Generalized Hunter–Saxton equations, optimal information transport, and factorization of diffeomorphisms
Journal article, 2015
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 generalizations of the μ-Hunter–Saxton equation, used to model liquid crystals under the 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 factorization of diffeomorphisms. Analogous to the polar factorization in optimal mass transport, this factorization solves an optimal information transport problem. It can be seen as an infinite-dimensional version of QR factorization of matrices.
Geometric statistics
Euler-Arnold equations
Riemannian submersion
Entropy differential metric
Calabi metric
QR factorization
Fisher-Rao metric
Cholesky factorization
Optimal transport
Fisher information metric
Descending metrics
Hunter-Saxton equation
Diffeomorphism groups
Polar factorization
Euler-Poincare equations
Information geometry
Author
Klas Modin
University of Gothenburg
Chalmers, Mathematical Sciences, Mathematics
Journal of Geometric Analysis
1050-6926 (ISSN)
Vol. 25 2 1306-1334Areas of Advance
Information and Communication Technology
Roots
Basic sciences
Subject Categories
Geometry
Mathematical Analysis
DOI
10.1007/s12220-014-9469-2