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Discrete-time gradient flows in Gromov hyperbolic spaces

Shin-ichi Ohta

TL;DR

The paper investigates discrete-time gradient flows for Lipschitz $K$-convex functions on proper geodesic Gromov hyperbolic spaces via the proximal operator $\mathsf{J}^f_{\tau}$. It establishes a concrete tendency-to-minimizer bound $d(p,y) \le d(p,x) - d(x,y) + \frac{4\sqrt{2\tau L \delta}}{\sqrt{K\tau+1}}$ and proves contraction-type estimates for the resolvent that reflect tree-like behavior up to the hyperbolicity parameter $\delta$, with sharper bounds when $K>0$ and $\tau>K^{-1}$. The results yield finite-step convergence guarantees and quantify how large-step proximal updates steer iterations toward minimizers, even in the presence of local perturbations from a tree. These findings broaden optimization theory to non-Riemannian spaces and suggest robustness of proximal methods under negative-curvature-like, large-scale perturbations such as those encountered in trees and their small perturbations. The work also outlines promising directions for extending the approach to discrete spaces, simulated annealing, and quasi-isometry-invariant convexity.

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.

Discrete-time gradient flows in Gromov hyperbolic spaces

TL;DR

The paper investigates discrete-time gradient flows for Lipschitz -convex functions on proper geodesic Gromov hyperbolic spaces via the proximal operator . It establishes a concrete tendency-to-minimizer bound and proves contraction-type estimates for the resolvent that reflect tree-like behavior up to the hyperbolicity parameter , with sharper bounds when and . The results yield finite-step convergence guarantees and quantify how large-step proximal updates steer iterations toward minimizers, even in the presence of local perturbations from a tree. These findings broaden optimization theory to non-Riemannian spaces and suggest robustness of proximal methods under negative-curvature-like, large-scale perturbations such as those encountered in trees and their small perturbations. The work also outlines promising directions for extending the approach to discrete spaces, simulated annealing, and quasi-isometry-invariant convexity.

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.
Paper Structure (9 sections, 9 theorems, 69 equations, 3 figures)

This paper contains 9 sections, 9 theorems, 69 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gromov hyperbolic spaces
  4. Proximal point algorithm
  5. Proofs of main results
  6. Proof of Theorem \ref{['th:conv']}
  7. Proof of Corollary \ref{['cr:conv']}
  8. Proof of Theorem \ref{['th:cont']}
  9. Further problems

Key Result

Theorem 1.1

Let $(X,d)$ be a proper $\delta$-hyperbolic geodesic space, and $f:X \longrightarrow \mathbb{R}$ be a $K$-convex $L$-Lipschitz function with $K \ge 0$, $L>0$ such that $\inf_X f$ is attained at some $p \in X$. Then, for any $x \in X$, $\tau>0$, and $y \in \mathsf{J}^f_{\tau}(x)$, we have In the case of $K>0$ and $\tau>K^{-1}$, we further obtain

Figures (3)

  • Figure 1: Gromov products in $\mathbb{R}^2$ and a tripod
  • Figure 2: Proximal operator in $0$-hyperbolic spaces
  • Figure 3: The case of $d(p,y_1) <(x_1|x_2)_p$

Theorems & Definitions (16)

  • Theorem 1.1: Tendency towards minimizer
  • Corollary 1.2
  • Theorem 1.3: Contraction estimates
  • Definition 2.1: Gromov hyperbolic spaces
  • Example 2.2
  • Lemma 2.3: Tripod lemma
  • Lemma 2.4
  • Lemma 3.1
  • Remark 3.2: A priori estimates
  • Proposition 3.3
  • ...and 6 more