论文标题
Hessian度量通过传输信息几何形状
Hessian metric via transport information geometry
论文作者
论文摘要
我们建议研究与Wasserstein $ 2 $ metric的概率措施空间的HESSIAN指标。我们将其命名为运输Hessian Metric，其中包含并扩展了古典Wasserstein- $ 2 $ Metric。我们制定了几种与运输Hessian指标相关的动态系统。发现了运输Hessian指标与数学物理方程之间的几个连接。例如，包括牛顿的流量在内的运输Hessian梯度流，用于制定平均场核Stein变化梯度流。 Boltzmann-Shannon熵的Hessian Hessian Hamiltonian流动形成了浅水方程； Fisher信息的运输Hessian梯度流导致热方程式。提出了几种用于运输Hessian距离的示例和封闭式解决方案。
We propose to study the Hessian metric of a functional on the space of probability measures endowed with the Wasserstein $2$-metric. We name it transport Hessian metric, which contains and extends the classical Wasserstein-$2$ metric. We formulate several dynamical systems associated with transport Hessian metrics. Several connections between transport Hessian metrics and mathematical physics equations are discovered. E.g., the transport Hessian gradient flow, including Newton's flow, formulates a mean-field kernel Stein variational gradient flow; The transport Hessian Hamiltonian flow of Boltzmann-Shannon entropy forms the Shallow water equation; The transport Hessian gradient flow of Fisher information leads to the heat equation. Several examples and closed-form solutions for transport Hessian distances are presented.
