Universal Approximation on the Hypersphere

Abstract

It is well known that any continuous probability density function on ${\mathbb{R}}^m$ 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 $S^m$ in ${\mathbb{R}}^{m+1}$, has properties that are analogous to those of the multivariate normal on ${\mathbb{R}}^{m+1}$. We prove that any continuous probability density function on $S^m$ can be approximated to arbitrary degrees of accuracy by a finite mixture of von-Mises-Fisher distributions.