Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles Entropy Year : 2015

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

Nina Miolane
  • Function : Author
  • PersonId : 951696
  • IdRef : 199540519

Abstract

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.
Fichier principal
Vignette du fichier
entropy-17-01850.pdf (332.58 Ko) Télécharger le fichier
Origin Publisher files allowed on an open archive
Loading...

Dates and versions

hal-01133922 , version 1 (19-06-2015)

Identifiers

Cite

Nina Miolane, Xavier Pennec. Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics. Entropy, 2015, 17 (4), pp.1850-1881. ⟨10.3390/e17041850⟩. ⟨hal-01133922⟩
585 View
411 Download

Altmetric

Share

Gmail Mastodon Facebook X LinkedIn More