On the Rate of Convergence of Iterative Methods for Nonexpansive Mappings in CAT(0) Spaces and Hyperbolic Optimization
Katherine Rossella Foglia, Vittorio Colao
TL;DR
This work analyzes fixed-point iterations for nonexpansive maps in complete $CAT(0)$ spaces, showing that classical rates of asymptotic regularity from linear settings carry over to the nonlinear geometric setting. It proves an $O\big(\frac{1}{\sqrt{\sum_{i=1}^n \lambda_i(1-\lambda_i)}}\big)$ bound for Krasnosel'skiĭ–Mann iterations and an $O(1/k)$ bound for Halpern/viscosity-type iterations, thereby extending these results to $CAT(0)$ spaces and enabling robust optimization in hyperbolic geometries. A key contribution is the Hyperbolic HalpernGD optimizer, which substitutes the Euclidean forward step with a proximal mapping in Hadamard spaces, providing a geometrically faithful counterpart to HalpernSGD for hyperbolic deep learning. These results bridge fixed-point theory, metric geometry, and hyperbolic optimization, offering both theoretical guarantees and practical algorithms for nonpositively curved spaces.
Abstract
The Krasnosel'skiĭ 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 \cite{foglia2024halpernsgd,colao2025optimizer} in Euclidean setting.