In this note, we introduce a family of "power sum" kernels and the corresponding Gaussian processes on symmetric groups $\mathrm{S}_n$. Such processes are bi-invariant: the action of $\mathrm{S}_n$ on itself from both sides does not change their finite-dimensional distributions. We show that the values of power sum kernels can be efficiently calculated, and we also propose a method enabling approximate sampling of the corresponding Gaussian processes with polynomial computational complexity. By doing this we provide the tools that are required to use the introduced family of kernels and the respective processes for statistical modeling and machine learning.

翻译：在本说明中, 我们引入了一个“ 电量和内核” 和相应的高斯进程, 用于对称组 $\ mathrm{ S ⁇ { s ⁇ n$ 。 这种过程是双变量的: 双方的 $ mathrm{ S ⁇ n$ 本身的行动不会改变其有限维分布 。 我们显示电量和内核的值可以有效计算, 我们还提出一种方法, 使相应的高斯进程能够以多元计算复杂度进行大致抽样。 通过这样做, 我们提供了使用引入的内核组合和相应的统计建模和机器学习程序所需的工具 。