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

arXiv: https://arxiv.org/abs/2606.07325 · HTML: https://mlsanity.com/paper/2606.07325/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 31 references that are in the map.
- First posted on arXiv 5 Jun 2026.

## Builds on (most cited first, 7 of 7)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Fast and Smooth Interpolation on Wasserstein Space](https://mlsanity.com/paper/2010.12101/) (2020)
- [Second order models for optimal transport and cubic splines on the Wasserstein space](https://mlsanity.com/paper/1801.04144/) (2018)
- [Wasserstein Autoregressive Models for Density Time Series](https://mlsanity.com/paper/2006.12640/) (2020)

## Abstract (arXiv)

We study the minimax rate of estimating a future value μtn+h of a curve tmapstoμt in the 2-Wasserstein space P2(ℝd) from finitely many noisy snapshots of its past, under an adiabatic bound |nablatk v|≤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 W2-risk with M-exponent γd(k+1)/(k+1+γd), γd=min(1/d,1/2) (M the total sample size). It follows from a temporal-to-spatial reduction: the smoothness budget defines a reachable W2-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 hk+1 -- the irreducible cost of an unobserved future, present even with the exact past -- and the spatial estimation curse M-γd, recovering the static distribution-estimation rate as k→∞. 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≥3) and, in a translation submodel, for all k; for k≥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.