Duality, Fréchet differentiability and Bregman distances in hyperbolic spaces

Abstract

For general hyperbolic metric spaces, we introduce a new notion of a dual system (extending the influential notion from the context of normed linear spaces) that allows for a uniform study of different notions of duality for these nonlinear spaces. Using this abstract notion of duality, we lift various notions from convex analysis into this nonlinear setting, including Fréchet differentiability and Bregman distances. Further, we introduce a notion of a monotone operator relative to a given dual system and, using the new Fréchet derivatives, we study corresponding resolvents relative to a given gradient, generalizing the seminal notion of Eckstein from the linear setting. These resolvents are then related to corresponding notions of Bregman nonexpansive mappings which are introduced relative to this generalization of the classical Bregman distance and we prove a convergence result of an analogue of the proximal point algorithm. For that, using methods from proof mining, we even provide quantitative results on its convergence in very general settings.