An algorithm for computing Fr\'echet means on the sphere

TL;DR

This paper introduces a Branch and Bound algorithm for computing Fréchet means on the sphere, addressing the challenge of non-convex, non-differentiable optimization without requiring vector space assumptions.

Contribution

The paper presents a novel global optimization algorithm specifically designed for the sphere to compute Fréchet means, overcoming limitations of traditional methods.

Findings

Algorithm successfully computes Fréchet means on the sphere

Performs well on both simulated and real data

Does not require convexity or differentiability of the objective

Abstract

For most optimisation methods an essential assumption is the vector space structure of the feasible set. This condition is not fulfilled if we consider optimisation problems over the sphere. We present an algorithm for solving a special global problem over the sphere, namely the determination of Fr\'echet means, which are points minimising the mean distance to a given set of points. The Branch and Bound method derived needs no further assumptions on the input data, but is able to cope with this objective function which is neither convex nor differentiable. The algorithm's performance is tested on simulated and real data.