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

Elsa Cazelles (), Vivien Seguy (), Jérémie Bigot (), Marco Cuturi () and Nicolas Papadakis ()
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Elsa Cazelles: Institut de Mathématiques de Bordeaux; CNRS; IMB-UMR5251; Université de Bordeaux
Vivien Seguy: Graduate School of Informatics; Kyoto University
Jérémie Bigot: Institut de Mathématiques de Bordeaux; CNRS; IMB-UMR5251; Université de Bordeaux
Marco Cuturi: CREST; ENSAE; Université Paris-Saclay
Nicolas Papadakis: Institut de Mathématiques de Bordeaux ; CNRS; IMB-UMR5251; Université de Bordeaux

No 2017-85, Working Papers from Center for Research in Economics and Statistics

Abstract: 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 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 bene ts and drawbacks of each method. We extend these ;Classification-JEL: 62-07, 68R10, 62H25

Keywords: Geodesic Principal Componant Analysis; Wasserstein Space; Non-convex optimization (search for similar items in EconPapers)
Pages: 32 pages
Date: 2017-08-27
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Citations: View citations in EconPapers (7)

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