Medians of probability measures in Riemannian manifolds and applications to radar target detection

Abstract : In this thesis, we study the medians of a probability measure in a Riemannian manifold. Firstly, the existence and uniqueness of local medians are proved. In order to compute medians in practical cases, we also 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 curvatures. Moreover, we show that, in compact Riemannian manifolds, the Fréchet medians of generic data points are always unique. Some stochastic and deterministic algorithms are proposed for computing Riemannian p-means. A connection between medians and fixed point problems are also given. Finally, we apply the medians and the Riemannian geometry of Toeplitz covariance matrices to radar target detection.
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Le Yang. Medians of probability measures in Riemannian manifolds and applications to radar target detection. Differential Geometry [math.DG]. Université de Poitiers, 2011. English. ⟨tel-00664188⟩

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