Information Geometry via the Q-Root Transform

2026/03/20 by Levin Maier · 1 voice
Computer Science · Mathematics · #cs.IT #math.DG #math.SG

paper · pdf · doi:10.48550/arxiv.2603.20081

Abstract

In this paper, we introduce ℓ^p-information geometry, an infinite-dimensional framework that shares key features with the geometry of the space of probability densities \( Dens(M) \) on a closed manifold, while also incorporating aspects of measure-valued information geometry. We define the ℓ²-probability simplex with a noncanonical differentiable structure induced via the q-root transform from an open subset of the \( ℓ^q \)-sphere. This choice makes the \(q\)-root transform an isometry and allows us to construct the \(ℓ²\)- and \(ℓ^q\)-Fisher--Rao geometries, including Amari--Čencov \(α\)-connections and a Chern connection in the \(ℓ^q\)-setting. We then apply this framework to an infinite-dimensional linear optimization problem. We show that the corresponding gradient flow with respect to the \(ℓ²\)--Fisher--Rao metric can be solved explicitly, converges to a maximizer under a natural monotonicity assumption, and admits an interpretation as the geodesic flow of an exponential connection. In particular, we prove that this \(e\)-connection is geodesically complete. We further relate these flows to a completely integrable Hamiltonian system through a momentum map associated with a Hamiltonian torus action on infinite-dimensional complex projective space. Finally, inspired by the \(ℓ²\)-theory, we outline an analogous Fisher--Rao geometry for \( Dens(M) \) on possibly noncompact Riemannian manifolds, showing that, with a suitable spherical differentiable structure, the square-root transform remains an isometry.

Citations

• On Mañé's Critical Value for Tonelli Lagrangians on Half Lie-Groups
• The Hopf--Rinow Theorem and Mañé's Critical Value for Magnetic Geodesics on Half Lie-Groups
• Information Geometry on the ℓ²-Simplex via the q-Root Transform
• Information geometry of diffeomorphism groups
• Riemannian geometry of the space of volume preserving immersions
• The Hopf-Rinow Theorem is false in infinite Dimensions

Discussions

• Levin Maier: Information Geometry via the Q-Root Transform https://arxiv.org/abs/2603.20081 https://arxiv.org/pdf/2603.20081 https://arxiv.org/html/2603.20081 [bsky, 0 points, 0 comments]

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