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

arXiv: https://arxiv.org/abs/2602.10640 · HTML: https://mlsanity.com/paper/2602.10640/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 36 references that are in the map.
- First posted on arXiv 11 Feb 2026.

## Builds on (most cited first, 9 of 9)

- [Simple, Robust and Optimal Ranking from Pairwise Comparisons](https://mlsanity.com/paper/1512.08949/) (2015)
- [Stochastically Transitive Models for Pairwise Comparisons: Statistical and Computational Issues](https://mlsanity.com/paper/1510.05610/) (2015)
- [A Structured Prediction Approach for Label Ranking](https://mlsanity.com/paper/1807.02374/) (2018)
- [Towards More Robust NLP System Evaluation: Handling Missing Scores in Benchmarks](https://mlsanity.com/paper/2305.10284/) (2023)
- [Top-K Ranking from Pairwise Comparisons: When Spectral Ranking is Optimal](https://mlsanity.com/paper/1603.04153/) (2016)
- [Reparameterizing the Birkhoff Polytope for Variational Permutation Inference](https://mlsanity.com/paper/1710.09508/) (2017)
- [Statistical Depth Functions for Ranking Distributions: Definitions, Statistical Learning and Applications](https://mlsanity.com/paper/2201.08105/) (2022)
- [Dimensionality Reduction and (Bucket) Ranking: a Mass Transportation Approach](https://mlsanity.com/paper/1810.06291/) (2018)
- [Robust Consensus in Ranking Data Analysis: Definitions, Properties and Computational Issues](https://mlsanity.com/paper/2303.12878/) (2023)

## Abstract (arXiv)

In this article we develop a new method for summarizing a ranking distribution, textiti.e. a probability distribution on the symmetric group mathfrakSn, beyond the classical theory of consensus and Kemeny medians. Based on the notion of textitlocal ranking median, we introduce the concept of textitconsensus ranking distribution (crd), a sparse mixture model of Dirac masses on mathfrakSn, in order to approximate a ranking distribution with small distortion from a mass transportation perspective. We prove that by choosing the popular Kendall τ 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 mathfrakSn. 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.