Empirical Brenier potentials converge in L1(μ) at rate n^{-1/2} for d≤3, n^{-1/2} log^{5/2} n for d=4, and n^{-2/d} log^{(d+2)/d} n for d≥5, with sharp polynomial exponents, an FCLT and consistent bootstrap for d≤3.
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Empirical optimal transport potentials: fast rates and a functional central limit theorem
Empirical Brenier potentials converge in L1(μ) at rate n^{-1/2} for d≤3, n^{-1/2} log^{5/2} n for d=4, and n^{-2/d} log^{(d+2)/d} n for d≥5, with sharp polynomial exponents, an FCLT and consistent bootstrap for d≤3.