  Motivated by applications from computer vision to bioinformatics, the field
of shape analysis deals with problems where one wants to analyze geometric
objects, such as curves, while ignoring actions that preserve their shape, such
as translations, rotations, or reparametrizations. Mathematical tools have been
developed to define notions of distances, averages, and optimal deformations
for geometric objects. One such framework, which has proven to be successful in
many applications, is based on the square root velocity (SRV) transform, which
allows one to define a computable distance between spatial curves regardless of
how they are parametrized. This paper introduces a supervised deep learning
framework for the direct computation of SRV distances between curves, which
usually requires an optimization over the group of reparametrizations that act
on the curves. The benefits of our approach in terms of computational speed and
accuracy are illustrated via several numerical experiments.
