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Minimal enclosing balls via geodesics

Ariel Goodwin, Adrian S. Lewis

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.

Minimal enclosing balls via geodesics

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.
Paper Structure (3 theorems, 18 equations, 1 figure, 1 algorithm)

This paper contains 3 theorems, 18 equations, 1 figure, 1 algorithm.

Key Result

Theorem 1

If Assumption assumptionA holds, then any initial sequence of points $x^0,x^1,\ldots,x^N$ generated by Algorithm GeodesicAlgorithm with stepsizes $t_k = \frac{2}{k+1}$ satisfies

Figures (1)

  • Figure 1: Computing the minimal enclosing ball on a CAT(1) space.

Theorems & Definitions (7)

  • Definition 1
  • Theorem 1
  • proof
  • lemma 1
  • proof
  • Theorem 2
  • proof