Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics
In Computational Anatomy, organ’s shapes are often modelled as deformations of a reference shape, i.e. as elements of a Lie group. To analyse 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 direct product of compact and abelian groups have no bi-invariant metric. But what about bi-invariant pseudo-metrics? In other words: could we remove the assumption of positivity of the metric and obtain consistent statistics on Lie groups through the pseudo-Riemannian framework? Our contribution is two folds. First, we present an algorithm that constructs bi-invariant pseudo-metrics on a given Lie group, in 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.