On power-sum kernels on symmetric groups

In this note, we introduce a family of “power-sum” covariance functions and the corresponding Gaussian processes on symmetric groups $S_n$. Such processes are bi-invariant: the action of $S_n$ on itself from both sides does not change their finite-dimensional distributions. We show that the values of power-sum covariance functions can be efficiently calculated, and we also propose a method enabling approximate modeling of the corresponding processes with polynomial computational complexity. By doing this we provide the tools that are required to use the introduced family of processes for statistical modeling.