Consistency and convergence rates under local smoothness

Determine consistency and convergence rates for the kernel-weighted local estimator of T_μ when the transport field {T_μ : μ ∈ P_2(ℝ^d)} satisfies local smoothness conditions, specifying the dependence of rates on sample size, dimension, and kernel bandwidth h.

Background

The proposed estimator minimizes a kernel-weighted Wasserstein-2 loss to learn locally defined transport maps around reference measures. The kernel bandwidth h controls the effective neighborhood and data localization, and the method draws analogy to classical kernel regression in Euclidean spaces.

While empirical results show strong performance in low dimensions and adequate data regimes, the authors explicitly note that theoretical guarantees such as consistency and convergence rates under local smoothness of the transport field remain open and are planned for future work.