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

Nina Miolane,Xavier Pennec

doi:10.3390/e17041850

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.

Highlights

IntroductionData can be modeled as elements of Lie groups in many different fields: computational anatomy, robotics, paleontology, etc
We investigate here a special case of Lie algebras that we gain by going from Riemannian to pseudo-Riemannian: the double extension of W = {0} by a compact simple Lie algebra K, which is an example of a Manin triple
We have shown that a double extension g = K ⊕ K ∗ of W = {0} by a compact simple K, endowed with a bi-invariant pseudo-metric, is isometrically isomorphic to a dual Lie algebra g with a bi-invariant metric

Summary

Introduction

Data can be modeled as elements of Lie groups in many different fields: computational anatomy, robotics, paleontology, etc. One can take examples in robotics or in computational anatomy. The positions of the arm can be modeled as the elements of the three-dimensional Lie group of rotations SO(3). The spine can be modeled as an articulated object. In this context, each vertebra is considered as an orthonormal frame that encodes the rigid body transformation from the previous vertebra. As the human spine has 24 vertebrae, a configuration of the spine can be modeled as an element of the Lie group SE(3) , where SE(3) is the Lie group of rigid body transformations in 3D, i.e., the Lie group of rotations and translations in R3 , called the special Euclidean group. Its differential (at g), DLh (g) : Tg G 7→ TLh g G is an isomorphism that connects tangent spaces of G

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