Minimax Rates for Estimating Powered Sliced Wasserstein Functionals

The Sliced Wasserstein (SW) distance has emerged as a computationally attractive alternative to the Wasserstein distance by leveraging one-dimensional optimal transport along random projections. Standard estimators of the SW distance rely on Monte Carlo averages of one-dimensional Wasserstein distances computed via quantile functions, which require sorting projected samples and access to full datasets. In this work, we introduce a new class of estimators for the Sliced Wasserstein distance based on cumulative distribution functions (CDFs) of projected measures, that avoid sorting and scale via massive dataset parallelism. This class includes several estimators, some of them being indexed by hyperparameters controlling their variance or smoothness. We show that they are especially well suited to scenarios in which CDFs are more tractable than quantile functions, such as mixtures of Gaussians, and moreover that they are also naturally compatible with federated learning, since CDFs of projected data can be computed and aggregated locally without requiring the exchange of raw samples.

We study the minimax rate of estimating a future value $μ_{t_n+h}$ of a curve $t\mapstoμ_t$ in the $2$-Wasserstein space $\mathcal{P}_2(\mathbb{R}^d)$ from finitely many noisy snapshots of its past, under an adiabatic bound $\|\nabla_t^k v\|\le\varepsilon$ on the $k$-th covariant derivative of the velocity field. Our central result is a unified temporal-spatial minimax lower bound: over regular, locally transport-rich subclasses, every estimator incurs $W_2$-risk with $M$-exponent $γ_d(k+1)/(k+1+γ_d)$, $γ_d=\min(1/d,1/2)$ ($M$ the total sample size). It follows from a temporal-to-spatial reduction: the smoothness budget defines a reachable $W_2$-ball into which a transport packing is embedded along the time axis, and the information of the entire snapshot experiment is controlled by a Fano argument -- the spatial packing is classical, but its smoothness-admissible temporal embedding and the full-window analysis are new. The bound interpolates a dimension-free extrapolation floor of order $\varepsilon h^{k+1}$ -- the irreducible cost of an unobserved future, present even with the exact past -- and the spatial estimation curse $M^{-γ_d}$, recovering the static distribution-estimation rate as $k\to\infty$. We state the lower bound in a design-dependent form -- with a design-weighted effective sample size -- valid for arbitrary observation times, and obtain the closed-form exponent in the dense (equispaced) regime. The matching upper bound is established at $k=0$ (rate $M^{-1/(d+1)}$, $d\ge3$) and, in a translation submodel, for all $k$; for $k\ge1$ a covariant estimator attains the rate conditionally on two estimates (a comparison-geometry bias bound and an optimal-transport map-estimation rate), leaving the unconditional general-$k$ upper bound as an open problem. Numerical experiments on synthetic curved and flat families corroborate the predicted exponents.

We propose sliced Orlicz-Wasserstein (SOW) distance which is a generalization of sliced Wasserstein (SW) distance. SOW replaces the $L^p$ norm in SW with a Luxemburg norm cost induced by an Orlicz function $φ$. First, we prove that SOW distance is a metric on the space of measures with finite Orlicz norm, and show that it recovers the SW distance when the Orlicz function is $φ(x)=x^p$. Next, we derive the topological properties of the SOW distance. In particular, we show that convergence under SOW implies weak convergence, and the converse is true under the compact support condition. We then present the theoretical results for estimating the SOW distance. We derive sample complexity for both the distance itself and the powered functional of the distance, and prove their minimax optimality. In addition, we discuss the computational algorithm for approximating the SOW distance by Monte-Carlo estimation and bisection search, as well as the associated approximation error and computational complexity analysis. Our experimental results reveal the superior computational efficiency of SOW compared with Orlicz-Wasserstein (OW) distance. Also, in the experiments, we demonstrate the favorable flexibility of SOW distance over SW in detecting differences between distributions by comparing their performance in two-sample tests and evaluating generative models on image datasets.