The Lᵖ-Fisher–Rao metric and Amari–C̆encov -Connections

Abstract We introduce a family of Finsler metrics, called the Lᵖ L p -Fisher–Rao metrics Fₚ F p, for p (1, ) p ∈ (1, ∞), which generalizes the classical Fisher–Rao metric F₂ F 2, both on the space of densities {Dens}_+ (M) Dens + (M) and probability densities {Prob} (M) Prob (M). We then study their relations to the Amari–C̆encov α -connections ^ () ∇ (α) from information geometry: on {Dens}_+ (M) Dens + (M), the geodesic equations of Fₚ F p and ^ () ∇ (α) coincide, for p = 2/ (1-) p = 2 / (1 - α). Both are pullbacks of canonical constructions on Lᵖ (M) L p (M), in which geodesics are simply straight lines. In particular, this gives a new variational interpretation of α -geodesics as being energy minimizing curves. On {Prob} (M) Prob (M), the Fₚ F p and ^ () ∇ (α) geodesics can still be thought as pullbacks of natural operations on the unit sphere in Lᵖ (M) L p (M), but in this case they no longer coincide unless p=2 p = 2. Using this transformation, we solve the geodesic equation of the α -connection by showing that the geodesic are pullbacks of projections of straight lines onto the unit sphere, and they always cease to exists after finite time when they leave the positive part of the sphere. This unveils the geometric structure of solutions to the generalized Proudman–Johnson equations, and generalizes them to higher dimensions. In addition, we calculate the associate tensors of Fₚ F p, and study their relation to ^ () ∇ (α).