Cartan-Schouten metrics for information geometry and machine learning
Andre Diatta, Bakary Manga, Fatimata Sy
TL;DR
The paper develops Cartan-Schouten metrics on Lie groups as a flexible, information-geometry–friendly generalization of biinvariant metrics, defined by a Levi-Civita connection $\nabla_x y=\tfrac12[x,y]$ and yielding geodesics along 1-parameter subgroups. It establishes a deep link between Lie-algebra metrics and group-level Cartan-Schouten metrics, proves completeness and rich structures (including Lorentzian variants) on notable families such as 2-nilpotent and oscillator groups, and provides explicit metric constructions in key cases like the Heisenberg and H-type Carnot groups. The work then connects these geometric structures to statistical modeling by formulating dual connections, Hessian statistics, and biinvariant duals, culminating in a new data-science model: a parametric, 2-nilpotent–based mean whose components blend arithmetic means with variance-like terms and evolve with a family of Lie-algebra parameters. A practical upshot is a robust, computation-friendly framework for information geometry on Lie groups, with an exponential barycenter interpretation and a versatile, tunable parametric mean for machine learning and data analysis on structured spaces.
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.