Minimax Rates for Estimating Powered Sliced Wasserstein Functionals
Abstract
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.
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BibTeX
@misc{tran2026minimax,
title = {{Minimax Rates for Estimating Powered Sliced Wasserstein Functionals}},
author = {Binh Thuan Tran},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12740},
eprint = {2610.12740},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12740},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Minimax Rates for Estimating Powered Sliced Wasserstein Functionals
AU - Binh Thuan Tran
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12740
UR - https://arxiv.org/abs/2610.12740
AB - 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.
ER -