Discrete-time gradient flows for unbounded convex functions on Gromov hyperbolic spaces

2025/03/25 by Shin‐ichi Ohta, Ohta, Shin-ichi · 1 voice
Mathematics · Computer Science · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Optimization and Variational Analysis

paper · doi:10.1142/s0219199726500033

openalex publication_date 2025/11/21 · openalex created_date 2025/11/23 · openalex updated_date 2026/05/21

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.

Citations

• A scaling characterization of nc-rank via unbounded gradient flow
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• Discrete-time gradient flows in Gromov hyperbolic spaces
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• The moment-weight inequality and the Hilbert-Mumford criterion
• The proximal point algorithm in metric spaces
• The proximal point algorithm in metric spaces
• Convex functions on symmetric spaces, side lengths of polygons and stability inequalities for weighted configurations at infinity
• Non-expanding maps and Busemann functions
• Curvature and rank of Teichmüller space
• Teichmuller Space is Not Gromov Hyperbolic

Discussions

• Shin-Ichi Ohta studies discrete-time gradient flows for unbounded convex functions on Gromov hyperbolic spaces, extending results on nonpositively curved spaces and semi-contractions. 🔗 doi.org/10.1 [bsky, 0 points, 0 comments]