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

التفاصيل البيبلوغرافية
العنوان: Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics
المؤلفون: Nina Miolane, Xavier Pennec
المصدر: Entropy, Vol 17, Iss 4, Pp 1850-1881 (2015)
بيانات النشر: MDPI AG, 2015.
سنة النشر: 2015
المجموعة: LCC:Science
LCC:Astrophysics
LCC:Physics
مصطلحات موضوعية: Lie group, Lie algebra, statistics, pseudo-Riemannian, Science, Astrophysics, QB460-466, Physics, QC1-999
الوصف: 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.
نوع الوثيقة: article
وصف الملف: electronic resource
اللغة: English
تدمد: 1099-4300
Relation: http://www.mdpi.com/1099-4300/17/4/1850; https://doaj.org/toc/1099-4300
DOI: 10.3390/e17041850
URL الوصول: https://doaj.org/article/9ea511ff289042c89936ecf23b5e4100
رقم الأكسشن: edsdoj.9ea511ff289042c89936ecf23b5e4100
قاعدة البيانات: Directory of Open Access Journals
الوصف
تدمد:10994300
DOI:10.3390/e17041850