Efficient estimator achieving optimal rate under Lipschitz optimal kernel

Develop a computationally efficient estimator that achieves the minimax-optimal n^{-1/(d ∨ 2p)} finite-sample rate for estimating a Markov kernel under the transportation error functional E_p when there exists an optimal Lipschitz kernel κ⋆ satisfying W_p(κ⋆_x, κ⋆_{x'}) ≤ L \|x − x'\| for all x, x'.

Background

Under the assumption that an optimal kernel is Hölder continuous, the authors provide an information-theoretic reduction from kernel estimation to distribution estimation, yielding rates that match W_p empirical convergence (e.g., \widetilde{O}_{p,d}(n^{-1/(d ∨ 2p)}) in the Lipschitz case). They propose a Wasserstein distributionally robust optimization (WDRO) estimator to achieve these rates, but note it is computationally inefficient.

The open question seeks a polynomial-time (computationally efficient) estimator that attains the optimal n^{-1/(d ∨ 2p)} rate under E_p when an optimal Lipschitz kernel exists. The authors highlight that their experiments show strong empirical performance of a nearest-neighbor estimator, suggesting it as a promising candidate for achieving the optimal rate efficiently.