  This paper is concerned by the statistical analysis of data sets whose
elements are random histograms. For the purpose of learning principal modes of
variation from such data, we consider the issue of computing the PCA of
histograms with respect to the 2-Wasserstein distance between probability
measures. To this end, we propose to compare the methods of log-PCA and
geodesic PCA in the Wasserstein space as introduced by Bigot et al. (2015) and
Seguy and Cuturi (2015). Geodesic PCA involves solving a non-convex
optimization problem. To solve it approximately, we propose a novel
forward-backward algorithm. This allows a detailed comparison between log-PCA
and geodesic PCA of one-dimensional histograms, which we carry out using
various data sets, and stress the benefits and drawbacks of each method. We
extend these results for two-dimensional data and compare both methods in that
setting.
