Approximation of Splines in Wasserstein Spaces

This paper investigates a time discrete variational model for splines in Wasserstein spaces to interpolate probability measures. Cubic splines in Euclidean space are known to minimize of the integrated squared acceleration subject to a set of interpolation constraints. As generalization on the manifold of probability measures the integral over the squared Riemannian acceleration is considered as a spline energy and adding the action functional a regularized spline energy is defined.
Both energies are then discretized in time using local Wasserstein-2 distances and the generalized Wasserstein barycenter. The existence of time discrete regularized splines for given interpolation conditions is established and On the subspace of Gaussian distributions, explicit notions for the time discrete and time continuous splines are investigated.
The implementation is based on classical optimal transport, entropy regularization and the Sinkhorn algorithm. A variant of the iPALM method is applied for the minimization of the fully discrete functional. A variety of numerical examples demonstrate the robustness of the approach and show striking characteristics of the approach.
As a particular application the spline interpolation for synthesized textures is presented.