An optimization problem on the sphere

TL;DR

This paper establishes the existence and uniqueness of a minimizer for the average geodesic distance on the sphere, leading to an optimal algorithm for halfspace learning with uniform data.

Contribution

It provides the first proof of existence and uniqueness of the minimizer for average geodesic distance on the sphere, with implications for learning algorithms.

Findings

Unique minimizer for average geodesic distance proven

Optimal halfspace learning algorithm derived from the result

Applicability to data and target functions from the uniform distribution

Abstract

We prove existence and uniqueness of the minimizer for the average geodesic distance to the points of a geodesically convex set on the sphere. This implies a corresponding existence and uniqueness result for an optimal algorithm for halfspace learning, when data and target functions are drawn from the uniform distribution.