A Temporal Spatial Minimax Rate for Smoothly-Varying Distributions in Wasserstein Space

Abstract

We study the minimax rate of estimating a future value $\hat{I}{¼}_{t_n+h}$ of a curve $t\mapsto \hat{I}{¼}_t$ in the $2$-Wasserstein space ${\mathcal{P}}_2({\mathbb{R}}^d)$ from finitely many noisy snapshots of its past, under an adiabatic bound $\Vert {\nabla }_t^{k}v\Vert \le \epsilon$ 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 ${\hat{I}}_d^3(k+1)/(k+1+{\hat{I}}_d^3)$, ${\hat{I}}_d^3=\min (1/d,1/2)$ ($M$ 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 $\epsilon h^{k+1}$ -- the irreducible cost of an unobserved future, present even with the exact past -- and the spatial estimation curse $M^{-{\hat{I}}_d^3}$, 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 $M^{-1/(d+1)}$, $d\ge 3$) and, in a translation submodel, for all $k$; for $k\ge 1$ 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.