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

Martin Bauer, Alice Le Brigant, Yuxiu Lu, Cy Maor

Research output: Contribution to journalArticlepeer-review


We introduce a family of Finsler metrics, called the Lp-Fisher–Rao metrics Fp, for p∈(1,∞), which generalizes the classical Fisher–Rao metric F2, both on the space of densities Dens+(M) and probability densities Prob(M). We then study their relations to the Amari–C̆encov α-connections ∇(α) from information geometry: on Dens+(M), the geodesic equations of Fp and ∇(α) coincide, for p=2/(1-α). Both are pullbacks of canonical constructions on Lp(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), the Fp and ∇(α) geodesics can still be thought as pullbacks of natural operations on the unit sphere in Lp(M), but in this case they no longer coincide unless 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, and study their relation to ∇(α).

Original languageAmerican English
Article number56
JournalCalculus of Variations and Partial Differential Equations
Issue number2
StatePublished - Mar 2024


  • 35Q35
  • 53C60
  • 58B20
  • 58D05

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics


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