Geodesic PCA in the Wasserstein space and applications to the statistical analysis of densities

Abstract

This work is motivated by the problem of defining non-linear PCA methods for the statistical analysis of densities. For this purpose, 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 give illustrative examples on simple statistical models to show the benefits of this approach for data analysis. We also discuss the connection between this approach and the statistical analysis of geometric deformations in signal and image processing.