Existence of optimal reparametrizations

Prove existence of an optimal reparametrization in the reparametrization group for the elastic alignment of two curves in the settings used by elastic shape regression.

Background

Elastic shape distance is defined through an infimum over rotations and curve reparametrizations. The alignment algorithm assumes that an optimal reparametrization is available, but existence is delicate: the appendix notes that optimal reparametrizations exist for C1 curves while examples of Lipschitz curves lack them. The paper therefore identifies a general unresolved issue concerning attainment of the reparametrization infimum.

References

The alignment step itself relies on the existence of an optimal reparametrization between two curves, which is still a delicate aspect in some cases.

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