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

2026-02-11

AI Summary

This work addresses the challenge of effectively modeling ranking distributions over the symmetric group by proposing the Consensus Ranking Distribution (CRD) model, which approximates the target distribution via a sparse mixture of Dirac measures. The approach introduces local ranking medians and employs the Kendall τ distance as the optimal transport cost, leveraging a top-down tree-structured algorithm to iteratively refine approximation accuracy. Theoretical analysis demonstrates that the transport distortion can be precisely expressed in terms of pairwise ranking probabilities, thereby circumventing the fundamental obstacle posed by the absence of a vector space structure on the symmetric group. Experimental results validate the superior efficiency and practical utility of the proposed algorithm.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

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.

Problem

• consensus ranking
• ranking distribution
• Kemeny median
• symmetric group
• Kendall tau distance

Innovation

• consensus ranking distribution
• local ranking median
• Kendall tau distance
• mass transportation
• tree-structured learning