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.
A PDE approach to a 2-dimensional matching problem
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abstract
We prove asymptotic results for 2-dimensional random matching problems. In particular, we obtain the leading term in the asymptotic expansion of the expected quadratic transportation cost for empirical measures of two samples of independent uniform random variables in the square. Our technique is based on a rigorous formulation of the challenging PDE ansatz by S.\ Caracciolo et al.\ (Phys. Rev. E, {\bf 90} 012118, 2014) that "linearise" the Monge-Amp\`ere equation.
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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.