Minimal enclosing balls via geodesics

TL;DR

This paper analyzes a simple geodesic-based algorithm for minimal enclosing balls, providing new complexity results and improved convergence rates in nonpositive curvature and bounded curvature geodesic spaces.

Contribution

It offers a simpler, more intuitive complexity analysis and the first complexity results for geodesic spaces with curvature bounds.

Findings

Improved convergence rate for the geodesic-based method in nonpositive curvature spaces.

First complexity analysis for the algorithm in geodesic spaces with curvature bounded above.

Simplified, self-contained proof of the algorithm's complexity.

Abstract

Algorithms for minimal enclosing ball problems are often geometric in nature. To highlight the metric ingredients underlying their efficiency, we focus here on a particularly simple geodesic-based method. A recent subgradient-based study proved a complexity result for this method in the broad setting of geodesic spaces of nonpositive curvature. We present a simpler, intuitive and self-contained complexity analysis in that setting, which also improves the convergence rate. We furthermore derive the first complexity result for the algorithm on geodesic spaces with curvature bounded above.