Establishes a unified temporal-spatial minimax lower bound of order M to the power of minus gamma_d times (k+1) over (k+1 plus gamma_d) for W2-risk of future distribution estimates under k-th order adiabatic smoothness on the velocity field.
Second order models for optimal transport and cubic splines on the Wasserstein space
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abstract
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.
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A Temporal Spatial Minimax Rate for Smoothly-Varying Distributions in Wasserstein Space
Establishes a unified temporal-spatial minimax lower bound of order M to the power of minus gamma_d times (k+1) over (k+1 plus gamma_d) for W2-risk of future distribution estimates under k-th order adiabatic smoothness on the velocity field.