From Generalized Arithmetic Means to Bregman Divergences and Back

Here we examine some connections between the notions of generalized arithmetic means, geodesics, Lagrange-Hamilton dynamics and Bregman divergences. The key ingredient for the relationship is the case in which a Riemannian metric has a square root that is the Jacobian of a diffeomorphism. In such case the geodesics of the of the Riemannian metric turn out be the pullback of straight lines by the diffeomorphism. This is interesting when the Riemann metric is the Hessian of a convex function because in this case we obtain comparison results between the Bregman divergence determined by the convex function and the geodesic distance determined by its square root