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Entropy 2015, 17(4), 1850-1881; doi:10.3390/e17041850

Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics

INRIA, Asclepios project-team, 2004 Route des Lucioles, BP93, Sophia Antipolis Cedex F-06902, France
This paper is an extended version of our paper published in MaxEnt 2014, Amboise, France, 21–26 September 2014.
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Received: 31 January 2015 / Revised: 19 March 2015 / Accepted: 20 March 2015 / Published: 31 March 2015
(This article belongs to the Special Issue Information, Entropy and Their Geometric Structures)
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Abstract

In computational anatomy, organ’s shapes are often modeled as deformations of a reference shape, i.e., as elements of a Lie group. To analyze the variability of the human anatomy in this framework, we need to perform statistics on Lie groups. A Lie group is a manifold with a consistent group structure. Statistics on Riemannian manifolds have been well studied, but to use the statistical Riemannian framework on Lie groups, one needs to define a Riemannian metric compatible with the group structure: a bi-invariant metric. However, it is known that Lie groups, which are not a direct product of compact and abelian groups, have no bi-invariant metric. However, what about bi-invariant pseudo-metrics? In other words: could we remove the assumption of the positivity of the metric and obtain consistent statistics on Lie groups through the pseudo-Riemannian framework? Our contribution is two-fold. First, we present an algorithm that constructs bi-invariant pseudo-metrics on a given Lie group, in the case of existence. Then, by running the algorithm on commonly-used Lie groups, we show that most of them do not admit any bi-invariant (pseudo-) metric. We thus conclude that the (pseudo-) Riemannian setting is too limited for the definition of consistent statistics on general Lie groups. View Full-Text
Keywords: Lie group; Lie algebra; statistics; pseudo-Riemannian Lie group; Lie algebra; statistics; pseudo-Riemannian
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Miolane, N.; Pennec, X. Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics. Entropy 2015, 17, 1850-1881.

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