection is beneficial only when the removable estimation error exceeds this defect.

The paper leaves unresolved how to make this comparison from data. In particular, it calls for a test or selector capable of detecting whether approximate symmetry is sufficiently accurate for projection to improve risk, especially at the theoretically relevant crossover amplitude An⋆A_n^\star.

References

Open questions. Three questions remain particularly relevant. First, if \psi is estimated from the data, as in registration, the transport defect must include the stochastic error of \hat\psi, including control of its derivatives; deriving a sharp joint bound is nontrivial. Second, approximate invariance calls for a data-driven decision rule comparing the symmetry defect A_G(C) with the anti-invariant estimation risk identified in Corollary~\ref{cor:expected-risk}; constructing a test or selector with power at the crossover scale A_n\star is a natural next problem.

— Equivariance, Curvature and Symmetry in Functional Covariance Estimation  (2609.03042 - Nembe, 2 Sep 2026) in Section 7, Discussion, paragraph “Open questions”