Identifiability of the local signal map T_mu

Establish identifiability conditions for the locally defined signal map T_mu in the nonparametric distribution-on-distribution regression model ν = T_ε#(T_μ#μ), given independent training pairs {(μ_i, ν_i)} and kernel weights K_h(μ, μ_i) based on the 2-Wasserstein distance, so that T_μ is uniquely recoverable from data in neighborhoods of a reference measure μ.

Background

The paper introduces Neural Local Wasserstein Regression, a nonparametric framework for distribution-on-distribution regression that models transport via locally defined maps rather than global optimal transport maps or tangent-space linearization. The regression model is ν = T_ε#(T_μ#μ), where the signal map T_μ depends on the source measure μ and is estimated locally using kernel weights based on W2 distances.

A central methodological component is a kernel-weighted empirical loss over pairs (μi, ν_i), which localizes the estimator and allows learning a separate transport map around each reference measure. While the approach is practically effective, theoretical properties such as identifiability of Tμ are explicitly stated as unresolved and deferred to future work.