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

Chapter

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.

Keywords

Probability Measure Riemannian Manifold Sectional Curvature Riemannian Geometry Compact Riemannian Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Laboratoire de Matheématiques et Applications, UMR 7348 du CNRSUniversité de PoitiersFuturoscope Chasseneuil CedexFrance
  2. 2.Anvanced Developments DepartmentThales Air Systems, Surface Radar, Technical DirectorateLimoursFrance

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