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Critical-Grid Debiasing and Transport Certificates for Supercritical Minimax Wasserstein Distance Estimation
Let P, Q ∈ P([0, 1]d) be observed through two independent samples of size N , and consider minimax estimation of
Wp(P, Q) in the supercritical regime d > 2p. The empirical plug-in estimator gives the scale N −1/d, while the Niles–Weed–
Rigollet lower-bound mechanism gives the smaller candidate scale
ηN = (N log N )−1/d.
The manuscript develops critical-grid debiasing, multiscale polynomial transport estimation, transport certificates, and
finite-LP curvature methods for the sharp law. The unrestricted upper bound is reduced, with constants and no loss of scale,
to an adaptive finite Kantorovich linear-program value on a Euclidean grid with ≍ N log N atoms. The target law is proved
on a large family of critical subclasses retaining the same N log N -alphabet difficulty as the lower-bound construction.
The first positive engine is an exact rooted total-variation skeleton principle. It converts the Wasserstein value into
finite weighted sums of large-alphabet L1 distances, where the effective N log N gain from functional estimation is available.
This yields exact minimax laws on paired Euclidean grids, finite-band and packed direct sums, dyadic pair-isolation models,
critical laminar hierarchies, hierarchical tree classes, sparse-shortcut graph classes, continuum blob lifts, partition lifts,
full-support paired cores, contiguous split shells, martingale/Haar shells, smooth and real-analytic lower cores, and local
block models.
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