Pairwise comparison matrices have received substantial attention in a variety of applications, especially in rank aggregation, the task of flattening items into a one-dimensional (and thus transitive) ranking. However, non-transitive preference cycles can arise in practice due to the fact that making a decision often requires a complex evaluation of multiple factors. In some applications, it may be important to identify and preserve information about the inherent non-transitivity, either in the pairwise comparison data itself or in the latent feature space. In this work, we develop structured models for non-transitive pairwise comparison matrices that can be exploited to recover such matrices from incomplete noisy data and thus allow the detection of non-transitivity. Considering that individuals' tastes and items' latent features may change over time, we formulate time-varying pairwise comparison matrix recovery as a dynamic skew-symmetric matrix recovery problem by modeling changes in the low-rank factors of the pairwise comparison matrix. We provide theoretical guarantees for the recovery and numerically test the proposed theory with both synthetic and real-world data.
翻译:在各种应用中,尤其是等级汇总中,对称比较矩阵在各种应用中受到大量注意,特别是将物品平整成一维(因而具有过渡性)等级的任务,然而,由于作出决定往往需要对多种因素进行复杂的评估,在实践中可能出现非过渡性优惠周期;在有些应用中,在对称比较数据本身或潜在特征空间中,确定和保存关于内在的非透明度的信息可能很重要;在这项工作中,我们为非透明对称比较矩阵制定了结构化模型,可以利用这些模型从不完整的噪音数据中恢复这种矩阵,从而能够发现非透明性;考虑到个人的口味和物品的潜在特征可能随时间变化而变化,我们通过模拟对称比较矩阵中低等级因素的变化,将矩阵的恢复作为动态的对称矩阵恢复问题,我们为恢复提供理论保证,并以合成数据和实际数据对拟议理论进行数字测试。