2025/08/01 by Martin Bauer, Bauer, Martin, Alice Le Brigant +3 · 1 citation
Mathematics · Physics and Astronomy · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Morphological variations and asymmetry #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.2508.00371
openalex publication_date 2025/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a new geometric interpretation of the Amari-Cencov α-connections ∇(α) from information geometry: On the space of densities Dens+(M), we show that there exist Riemannian metrics Gα, which we call α-Fisher-Rao metrics, whose Levi-Civita connections are ∇(α). With the exception of α=0 (the Fisher-Rao 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 α∈ ℝ. In addition, we prove analogous results for appropriate metrics and connections on Diff(M), which, for the case M=ℝ, imply that the generalized Proudman-Johnson equations on the real line are the Euler-Arnold equations of non-right invariant metrics. Finally, in the finite-dimensional case, we show that ∇(α) can be metric or non-metric depending on the considered statistical model.