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

arXiv2007.15473MaRDI QIDQ6346178

Henryk Gzyl

Publication date: 29 July 2020

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.

Mathematics Subject Classification ID

Computational aspects related to convexity (52B55) Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics (70H06) Methods of global Riemannian geometry, including PDE methods; curvature restrictions (53C21) Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics (70H15) Other special differential geometries (53A40)

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