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

arXiv:2606.07325v1 Announce Type: cross
Abstract: We study the minimax rate of estimating a future value $mu_{t_n+h}$ of a curve $tmapstomu_t$ in the $2$-Wasserstein space $mathcal{P}_2(mathbb{R}^d)$ from finitely many noisy snapshots of its past, under an adiabatic bound $|nabla_t^k v|levarepsilon$ 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 $gamma_d(k+1)/(k+1+gamma_d)$, $gamma_d=min(1/d,1/2)$ ($M$ 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 $M^{-gamma_d}$, recovering the static distribution-estimation rate as $ktoinfty$. 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 $M^{-1/(d+1)}$, $dge3$) and, in a translation submodel, for all $k$; for $kge1$ 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.