对偶平坦流形中正则发散诱导的动力系统
Dynamical Systems induced by Canonical Divergence in dually flat manifolds
摘要：
经典力学的原理已经表明，质量的惯性质量是由动能来表征的。这反过来又建立了几何学和力学之间的联系。我们的目标是利用这样一个基本原则的信息几何进入力学领域。根据Amari和Nagaoka对具有对偶结构$(\mathrm{g},\nabla,\nabla^*)$的光滑流形$\mathrm{M}$的曲线能量的修正，我们考虑了$\nabla$和$\nabla^*$的动能.然后，证明了当$(\mathrm{M},\mathrm{g},\nabla,\nabla^*)$是对偶平坦的时，最近引入的一个正则发散及其对偶函数与Hamilton主函数和适当的拉格朗日函数相一致.相应的动力系统的研究和切线动力学概述的正则发散的黎曼梯度。证明了这种动力学的解是连接任意两个足够接近的点的$\nabla$和$\nabla^*$测地线。标准高斯模型的应用也进行了研究。
英文摘要：
The principles of classical mechanics have shown that the inertial quality of mass is characterized by the kinetic energy. This, in turn, establishes the connection between geometry and mechanics. We aim to exploit such a fundamental principle for information geometry entering the realm of mechanics. According to the modification of curve energy stated by Amari and Nagaoka for a smooth manifold $\mathrm{M}$ endowed with a dual structure $(\mathrm{g},\nabla,\nabla^*)$, we consider $\nabla$ and $\nabla^*$ kinetic energies. Then, we prove that a recently introduced canonical divergence and its dual function coincide with Hamilton principal functions associated with suitable Lagrangian functions when $(\mathrm{M},\mathrm{g},\nabla,\nabla^*)$ is dually flat. Corresponding dynamical systems are studied and the tangent dynamics is outlined in terms of the Riemannian gradient of the canonical divergence. Solutions of such dynamics are proved to be $\nabla$ and $\nabla^*$ geodesics connecting any two points sufficiently close to each other. Application to the standard Gaussian model is also investigated.
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