A Riemannian viewpoint on the Amari-Cencov $α$-connections and Proudman-Johnson equations
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| Format: | Preprint |
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2025
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| _version_ | 1866908475214266368 |
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| author | Bauer, Martin Brigant, Alice Le Maor, Cy |
| author_facet | Bauer, Martin Brigant, Alice Le Maor, Cy |
| contents | We give a new geometric interpretation of the Amari-Cencov $α$-connections $\nabla^{(α)}$ from information geometry: On the space of densities $\operatorname{Dens}_+(M)$, we show that there exist Riemannian metrics $G^α$, which we call $α$-Fisher-Rao metrics, whose Levi-Civita connections are $\nabla^{(α)}$. With the exception of $α=0$ (the Fisher-Rao metric), these metrics are non-invariant to the action of the diffeomorphism group $\operatorname{Diff}(M)$, even though the connections are invariant. This gives a new way of interpreting the geodesics of the $\nabla^{(α)}$ as energy-minimizing curves. On the space of probability densities $\operatorname{Prob}(M)$, we show that the same phenomenon holds for $α\in \{-1,0,1\}$ and that the $α$-connections are not metric otherwise. We show that $\nabla^{(α)}$-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 $α\in \mathbb{R}$. In addition, we prove analogous results for appropriate metrics and connections on $\operatorname{Diff}(M)$, which, for the case $M=\mathbb{R}$, 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 $\nabla^{(α)}$ can be metric or non-metric depending on the considered statistical model. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_00371 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Riemannian viewpoint on the Amari-Cencov $α$-connections and Proudman-Johnson equations Bauer, Martin Brigant, Alice Le Maor, Cy Differential Geometry We give a new geometric interpretation of the Amari-Cencov $α$-connections $\nabla^{(α)}$ from information geometry: On the space of densities $\operatorname{Dens}_+(M)$, we show that there exist Riemannian metrics $G^α$, which we call $α$-Fisher-Rao metrics, whose Levi-Civita connections are $\nabla^{(α)}$. With the exception of $α=0$ (the Fisher-Rao metric), these metrics are non-invariant to the action of the diffeomorphism group $\operatorname{Diff}(M)$, even though the connections are invariant. This gives a new way of interpreting the geodesics of the $\nabla^{(α)}$ as energy-minimizing curves. On the space of probability densities $\operatorname{Prob}(M)$, we show that the same phenomenon holds for $α\in \{-1,0,1\}$ and that the $α$-connections are not metric otherwise. We show that $\nabla^{(α)}$-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 $α\in \mathbb{R}$. In addition, we prove analogous results for appropriate metrics and connections on $\operatorname{Diff}(M)$, which, for the case $M=\mathbb{R}$, 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 $\nabla^{(α)}$ can be metric or non-metric depending on the considered statistical model. |
| title | A Riemannian viewpoint on the Amari-Cencov $α$-connections and Proudman-Johnson equations |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2508.00371 |