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 $N=n\wedge m$, 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.

Statistics Theory (math.ST); Machine Learning (cs.LG); Machine Learning (stat.ML)

arXiv:2610.12740 [math.ST]

(or arXiv:2610.12740v1 [math.ST] for this version)

https://doi.org/10.48550/arXiv.2610.12740 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.