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 $τ$ 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 $\mathfrak{S}_n$. 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.

翻译：本文提出了一种在对称群$\mathfrak{S}_n$上概率分布的排序分布汇总新方法，超越了经典的共识理论与Kemeny中位数框架。基于局部排序中位数概念，我们引入共识排序分布概念，即$\mathfrak{S}_n$上狄拉克质量的稀疏混合模型，旨在从质量传输视角以较小失真逼近排序分布。我们证明当选择流行的Kendall $τ$距离作为成本函数时，最优失真可表示为成对概率的函数，这为开发不受$\mathfrak{S}_n$缺乏向量空间结构影响的高效学习方法开辟了道路。特别地，我们提出一种自上而下的树形统计算法，该算法支持基于排序数据逐步细化CRD——从树根处Kemeny中位数的狄拉克质量开始，到树完全生长末端处的经验排序数据分布本身结束。除理论论证外，该算法的实际意义通过多项数值实验得到了实证支持。