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 the integrated squared acceleration subject to a set of interpolation constraints. As generalization on the space of probability measures the integral over the squared acceleration is considered as a spline energy and regularized by addition of the usual action functional. 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. On the subspace of Gaussian distributions, the spline interpolation problem is solved explicitly and consistency in the discrete to continuous limit is shown. The computation of time discrete splines is implemented numerically, based on 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 method. As a particular application the spline interpolation for synthesized textures is presented.

翻译：本文研究Wasserstein空间中用于插值概率测度的样条时间离散变分模型。欧氏空间中的三次样条已知能在满足一组插值约束条件下最小化加速度的平方积分。作为对概率测度空间的推广，加速度平方积分被视作样条能量，并通过添加通常的作用泛函进行正则化。随后利用局部Wasserstein-2距离和广义Wasserstein重心对这两种能量进行时间离散化。针对给定插值条件，建立了时间离散正则化样条的存在性。在高斯分布子空间上，样条插值问题被显式求解，并证明了离散到连续极限的一致性。基于熵正则化和Sinkhorn算法，实现了时间离散样条的数值计算。采用iPALM方法的变体对完全离散泛函进行最小化。大量数值实例展示了该方法的鲁棒性及其显著特征。特别地，针对合成纹理的样条插值作为具体应用进行了展示。