{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T07:25:45Z","timestamp":1791271545065,"version":"4.1.0"},"reference-count":52,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2015,3,31]],"date-time":"2015-03-31T00:00:00Z","timestamp":1427760000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In computational anatomy, organ\u2019s 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.<\/jats:p>","DOI":"10.3390\/e17041850","type":"journal-article","created":{"date-parts":[[2015,4,7]],"date-time":"2015-04-07T03:47:46Z","timestamp":1428378466000},"page":"1850-1881","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics"],"prefix":"10.3390","volume":"17","author":[{"given":"Nina","family":"Miolane","sequence":"first","affiliation":[{"name":"INRIA, Asclepios project-team, 2004 Route des Lucioles, BP93, Sophia Antipolis Cedex F-06902, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xavier","family":"Pennec","sequence":"additional","affiliation":[{"name":"INRIA, Asclepios project-team, 2004 Route des Lucioles, BP93, Sophia Antipolis Cedex F-06902, France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2015,3,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Thompson, D.W., and Bonner, J.T. 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