Year: 2025
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Ohta, Shin-ichi
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6979231764
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