Generalized Hunter-Saxton Equations, Optimal Information Transport, and Factorization of Diffeomorphisms

作者：Modin Klas^*

来源：Journal of Geometric Analysis, 2015, 25(2): 1306-1334.

DOI：10.1007/s12220-014-9469-2

• Euler-Arnold equations
• Euler-Poincare equations
• Descending metrics
• Riemannian submersion
• Diffeomorphism groups
• Fisher information metric
• Fisher-Rao metric
• Entropy differential metric
• Geometric statistics
• Hunter-Saxton equation
• Information geometry
• Optimal transport
• Polar factorization
• QR factorization
• Cholesky factorization
• Calabi metric

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 mu-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.

• 出版日期2015-4