Gaussian Processes indexed on the symmetric group: prediction and learning

TL;DR

This paper extends Gaussian process modeling to the non-commutative finite permutation group, developing harmonic analysis tools for covariance operators to address statistical ranking problems.

Contribution

It introduces a novel harmonic analysis framework for Gaussian processes indexed on the symmetric group, enabling prediction and learning in non-Euclidean settings.

Findings

Developed harmonic analysis of covariance operators on the symmetric group

Enabled Gaussian process modeling for permutation-based data

Provided insights into forecasting in statistical ranking applications

Abstract

In the framework of the supervised learning of a real function defined on a space X , the so called Kriging method stands on a real Gaussian field defined on X. The Euclidean case is well known and has been widely studied. In this paper, we explore the less classical case where X is the non commutative finite group of permutations. In this setting, we propose and study an harmonic analysis of the covariance operators that enables to consider Gaussian processes models and forecasting issues. Our theory is motivated by statistical ranking problems.