Averaged alternating reflections in geodesic spaces

TL;DR

This paper investigates the properties of reflection mappings in geodesic spaces and demonstrates how the averaged alternating reflection algorithm can be effectively applied to convex feasibility problems in nonlinear, constant curvature spaces, achieving strong convergence.

Contribution

It extends weak convergence results from Hilbert spaces to geodesic spaces of constant curvature, establishing strong convergence in this nonlinear setting.

Findings

Reflection mappings are nonexpansive in geodesic spaces.

The averaged alternating reflection algorithm converges strongly in constant curvature spaces.

Weak convergence results from Hilbert spaces have natural counterparts in nonlinear geodesic spaces.

Abstract

We study the nonexpansivity of reflection mappings in geodesic spaces and apply our findings to the averaged alternating reflection algorithm employed in solving the convex feasibility problem for two sets in a nonlinear context. We show that weak convergence results from Hilbert spaces find natural counterparts in spaces of constant curvature. Moreover, in this particular setting, one obtains strong convergence.