arXiv：cs.LG· Binh Thuan Tran·· 13 小时前
Powered Sliced Wasserstein 泛函估计的极小极大速率
Minimax Rates for Estimating Powered Sliced Wasserstein Functionals
AI 导读
研究在投影 r 阶矩假设下 powered Sliced Wasserstein 泛函的双样本估计，设 N=n∧m。非临界情形下，p<r<2p 时极小极大风险阶为 N^{-(r-p)/p}，r>2p 时为 N^{-1/2}，两种情形下自然 plug-in 估计量均达到极小极大最优；端点 r=p 时极小极大风险有下界，无法一致估计。
正文
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 $p<r<2p$, 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.
| Subjects: | Statistics Theory (math.ST); Machine Learning (cs.LG); Machine Learning (stat.ML) |
| Cite as: | arXiv:2610.12740 [math.ST] |
| (or arXiv:2610.12740v1 [math.ST] for this version) | |
| https://doi.org/10.48550/arXiv.2610.12740 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Binh Thuan Tran [view email]
[v1]
Thu, 8 Oct 2026 21:37:03 UTC (276 KB)
来源：arXiv：cs.LG · arxiv.org