On power sum kernels on symmetric groups

TL;DR

This paper introduces a new family of bi-invariant kernels and Gaussian processes on symmetric groups, with efficient computation and sampling methods, facilitating their use in statistical modeling and machine learning.

Contribution

The paper presents a novel family of power sum kernels on symmetric groups, along with efficient computation and sampling techniques for Gaussian processes based on these kernels.

Findings

Power sum kernels can be efficiently computed.

Approximate sampling of Gaussian processes is feasible with polynomial complexity.

Tools for applying these kernels in machine learning are provided.

Abstract

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.