Generalised Hunter–Saxton equations and optimal information transport
Preprint, 2012

A right invariant H1–type Riemannian metric on the group of diffeomorphisms of a compact manifold is studied. The significance of this metric is that it descends, by a Riemannian submersion, to the constant curvature Fisher metric on the space of smooth probability densities. The right reduced geodesic equation is a higher dimensional generalisation of the μ–Hunter–Saxton equation, which describes liquid crystals under influence of an external magnetic field. A local existence and uniqueness result is obtained by proving that the geodesic spray is smooth with respect to Hs Banach topologies. Based on the descending property of the metric, a polar factorisation result for diffeomorphisms is given. Analogous to the polar factorisation used in optimal mass transport, this factorisation solves a corresponding optimal information transport problem with respect to the Fisher metric. It can be seen as an infinite dimensional version of the classical QR factorisation of matrices.


Klas Modin

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Matematik