  On the space of probability densities, we extend the Wasserstein geodesics to
the case of higher-order interpolation such as cubic spline interpolation.
After presenting the natural extension of cubic splines to the Wasserstein
space, we propose a simpler approach based on the relaxation of the variational
problem on the path space. We explore two different numerical approaches, one
based on multi-marginal optimal transport and entropic regularization and the
other based on semi-discrete optimal transport.
