An algorithm for computing Fréchet means on the sphere

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 Fréchet 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.

An Invitation to Intrinsic Compositional Data Analysis Using Projective Geometry and Hilbert’s Metric

“…[28] [31]. Efficient computations of the Fréchet mean or its smooth surrogate ( [23]) is another topic of further research. Another obvious candidate for an interesting alternative to the log-sum-exp smooth approximation of Hilbert's metric would be the Moreau-Yosida regularisation based on infimal convolution (see e.g.…”

“…Definition. The non-differentiability of Hilbert's metric comes from the max operation in the expressions (23) or (24) defining it. This suggests to use a smooth approximation of the maximum in order to obtain a differentiable approximate proxy of the Hilbert metric.…”

Nonparametric regression in nonstandard spaces

“…We use R's general purpose optimizers stats::optim(method = "L-BFGS-B") and stats::optimize(), both without explicit implementation of derivatives, but with several starting points. The implementations could potentially be improved by using the algorithm presented in [9]. For alternative implementation of geodesic regression, see [22].…”

Nonparametric Regression in Nonstandard Spaces

“…We use R's general purpose optimizers stats::optim(method = "L-BFGS-B") and stats::optimize(), both without explicit implementation of derivatives, but with several starting points. The implementations could potentially be improved by using the algorithm presented in [EHW19]. For alternative implementation of geodesic regression, see [SO20].…”