Geodesic PCA versus Log-PCA of Histograms in the Wasserstein Space

Elsa Cazelles, Vivien Seguy, Jérémie Bigot, Marco Cuturi, Nicolas Papadakis
2018 SIAM Journal on Scientific Computing  
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 in [BGKL15, SC15]. Geodesic PCA involves solving a non-convex optimization
more » ... x 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 datasets, and stress the benefits and drawbacks of each method. We extend these results for two-dimensional data and compare both methods in that setting.
doi:10.1137/17m1143459 fatcat:dgm26znm2rc6dcqbesuioidqie