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Minimax Rate for Wasserstein Distance Estimation in High Dimensions
§ Problem Statement
Setup
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§ Discussion
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§ Significance & Implications
§ Known Partial Results
§ References
[1]
Estimation of Wasserstein distances in the spiked transport model
Jonathan Niles-Weed, Philippe Rigollet (2022)
Bernoulli
📍 Section 3.3 (Lower bounds), paragraph immediately after Theorem 3 (explicitly: “Closing this gap in the context of estimation of the Wasserstein distance is an interesting and fundamental question”).
[2]
Sharp convergence rates for empirical optimal transport with smooth costs
Tudor Manole, Jonathan Niles-Weed, Philippe Rigollet (2024)
Annals of Applied Probability
[3]
Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance
Jonathan Weed, Francis Bach (2019)
Bernoulli
📍 Section 3 (Main results), Theorem 2 (minimax rate n^{-2/d} for W_p on [0,1]^d when d ≥ 2p+1), p. 2889.