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

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.

30 pages, published in J. Geom. Anal

References in corpus (4)

• Classical and Quantum Fisher Information in the Geometrical Formulation of Quantum Mechanics
• Integrable evolution equations on spaces of tensor densities and their peakon solutions
• Ricci flow and the metric completion of the space of Kahler metrics
• A note on multi-dimensional Camassa-Holm type systems on the torus

Cited by in corpus (14)

• Geometric Hydrodynamics via Madelung Transform
• Geometric hydrodynamics and infinite-dimensional Newton's equations
• Diffeomorphic density matching by optimal information transport
• Real-Time 2D-3D Deformable Registration with Deep Learning and Application to Lung Radiotherapy Targeting
• Geometry of Matrix Decompositions Seen Through Optimal Transport and Information Geometry
• Semi-invariant Riemannian metrics in hydrodynamics
• Gaussian distributions, Jacobi group and Siegel-Jacobi space
• The $L^p$-Fisher-Rao metric and Amari-Cencov $\alpha$-connections
• Diffeomorphic random sampling using optimal information transport
• Geometry of discrete-time spin systems
• Shape analysis via gradient flows on diffeomorphism groups
• The Toda flow as a porous medium equation
• On Geodesic Completeness for Riemannian Metrics on Smooth Probability Densities
• Rank Constrained Diffeomorphic Density Motion Estimation for Respiratory Correlated Computed Tomography