A Riemannian viewpoint on the Amari-Cencov $\hat{I}\pm$-connections and Proudman-Johnson equations

Abstract

We give a new geometric interpretation of the Amari-Cencov $\hat{I}\pm$-connections ${\nabla }^{(\hat{I}\pm )}$ from information geometry: On the space of densities ${\mathrm{Dens}}_{+}(M)$, we show that there exist Riemannian metrics $G^{\hat{I}}\pm$, which we call $\hat{I}\pm$-Fisher-Rao metrics, whose Levi-Civita connections are ${\nabla }^{(\hat{I}\pm )}$. With the exception of $\hat{I}\pm =0$ (the Fisher-Rao metric), these metrics are non-invariant to the action of the diffeomorphism group $\mathrm{Diff}(M)$, even though the connections are invariant. This gives a new way of interpreting the geodesics of the ${\nabla }^{(\hat{I}\pm )}$ as energy-minimizing curves. On the space of probability densities $\mathrm{Prob}(M)$, we show that the same phenomenon holds for $\hat{I}\pm \in \{-1,0,1\}$ and that the $\hat{I}\pm$-connections are not metric otherwise. We show that ${\nabla }^{(\hat{I}\pm )}$-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 $\hat{I}\pm \in \mathbb{R}$. In addition, we prove analogous results for appropriate metrics and connections on $\mathrm{Diff}(M)$, 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 ${\nabla }^{(\hat{I}\pm )}$ can be metric or non-metric depending on the considered statistical model.