Lagrangian and Hamiltonian Mechanics for Probabilities on the Statistical Manifold
This provides a novel mathematical framework for applying classical mechanics concepts to probability distributions, which could benefit researchers in optimization, game theory, and neural networks.
The authors developed an information-geometric formulation of classical mechanics on the Riemannian manifold of probability distributions, creating a coherent framework for Lagrangian and Hamiltonian mechanics on the statistical bundle. They demonstrated how this formalism enables accelerated natural gradient dynamics on the probability simplex, with potential applications in optimization, game theory, and neural networks.
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.