论文标题
Lagrangian和Hamiltonian机制在统计歧管上的概率
Lagrangian and Hamiltonian Mechanics for Probabilities on the Statistical Manifold
论文作者
论文摘要
我们在概率分布的Riemannian歧管上提供了经典力学的信息几何公式，该分布是具有双静电连接的仿射歧管。在非参数形式主义中，我们考虑了有限的样本空间上的全套正概率函数，并且就统计束的希尔伯特束结构而言，我们为统计歧管上的切线和cotangent空间提供了特定的表达。在这种情况下，我们使用规范双对平行传输来计算一维统计模型的速度和加速度，并在捆绑包上定义了Lagrangian和Hamiltonian力学的连贯形式主义。最后，在一系列示例中，我们展示了我们的形式主义如何为概率单纯性加速自然梯度动力学提供一个一致的框架，为在优化，游戏理论和神经网络中的直接应用铺平了道路。
We provide an Information-Geometric formulation of Classical Mechanics on the Riemannian manifold of probability distributions, which is an affine manifold endowed with a dually-flat connection. In a non-parametric formalism, we consider the full set of positive probability functions on a finite sample space, and we provide a specific expression for the tangent and cotangent spaces over the statistical manifold, in terms of a Hilbert bundle structure that we call the Statistical Bundle. In this setting, we compute velocities and accelerations of a one-dimensional statistical model using the canonical dual pair of parallel transports and define a coherent formalism for Lagrangian and Hamiltonian mechanics on the bundle. Finally, in a series of examples, we show how our formalism provides a consistent framework for accelerated natural gradient dynamics on the probability simplex, paving the way for direct applications in optimization, game theory and neural networks.
