Lagrangian and Hamiltonian Mechanics for Probabilities on the Statistical Manifold

TL;DR

This paper develops a geometric framework for classical mechanics on the space of probability distributions, enabling new insights into optimization and neural network dynamics through a formal Lagrangian and Hamiltonian approach.

Contribution

It introduces an information-geometric formulation of mechanics on the statistical manifold, including tangent/cotangent structures and applications to natural gradient dynamics.

Findings

Provides a formalism for velocities and accelerations in probability spaces

Defines Lagrangian and Hamiltonian mechanics on the statistical bundle

Demonstrates applications to accelerated natural gradient methods

Abstract

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.