Universal Approximation on the Hypersphere

It is well known that any continuous probability density function on ℝm can be approximated arbitrarily well by a finite mixture of normal distributions, provided that the number of mixture components is sufficiently large. The von-Mises-Fisher distribution, defined on the unit hypersphere Sm in ℝm+1, has properties that are analogous to those of the multivariate normal on ℝm+1. We prove that any continuous probability density function on Sm can be approximated to arbitrary degrees of accuracy by a finite mixture of von-Mises-Fisher distributions.