Universal Approximation on the Hypersphere
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It is well known that any continuous probability density function on 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 R^m+1, has properties that are analogous to those of the multivariate normal on 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.
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