Discrete-time gradient flows in Gromov hyperbolic spaces

TL;DR

This paper studies the behavior of the proximal point algorithm for convex functions in Gromov hyperbolic spaces, showing convergence properties and contraction estimates that extend results known for trees.

Contribution

It introduces new convergence and contraction results for the proximal point algorithm in Gromov hyperbolic spaces, generalizing tree-based properties.

Findings

Proximal point algorithm finds points near minimizers in hyperbolic spaces.

Establishes contraction estimates similar to those in tree structures.

Results applicable to small perturbations of trees.

Abstract

We investigate fundamental properties of the proximal point algorithm for Lipschitz convex functions on (proper, geodesic) Gromov hyperbolic spaces. We show that the proximal point algorithm from an arbitrary initial point can find a point close to a minimizer of the function. Moreover, we establish contraction estimates (akin to trees) for the proximal (resolvent) operator. Our results can be applied to small perturbations of trees.