On the Rate of Convergence of Iterative Methods for Nonexpansive Mappings in CAT(0) Spaces and Hyperbolic Optimization

2025/10/29 by Foglia, Katherine Rossella, Colao, Vittorio
#FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2510.25363

Abstract

The Krasnosel'ski\uı Mann and Halpern iterations are classical schemes for approximating fixed points of nonexpansive mappings in Banach spaces, and have been widely studied in more general frameworks such as CAT(κ) and, more generally, geodesic spaces. Convergence results and convergence rate estimates in these nonlinear settings are already well established. The contribution of this paper is to extend to complete CAT(0) spaces the proof techniques originally developed in the linear setting of Banach and Hilbert spaces, thereby recovering the same asymptotic regularity bounds and to introduce a novel optimizer for Hyperbolic Deep learning based on Halpern Iteration similarly to HalpernSGD \citefoglia2024halpernsgd,colao2025optimizer in Euclidean setting.

Citations

• Rates of asymptotic regularity for the alternating Halpern-Mann iteration
• Optimal error bounds for nonexpansive fixed-point iterations in normed spaces
• Hyperbolic Deep Neural Networks: A Survey
• Quantitative results on a Halpern-type proximal point algorithm
• Hyperbolic Neural Networks
• Representation Tradeoffs for Hyperbolic Embeddings
• A First Order Method for Solving Convex Bi-Level Optimization Problems
• Sharp convergence rates for averaged nonexpansive maps
• The Baillon-Haddad Theorem Revisited
• Monotone Operators and the Proximal Point Algorithm
• Fast Krasnosel'skii-Mann algorithm with a convergence rate of the fixed point iteration of o((1)/(k))

Related

• A quadratic rate of asymptotic regularity for CAT(0)-spaces
• Strong convergence for the alternating Halpern-Mann iteration in CAT(0) spaces
• Rates of asymptotic regularity for the alternating Halpern-Mann iteration
• Effective metastability of Halpern iterates in CAT(0) spaces
• Rates of convergence for inexact Krasnosel'skii-Mann iterations in\n Banach spaces