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Open AccessArticle

A Bi-Invariant Statistical Model Parametrized by Mean and Covariance on Rigid Motions

1
Institut Fresnel, Aix-Marseille University, CNRS, Centrale Marseille, 13013 Marseille, France
2
Université Côte d’Azur, Inria Epione, 06902 Sophia Antipolis, France
*
Author to whom correspondence should be addressed.
Entropy 2020, 22(4), 432; https://doi.org/10.3390/e22040432
Received: 10 March 2020 / Revised: 7 April 2020 / Accepted: 9 April 2020 / Published: 10 April 2020
This paper aims to describe a statistical model of wrapped densities for bi-invariant statistics on the group of rigid motions of a Euclidean space. Probability distributions on the group are constructed from distributions on tangent spaces and pushed to the group by the exponential map. We provide an expression of the Jacobian determinant of the exponential map of S E ( n ) which enables the obtaining of explicit expressions of the densities on the group. Besides having explicit expressions, the strengths of this statistical model are that densities are parametrized by their moments and are easy to sample from. Unfortunately, we are not able to provide convergence rates for density estimation. We provide instead a numerical comparison between the moment-matching estimators on S E ( 2 ) and R 3 , which shows similar behaviors. View Full-Text
Keywords: wrapped distributions; rigid motions; Euclidean groups; differential of the exponential; moment-matching estimator; density estimation; sampling wrapped distributions; rigid motions; Euclidean groups; differential of the exponential; moment-matching estimator; density estimation; sampling
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Chevallier, E.; Guigui, N. A Bi-Invariant Statistical Model Parametrized by Mean and Covariance on Rigid Motions. Entropy 2020, 22, 432.

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