Year: 2026
Type: article
Abstract: We give a new geometric interpretation of the Amari–Čencov α -connections ∇ ( α ) from information geometry: On the space of densities Dens + ( M ) , we show that there exist Riemannian metrics G α , which we call α -Fisher–Rao (FR) metrics, whose Levi-Civita connections are ∇ ( α ) . With the exception of α = 0 (the FR metric), these metrics are non-invariant to the action of the diffeomorphism group Diff ( M ) , even though the connections are invariant. This gives a new way of interpreting the geodesics of the ∇ ( α ) as energy-minimizing curves. On the space of probability densities Prob ( M ) , we show that the same phenomenon holds for α ∈ { − 1 , 0 , 1 } and that the α -connections are not metric otherwise. We show that ∇ ( α ) -geodesics on this space can be interpreted as radial projections of straight lines on appropriate hyper-surfaces, and use this geometric picture to obtain geodesic convexity for any
Source: Nonlinearity
Authors: Martin Bauer, Alice Le Brigant, Cy Maor
Institutions: Florida State University, Hebrew University of Jerusalem
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,325
Funders: National Science Foundation, United States - Israel Binational Science Foundation, Erwin Schrödinger International Institute for Mathematics and Physics, University of Toronto, Israel Science Foundation, Directorate for Computer and Information Science and Engineering, Division of Computing and Communication Foundations
OpenAlex ID: W7167287728