Geodesic PCA in the Wasserstein space

We introduce the method of Geodesic Principal Component Analysis (GPCA) analysis on the space of probability measures on the line, with finite second moments, endowed with the Wasserstein metric. We discuss the advantages of this approach over a standard functional PCA of probability densities in the Hilbert space of square-integrable functions. We establish the consistency of the method by showing that the empirical GPCA converges to its population counterpart as the sample size tends to infinity. We also give illustrative examples on simple statistical models to show the benefits of this approach for data analysis.