Article
Version 1
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Cartan-Schouten Metrics for Information Geometry and Machine Learning
Version 1
: Received: 28 August 2024 / Approved: 29 August 2024 / Online: 29 August 2024 (08:30:04 CEST)
How to cite: Diatta, A.; Manga, B.; Sy, F. Cartan-Schouten Metrics for Information Geometry and Machine Learning. Preprints 2024, 2024082120. https://doi.org/10.20944/preprints202408.2120.v1 Diatta, A.; Manga, B.; Sy, F. Cartan-Schouten Metrics for Information Geometry and Machine Learning. Preprints 2024, 2024082120. https://doi.org/10.20944/preprints202408.2120.v1
Abstract
We study Cartan-Schouten metrics, explore invariant dual connections, and propose them as models for Information Geometry. Based on the underlying Riemannian barycenter and the biinvariant mean of Lie groups, we subsequently propose a new parametric mean for data science and machine learning which comes with several advantages compared to traditional tools such as the arithmetic mean, median, mode, expectation, least square method, maximum likelihood, linear regression. We call a metric on a Lie group, a Cartan-Schouten metric, if its Levi-Civita connection is biinvariant, so every 1-parameter subgroup through the unit is a geodesic. Except for not being left or right invariant in general, Cartan-Schouten metrics enjoy the same geometry as biinvariant metrics, since they share the same Levi-Civita connection. To bypass the non-invariance apparent drawback, we show that Cartan-Schouten metrics are completely determined by their value at the unit. We give an explicit formula for recovering them from their value at the unit, thus making them much less computationally demanding, compared to general metrics on manifolds. Furthermore, Lie groups with Cartan-Schouten metrics are complete Riemannian or pseudo-Riemannian manifolds. We give a complete characterization of Lie groups with Riemannian or Lorentzian Cartan-Schouten metrics. Cartan-Schouten metrics are in abundance on 2-nilpotent Lie groups. Namely, on every 2-nilpotent Lie group, there is a 1-1 correspondence between the set of left invariant metrics and that of Cartan-Schouten metrics.
Keywords
Cartan-Schouten metric; dual connections; $\alpha$-connections; Fisher information matrix; biinvariant metric; Lorentz metric; exponential barycenter; center of mass; Amari-Chentov 3-tenor; Machine Learning; information geometry
Subject
Computer Science and Mathematics, Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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