Semiparametric Efficiency for Entropic Optimal Transport under Structured Distribution Shift

Entropic optimal transport (EOT) is widely used to compare distributions and learned representations, but its empirical estimate can be noisy when target data are scarce. We study whether a low-dimensional structural relation between source and target distributions can improve estimation of a fixed-regularization EOT functional. Under an exponential density-ratio model with a nonparametric baseline, we derive the canonical gradient and an exact efficiency identity: the attainable variance reduction is characterized by a weighted projection residual of the unrestricted transport influence contrast. We construct an empirical-likelihood estimator, DRM-EOT, that attains the semiparametric efficiency bound on compact support, together with consistent analytic variance estimation and a valid stratified refit bootstrap. The same residual also governs first-order sensitivity to local misspecification, yielding an explicit efficiency–robustness trade-off. Controlled simulations and exact semi-synthetic shifts on OfficeHome representations confirm the predicted variance gains. Under natural OfficeHome shifts, low-dimensional DRM can reduce target-scarce RMSE mainly through bias reduction, while increasing model complexity exposes separation and persistent misspecification bias. These results distinguish exact-model efficiency from structural regularization and characterize when information sharing across domains improves EOT estimation.