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.
Due to modifications of the definition of the bbob-largescale testbed of COCO [2] and of the corresponding source code, a second, updated version of the article [1] is presented here, with corrections/additions indicated by the colored text. Additionally to this, the following corrections/modifications have been done:• All figures have been updated: they present post processed results after benchmarking the solvers on the updated suite.• The links to plots of specific testbed functions within the text redirect to an updated repository, containing the whole new dataset of post processed results.
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