Minimax finite-sample risk for E_p-based kernel estimation

Determine the exact minimax finite-sample risk rate for estimating a Markov kernel (stochastic optimal transport map) κ from a source distribution μ to a target distribution ν under the transportation error functional E_p, when μ and ν are supported on [0,1]^d. Here E_p(κ; μ, ν) is defined as the sum of the nonnegative difference between the kernel’s transportation cost and W_p(μ, ν), and the W_p distance between κ_♯μ and ν. Ascertain whether the optimal rate is n^{-1/(d ∨ 2p)} or n^{-1/(d + 2p)}, and investigate whether a multi-scale sampling analysis can extend to kernel estimation to improve the current upper bound.

Background

The paper introduces the transportation error functional E_p(κ; μ, ν) to evaluate stochastic optimal transport kernels without requiring uniqueness or existence of deterministic optimal maps. Using this metric, the authors develop estimators with provable rates, including a rounding-based estimator that achieves E[E_p] = \widetilde{O}_{p,d}(n^{-1/(d+2p)}), and establish a minimax lower bound of \Omega(n^{-1/(d ∨ 2p)}), leaving a gap between upper and lower bounds.

The authors explicitly pose determining the precise minimax rate as an open question, suggesting that a multi-scale approach (which improves empirical Wasserstein convergence) might extend to kernel estimation. They note a one-dimensional refinement achieving n^{-1/2} and ask whether analogous improvements are possible in higher dimensions to close the current gap.