Generalized Hunter–Saxton Equations, Optimal Information Transport, and Factorization of Diffeomorphisms
Generalized Hunter–Saxton Equations, Optimal Information Transport, and Factorization of Diffeomorphisms
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广义 Hunter-Saxton 方程、最优信息传输和微分同胚因式分解
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发表时间：
2012
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通讯作者：
K. Modin
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作者：
K. Modin
我们研究紧致流形的微分同胚群上的一族右不变黎曼度量的测地方程。这些度量下降到光滑概率密度空间上的Fisher信息度量。右简化测地线方程是μ-Hunter-Saxton方程的高维推广，用于模拟磁场影响下的液晶。通过证明测地线喷雾的光滑性，建立了局部存在唯一性结果，并利用度量的下降性得到了一种新的微分同态分解.类似于最优质量传输中的极因式分解，这种因式分解解决了最优信息传输问题。它可以被视为矩阵因式分解的无穷维QR版本。
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.