  Given a family of probability measures in P(X), the space of probability
measures on a Hilbert space X, our goal in this paper is to highlight one ore
more curves in P(X) that summarize efficiently that family. We propose to study
this problem under the optimal transport (Wasserstein) geometry, using curves
that are restricted to be geodesic segments under that metric. We show that
concepts that play a key role in Euclidean PCA, such as data centering or
orthogonality of principal directions, find a natural equivalent in the optimal
transport geometry, using Wasserstein means and differential geometry. The
implementation of these ideas is, however, computationally challenging. To
achieve scalable algorithms that can handle thousands of measures, we propose
to use a relaxed definition for geodesics and regularized optimal transport
distances. The interest of our approach is demonstrated on images seen either
as shapes or color histograms.
