Discrete-time gradient flows for unbounded convex functions on Gromov hyperbolic spaces
Shin-ichi Ohta
TL;DR
The article develops a theory of discrete-time gradient flows for unbounded Lipschitz convex functions on proper geodesic Gromov hyperbolic spaces, using giant-step proximal iterations. Under a negative asymptotic slope condition, it proves the existence and uniqueness of a negative boundary direction $v_*$ in $\partial X$ toward which the discrete gradient flow diverges, and provides a quantitative rate for convergence to $v_*$ based on a contraction-type bound for the proximal operator. This work extends convergence phenomena known in nonpositively curved spaces to hyperbolic and non-Riemannian settings, connecting boundary behavior with optimization dynamics and suggesting avenues for discrete and quasi-isometric extensions. The results unify variational gradient-flow ideas with hyperbolic geometry, offering tools for convex optimization in spaces with large-scale negative curvature and potential applications to discrete geometries and geometry-inspired optimization problems.
Abstract
In proper, geodesic Gromov hyperbolic spaces, we investigate discrete-time gradient flows via the proximal point algorithm for unbounded Lipschitz convex functions. Assuming that the target convex function has negative asymptotic slope along some ray (thus unbounded below), we first prove the uniqueness of such a negative direction in the boundary at infinity. Then, we show that the discrete-time gradient flow from an arbitrary initial point diverges to that unique direction of negative asymptotic slope. This is inspired by and generalizes results of Karlsson-Margulis and Hirai-Sakabe on nonpositively curved spaces and a result of Karlsson concerning semi-contractions on Gromov hyperbolic spaces. We also give an estimate of the rate of convergence based on a contraction property for the proximal (resolvent) operator established in our previous work.