提出新方法用稀疏混合模型逼近排名分布，解决非向量空间下的统计学习难题。
Beyond Kemeny Medians: Consensus Ranking Distributions Definition, Properties and Statistical Learning
- 基于局部中位排名定义共识排名分布，用狄拉克质量混合建模
- 以凯尔德诺τ距离为代价函数，最优失真可由成对概率表达
- 设计树状逐级优化算法，从肯梅尼中位开始逐步逼近真实分布
本文提出一种新的排名分布摘要方法，突破传统共识与肯梅尼中位的局限。基于局部排名中位概念，引入共识排名分布（CRD），即在对称群$ rak{S}_n$上由狄拉克质量构成的稀疏混合模型，以最小化质量传输视角下的失真。证明当采用流行的凯尔德诺τ距离作为代价函数时，最优失真可表示为成对概率的函数，从而实现无需向量空间结构的高效学习。进一步提出一种自顶向下的树结构统计算法，从根节点的肯梅尼中位出发，通过树的完全生长逐步精细逼近经验排名分布。理论分析结合多项数值实验验证了方法的有效性。
原文摘要 · Abstract (English)
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.
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