Generalised Hunter-Saxton equations, optimal information transport, and factorisation of diffeomorphisms

TL;DR

This paper explores geodesic equations on the diffeomorphism group related to Fisher's information metric, leading to a new factorisation of diffeomorphisms that solves an optimal information transport problem.

Contribution

It introduces a higher-dimensional generalisation of the Hunter-Saxton equation and a novel diffeomorphism factorisation akin to QR decomposition, linked to optimal information transport.

Findings

Established local existence and uniqueness of geodesics.

Derived a new factorisation of diffeomorphisms.

Connected the metric to Fisher's information and optimal transport.

Abstract

We study geodesic equations for a family of right-invariant Riemannian metrics on the group of diffeomorphisms of a compact manifold. The metrics descend to Fisher's information metric on the space of smooth probability densities. The right reduced geodesic equations are higher-dimensional generalisations of the $\mu$--Hunter--Saxton equation, used to model liquid crystals under influence of magnetic fields. Local existence and uniqueness results are established by proving smoothness of the geodesic spray. The descending property of the metrics is used to obtain a novel factorisation of diffeomorphisms. Analogous to the polar factorisation in optimal mass transport, this factorisation solves an optimal information transport problem. It can be seen as an infinite-dimensional version of $QR$ factorisation of matrices.