On March 20, 2024 at 16:00 EET, Nicholas Pischke (TU Darmstadt) will give a talk in the Proof Mining Seminar.
Title: Duality, Fréchet Differentiability and Bregman Distances in Hyperbolic Space
Abstract:
I discuss recent results which provide quantitative and abstract strong convergence results for sequences from a cIn the context of general hyperbolic metric spaces, I discuss a new notion of a dual system (akin to the influential notion from the context of topological vector spaces) that allows for a uniform study of different notions of duality for these nonlinear spaces. Over 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 quasi-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 (together with quantitative results on its convergence in form of a rate of metastability) using the abstract convergence results for generalized Fejér monotone sequences in metric spaces introduced in the preceding talk.
Google Meet link: https://meet.google.com/jpw-hrwe-evu