Local linear convergence of alternating projections in metric spaces with bounded curvature

TL;DR

This paper analyzes the local linear convergence of the alternating projections method in metric spaces with bounded curvature, emphasizing geometric conditions like transversality and convexity-like properties.

Contribution

It extends the understanding of alternating projections to metric spaces with bounded curvature, identifying key geometric conditions for convergence.

Findings

Established local linear convergence under geometric conditions

Identified transversality and convexity-like conditions as crucial

Extended classical results to metric spaces with bounded curvature

Abstract

We consider the popular and classical method of alternating projections for finding a point in the intersection of two closed sets. By situating the algorithm in a metric space, equipped only with well-behaved geodesics and angles (in the sense of Alexandrov), we are able to highlight the two key geometric ingredients in a standard intuitive analysis of local linear convergence. The first is a transversality-like condition on the intersection; the second is a convexity-like condition on one set: "uniform approximation by geodesics."