Symmetry-assisted covariance estimation in the ultra-sparse one-pair regime

Determine whether symmetry still improves covariance estimation when every subject contributes exactly one observation pair, corresponding to N_i\equiv2, and characterize the form of any resulting improvement.

Background

The main rate and threshold results rely on the factor m2, which represents the expected number of within-subject observation pairs. At the boundary N_i\equiv2, each subject supplies only one pair, so the pair-count mechanism underlying the claimed symmetry gain degenerates.

Although the saturation lemma remains valid in this setting, the paper does not establish whether group averaging can still reduce the estimation risk or what an appropriate threshold formulation would be.

References

Whether symmetry still helps in that regime, and in what form, is not settled by the present analysis.

— How far can symmetry help? Phase transitions and symmetry selection in sparse functional data analysis  (2608.27055 - Nembe, 27 Aug 2026) in Section 5, Discussion

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. Third, the ultra-sparse boundary N_i\equiv2 leaves the deterministic projection identities intact but makes sharp quantitative variance reduction especially sensitive to within-subject dependence and orbit overlap.

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