Log-PCA versus Geodesic PCA of histograms in the Wasserstein space

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 meas...

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Bibliographic Details
Published inarXiv.org
Main Authors Cazelles, Elsa, Seguy, Vivien, Bigot, Jérémie, Cuturi, Marco, Papadakis, Nicolas
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 27.08.2017
Subjects
Algorithms
Computational geometry
Convexity
Datasets
Histograms
Optimization
Statistical analysis
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Summary: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.
ISSN:2331-8422