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

Principal Geodesic Analysis for Probability Measures under the Optimal Transport Metric