Statistical Learning in Wasserstein Space

We seek a generalization of regression and principle component analysis (PCA) in metric space where data points are distributions metrized by the Wasserstein metric. recast these analyses as multimarginal optimal transport problems. The particular formulation allows efficient computation, ensures existence solutions, admits probabilistic interpretation over paths (line segments). Application theory to interpolation empirical distributions, images, power spectra, well assessing uncertainty experimental designs, is envisioned.