From generalized arithmetic means to geodesics to Hamilton dynamics to Bregman divergences

TL;DR

This paper explores the geometric relationships between generalized arithmetic means, geodesic distances, Hamiltonian dynamics, and Bregman divergences, revealing how these concepts interconnect through convex functions and Riemannian metrics.

Contribution

It introduces a geometric perspective on generalized arithmetic means as minimizers of geodesic distances, linking them to Hessian metrics and Bregman divergences.

Findings

Generalized arithmetic means minimize geodesic distances in certain metrics.

Pull-backs of Euclidean metrics can coincide with Hessian metrics of convex functions.

Comparison established between Bregman divergence and geodesic distance in Hessian metrics.

Abstract

Here we examine some connections between the notions of generalized arithmetic means, geodesics, Lagrange-Hamilton dynamics and Bregman divergences. In a previous paper we developed a predictive interpretation of generalized arithmetic means. That work was more probabilistically oriented. Here we take a geometric turn, and see that generalized arithmetic means actually minimize a geodesic distance on $\mathbb{R}^n.$ Such metrics might result from pull-backs of the Euclidean metric in $\mathbb{R}^n.$ We shall furthermore see that in some cases these pull-backs might coincide with the Hessian of a convex function. This occurs when the Hessian of a convex function has a square root that is the Jacobian of a diffeomorphism in $\mathbb{R}^n.$ In this case we obtain a comparison between the Bregman divergence defined by the convex function and the geodesic distance in the metric defined by its Hessian.