Second order models for optimal transport and cubic splines on the Wasserstein space
Résumé
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 simpler approach, similarly to Brenier's generalized Euler solutions. Our method is based on the relaxation of the variational problem on the path space. We propose an efficient implementation based on multimarginal optimal transport and entropic regularization in 1D and 2D. Our framework also enables extrapolation in the Wasserstein geodesic via a natural convex relaxation.
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