Generalized Hunter–Saxton equations, optimal information transport, and factorization of diffeomorphisms
Artikel i vetenskaplig tidskrift, 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

Författare

Klas Modin

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Matematik

Journal of Geometric Analysis

1050-6926 (ISSN)

Vol. 25 1306-1334

Styrkeområden

Informations- och kommunikationsteknik

Fundament

Grundläggande vetenskaper

Ämneskategorier

Geometri

Matematisk analys

DOI

10.1007/s12220-014-9469-2