A Riemannian viewpoint on the Amari–Cencov α-connections and Proudman–Johnson equations
Abstract
We give a new geometric interpretation of the Amari--Cencov α-connections ∇(α) from information geometry: On the space of densities, 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, 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, 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 the space of diffeomorphisms, which, for the case M=R, imply that the generalized Proudman--Johnson equations on the real line are the Euler--Arnold-type 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 statistical model considered.
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BibTeX
@article{bauer2026riemannian,
title = {{A Riemannian viewpoint on the Amari–Cencov $\alpha$-connections and Proudman–Johnson equations}},
author = {Martin Bauer and Alice Le Brigant and Cy Maor},
journal = {Nonlinearity},
year = {2026},
month = jul,
doi = {10.1088/1361-6544/ae806f},
url = {https://hal.science/hal-05791793v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - A Riemannian viewpoint on the Amari–Cencov α-connections and Proudman–Johnson equations
AU - Martin Bauer
AU - Alice Le Brigant
AU - Cy Maor
PY - 2026
DA - 2026/07/03
JO - Nonlinearity
DO - 10.1088/1361-6544/ae806f
UR - https://hal.science/hal-05791793v1
AB - We give a new geometric interpretation of the Amari--Cencov α-connections ∇^(α) from information geometry: On the space of densities, we show that there exist Riemannian metrics G^α, 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, 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, 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 the space of diffeomorphisms, which, for the case M=R, imply that the generalized Proudman--Johnson equations on the real line are the Euler--Arnold-type 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 statistical model considered.
SN - 0951-7715
SN - 1361-6544
ER -