Marginal Optimal Transport for Missing Value Imputation

Optimal-transport methods for missing value imputation assume that compared batches share a common joint support. Heterogeneous masks violate this assumption: samples can have no jointly observed coordinates and therefore represent different marginals. We formulate Marginal Optimal Transport (MOT), which matches one Wasserstein marginal per missingness pattern. Under pattern coverage and chordal Markov regularity, the objective identifies the true joint; we also derive a finite-sample rate of order ${\mathcal{O}}_p(\min (n,r{)}^{-1/2})$ for an importance-weighted estimator and develop a scalable Kantorovich-dual estimator. To handle non-random missingness, an inverse-propensity-weighted variant is introduced as an empirical heuristic. Experiments confirm imputation, distributional, covariance, source, and graph recovery across synthetic and tabular benchmarks. We further prove that unprojected joint-support matching violates metric axioms and leaves its Kantorovich potential unidentified on unobserved coordinates.