Consistency and convergence rates for elastic shape regression

Establish consistency and convergence rates for estimators of the conditional Fréchet mean in elastic shape space when the shape distance is defined as the value of an optimization problem.

Background

The paper develops a nonparametric estimator of conditional Fréchet means for planar curves modulo translation, rotation, scaling, and reparametrization. Existing consistency results cited in the paper concern spherical regression, whereas the elastic shape setting introduces an additional optimization over alignments into the distance. The authors explicitly leave the corresponding statistical theory unresolved.

References

Finally, consistency and rates of convergence for estimators of the conditional Fréchet mean in the elastic shape space, where the distance is itself the value of an optimization problem, remain open problems.

— Elastic kernel Ridge regression, with applications in phonetics  (2610.08386 - Matteo et al., 6 Oct 2026) in Section Discussion