# Universal Approximation on the Hypersphere

arXiv: https://arxiv.org/abs/2004.06328 · HTML: https://mlsanity.com/paper/2004.06328/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Apr 2020.

## Built on by (most cited first, 1 of 1)

- [Risk Bounds for Mixture Density Estimation on Compact Domains via the h-Lifted Kullback--Leibler Divergence](https://mlsanity.com/paper/2404.12586/) (2024)

## Abstract (arXiv)

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.