Beyond Kemeny Medians: Consensus Ranking Distributions Definition, Properties and Statistical Learning

TL;DR

This paper introduces a novel framework for summarizing ranking distributions using consensus ranking distributions and local median concepts, enabling efficient statistical learning and refinement of ranking models.

Contribution

It develops the concept of consensus ranking distributions and a tree-structured algorithm for progressive refinement of ranking models based on pairwise probabilities.

Findings

Optimal distortion expressed via pairwise probabilities

Efficient learning methods without vector space structure

Empirical validation through numerical experiments

Abstract

In this article we develop a new method for summarizing a ranking distribution, \textit{i.e.} a probability distribution on the symmetric group $\mathfrak{S}_n$, beyond the classical theory of consensus and Kemeny medians. Based on the notion of \textit{local ranking median}, we introduce the concept of \textit{consensus ranking distribution} ($\crd$), a sparse mixture model of Dirac masses on $\mathfrak{S}_n$, in order to approximate a ranking distribution with small distortion from a mass transportation perspective. We prove that by choosing the popular Kendall $\tau$ distance as the cost function, the optimal distortion can be expressed as a function of pairwise probabilities, paving the way for the development of efficient learning methods that do not suffer from the lack of vector space structure on $\mathfrak{S}_n$. In particular, we propose a top-down tree-structured statistical algorithm that allows for the progressive refinement of a CRD based on ranking data, from the Dirac mass at a Kemeny median at the root of the tree to the empirical ranking data distribution itself at the end of the tree's exhaustive growth. In addition to the theoretical arguments developed, the relevance of the algorithm is empirically supported by various numerical experiments.