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