Minimax Rates for Estimating Powered Sliced Wasserstein Functionals

We study the two-sample estimation of the powered Sliced Wasserstein functional under projected r-th moment assumptions. Let N=n m, where n and m are two sample sizes. We establish the minimax rates in the noncritical regimes. When p<r<2p, the minimax risk is of order N-(r-p)/r, while for r>2p it is of order N-1/2. In both cases, the natural plug-in estimator is minimax rate-optimal. At the endpoint r=p, the minimax risk is bounded away from zero, so uniformly consistent estimation is impossible. At the critical order r=2p, we obtain the exact worst-case rate of the plug-in estimator and a lower bound for the minimax risk, leaving a logarithmic gap between the lower and upper bounds. The exact minimax rate in this critical case remains open.