论文标题
超级球的通用近似
Universal Approximation on the Hypersphere
论文作者
论文摘要
众所周知，只要有限分布的有限混合物，如果混合组件的数量足够大，则可以通过有限的混合物来任意地近似于$ \ mathbb {r}^m $上的任何连续概率密度函数。在$ \ mathbb {r}^{m+1} $中定义的von-mises-fisher分布，在$ \ mathbb {r}^{m+1} $中定义的属性类似于$ \ mathbb {r}^{m+1} $的多变量normal的属性。我们证明，通过von-mises-fisher分布的有限混合物，可以将$ S^m $上的任何连续概率密度函数近似于任意程度。
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.
