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

Author

Listed:
  • 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)

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

Suggested Citation

  • Elsa Cazelles & Vivien Seguy & Jérémie Bigot & Marco Cuturi & Nicolas Papadakis, 2017. "Log-PCA versus Geodesic PCA of histograms in the Wasserstein space," Working Papers 2017-85, Center for Research in Economics and Statistics.
    • Handle: RePEc:crs:wpaper:2017-85
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    Cited by:

    1. Betancourt, José & Bachoc, François & Klein, Thierry & Idier, Déborah & Pedreros, Rodrigo & Rohmer, Jérémy, 2020. "Gaussian process metamodeling of functional-input code for coastal flood hazard assessment," Reliability Engineering and System Safety, Elsevier, vol. 198(C).
    2. Le Brigant, Alice & Puechmorel, Stéphane, 2019. "Quantization and clustering on Riemannian manifolds with an application to air traffic analysis," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 685-703.
    3. Chao Zhang & Piotr Kokoszka & Alexander Petersen, 2022. "Wasserstein autoregressive models for density time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(1), pages 30-52, January.
    4. Florian Gunsilius & Meng Hsuan Hsieh & Myung Jin Lee, 2022. "Tangential Wasserstein Projections," Papers 2207.14727, arXiv.org, revised Aug 2022.
    5. Petersen, Alexander & Zhang, Chao & Kokoszka, Piotr, 2022. "Modeling Probability Density Functions as Data Objects," Econometrics and Statistics, Elsevier, vol. 21(C), pages 159-178.

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