Universal approximation on the hypersphere

Abstract

The approximation properties of finite mixtures of location-scale distributions on Euclidean space have been well studied. It has been shown that mixtures of location-scale distributions can approximate arbitrary probability density function up to any desired level of accuracy provided the number of mixture components is sufficiently large. However, analogous results are not available for probability density functions defined on the unit hypersphere. The von-Mises-Fisher distribution, defined on the unit hypersphere Sm in plays the central role in directional statistics. We prove that any continuous probability density function on Sm can be approximated to arbitrary degrees of accuracy in sup norm by a finite mixture of von-Mises-Fisher distributions. Our proof strategy and result are also useful in studying the approximation properties of other finite mixtures of directional distributions.