Minimax Rates for Estimating Powered Sliced Wasserstein Functionals

arXiv:2610.12740v1 Announce Type: cross Abstract: We study the two-sample estimation of the powered Sliced Wasserstein functional under projected r-th moment assumptions. Let N=nwedge m, where n and m are two sample sizes. We establish the minimax rates in the noncritical regimes. When p2p 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.

Source: arXiv cs.LG | 2026-10-12