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

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• 作者：Klas Modin
• 关键词：Euler–Arnold equations ; Euler–Poincaré 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 ; 58D05 ; 58D15 ; 35Q31 ; 53C21 ; 58B20 ; 94A17 ; 65F99
• 刊名：Journal of Geometric Analysis
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：25
• 期：2
• 页码：1306-1334
• 全文大小：439 KB
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Differential Geometry
  Convex and Discrete Geometry
  Fourier Analysis
  Abstract Harmonic Analysis
  Dynamical Systems and Ergodic Theory
  Global Analysis and Analysis on Manifolds
  
• 出版者：Springer New York
• ISSN：1559-002X

文摘

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.