Эта статья является препринтом и не была отрецензирована.
О результатах, изложенных в препринтах, не следует сообщать в СМИ как о проверенной информации.
Exact Fixed-Dimensional Minimax Theory for Wasserstein Distance Estimation
2026-04-03
We establish the unrestricted fixed-dimensional minimax theory for estimating the two-sample Wasserstein distance $W_{p}(P,Q)$ over the full class of Borel probability measures on $[0,1]^{d}$. For every $p \ge 1$, every $d > 2p$, and arbitrary sample sizes $n, m$ with $N := n \wedge m$, we prove that the exact minimax absolute and squared risks are $(N \log N)^{-1/d}$ and $(N \log N)^{-2/d}$.The argument is entirely Euclidean. First, we show that the lower-envelope scale is already exact on a full diagonal neighborhood and that empirical optimal transport remains locally suboptimal there. Second, we identify the critical finite model by reducing the balanced supercritical problem to the Euclidean problem on the dyadic grid with $|X_{J}| \approx N \log N$ atoms, and we determine the generic off-diagonal geometry through full positive-distance neighborhoods, mixed-scale two-ball classes, and translated thin-annulus examples. Third, a critical linearization of the $p$-cost separates a parametric first-order affine layer from normalized second-order residuals; semiconcave branching bounds and support-complexity eliminations then reduce the unresolved part of the problem to a finite nearest-neighbor macrograph. Finally, that terminal layer is closed by three different mechanisms: uniform semiconcavity for $p \ge 2$, canonical corner kernels for $p < 2$, and a sharp $L_1$-covering argument for the singular limit.
Ссылка для цитирования:
Чурилов М. В. 2026. Exact Fixed-Dimensional Minimax Theory for Wasserstein Distance Estimation. PREPRINTS.RU. https://doi.org/10.24108/preprints-3114797
Список литературы
1. [1] S. Chewi, J. Niles-Weed, and P. Rigollet. Estimation of Wasserstein distances. In
2. Statistical Optimal Transport, Lecture Notes in Mathematics, vol. 2364, pages 37–76.
3. Springer, Cham, 2025.
4. [2] N. Fournier and A. Guillin. On the rate of convergence in Wasserstein distance of the
5. empirical measure. Probability Theory and Related Fields, 162(3–4):707–738, 2015.
6. [3] J. Jiao, Y. Han, and T. Weissman. Minimax estimation of the L1 distance. IEEE
7. Transactions on Information Theory, 64(10):6672–6706, 2018.
8. [4] E. M. Bronshtein. ε-entropy of convex sets and functions. Siberian Mathematical Journal,
9. 17:393–398, 1976.
10. [5] A. Guntuboyina and B. Sen. L1 covering numbers for uniformly bounded convex functions.
11. In Proceedings of the 25th Annual Conference on Learning Theory, volume 23 of JMLR
12. Workshop and Conference Proceedings, pages 12.1–12.13, 2012.
13. [6] T. Manole and J. Niles-Weed. Sharp convergence rates for empirical optimal transport
14. with smooth costs. Annals of Applied Probability, 34(1B):1108–1135, 2024.
15. [7] J. Niles-Weed and P. Rigollet. Estimation of Wasserstein distances in the spiked transport
16. model. Bernoulli, 28(4):2663–2688, 2022.
17. [8] S. Wang, T. T. Cai, and H. Li. Optimal estimation of Wasserstein distance on a tree with
18. an application to microbiome studies. Journal of the American Statistical Association,
19. 116(535):1237–1253, 2021.
20. [9] J. Weed and F. Bach. Sharp asymptotic and finite-sample rates of convergence of
21. empirical measures in Wasserstein distance. Bernoulli, 25(4A):2620–2648, 2019.
