Intransitivity is a critical issue in pairwise preference modeling. It refers to the intransitive pairwise preferences between a group of players or objects that potentially form a cyclic preference chain and has been long discussed in social choice theory in the context of the dominance relationship. However, such multifaceted intransitivity between players and the corresponding player representations in high dimensions is difficult to capture. In this paper, we propose a probabilistic model that jointly learns each player's d-dimensional representation (d>1) and a dataset-specific metric space that systematically captures the distance metric in Rd over the embedding space. Interestingly, by imposing additional constraints in the metric space, our proposed model degenerates to former models used in intransitive representation learning. Moreover, we present an extensive quantitative investigation of the vast existence of intransitive relationships between objects in various real-world benchmark datasets. To our knowledge, this investigation is the first of this type. The predictive performance of our proposed method on different real-world datasets, including social choice, election, and online game datasets, shows that our proposed method outperforms several competing methods in terms of prediction accuracy.

该研究解决了在成对偏好建模中无序性的问题，特别是高维情况下玩家间复杂无序性难以捕捉的挑战。论文提出了一种概率模型，能够共同学习每个玩家的高维表示及其特定的数据度量空间，并且通过额外的约束，模型能退化为先前的无序表示学习模型。研究表明，该方法在多个真实世界数据集上的预测性能优于现有竞争方法，具有重要的应用潜力。

多维无序性的广义模型