An Algorithmic Approach to Compute Principal Geodesics in the Wasserstein Space

We consider in this work the space of probability measures P(X) on a Hilbert space X endowed with the 2-Wasserstein metric. Given a family of probability measures in P(X), we propose an algorithm to compute curves that summarize efficiently that family in the 2-Wasserstein metric sense. To do so, we adapt the intuitive approach laid out by standard principal component analysis to the 2-Wasserstein metric, by using the Riemannian structure and associated concepts (Fr\'echet mean, geodesics, tangent vectors) that this metric defines on P(X). The curves we consider are generalized geodesics, which can be parameterized by two velocity fields defined on the support of the Wasserstein mean of the family of measures, each pointing towards an ending point of the generalized geodesic. We propose several approximations to optimize efficiently such velocity fields. Experiment re- sults show the ability of the computed principal components to capture axes of variability on histograms and probability measures data.