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Medians and Means in Riemannian Geometry: Existence, Uniqueness and Computation

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Matrix Information Geometry

Abstract

This paper is a short summary of our recent work on the medians and means of probability measures in Riemannian manifolds. Firstly, the existence and uniqueness results of local medians are given. In order to compute medians in practical cases, we propose a subgradient algorithm and prove its convergence. After that, Fréchet medians are considered. We prove their statistical consistency and give some quantitative estimations of their robustness with the aid of upper curvature bounds. We also show that, in compact Riemannian manifolds, the Fréchet medians of generic data points are always unique. Stochastic and deterministic algorithms are proposed for computing Riemannian p-means. The rate of convergence and error estimates of these algorithms are also obtained. Finally, we apply the medians and the Riemannian geometry of Toeplitz covariance matrices to radar target detection.

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Authors and Affiliations

  1. Laboratoire de Matheématiques et Applications, UMR 7348 du CNRS, Université de Poitiers, Téléport 2, rue Marie et Pierre Curie, BP 30179, 86962, Futuroscope Chasseneuil Cedex, France

    Marc Arnaudon & Le Yang

  2. Anvanced Developments Department, Thales Air Systems, Surface Radar, Technical Directorate, 91470, Limours, France

    Frédéric Barbaresco

Authors
  1. Marc Arnaudon
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  2. Frédéric Barbaresco
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  3. Le Yang
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Corresponding author

Correspondence to Marc Arnaudon .

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Editors and Affiliations

  1. Higashi Gotanda 3-14-13, Shinagawa-Ku, 141-0022, Japan

    Frank Nielsen

  2. S.J.S. Sansanwal Marg 7, Delhi, 110016, India

    Rajendra Bhatia

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Arnaudon, M., Barbaresco, F., Yang, L. (2013). Medians and Means in Riemannian Geometry: Existence, Uniqueness and Computation. In: Nielsen, F., Bhatia, R. (eds) Matrix Information Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30232-9_8

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  • DOI: https://doi.org/10.1007/978-3-642-30232-9_8

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-30231-2

  • Online ISBN: 978-3-642-30232-9

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