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.