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arXiv · 2610.12740

Minimax Rates for Estimating Powered Sliced Wasserstein Functionals

Binh Thuan Tran
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Abstract

We study the two-sample estimation of the powered Sliced Wasserstein functional under projected $r$-th moment assumptions. Let $N=n\wedge m$, where $n$ and $m$ are two sample sizes. We establish the minimax rates in the noncritical regimes. When $p 2p$ 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.

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Binh Thuan Tran. 2026-10-08. Minimax Rates for Estimating Powered Sliced Wasserstein Functionals. https://arxiv.org/abs/2610.12740

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