Joint bounds for estimated registration maps

Derive a sharp joint transport-defect bound for sparse functional covariance estimation when the reparametrisation is estimated from the data, including the stochastic error of the estimated map and suitable control of its derivatives.

Background

The paper establishes curvature-controlled equivariance when the reparametrisation psi is known. In applications such as registration, however, the transformation is estimated from the observations, so its stochastic error propagates through the covariance smoother and may also involve errors in estimated derivatives.

The authors identify the derivation of a sharp bound that jointly accounts for covariance-smoothing error and registration-map estimation error as unresolved. Such a result would extend the deterministic-map theory to practically relevant estimated registrations.

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.

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