In this paper, we propose a new constraint, called shift-consistency, for solving matrix/tensor completion problems in the context of recommender systems. Our method provably guarantees several key mathematical properties: (1) satisfies a recently established admissibility criterion for recommender systems; (2) satisfies a definition of fairness that eliminates a specific class of potential opportunities for users to maliciously influence system recommendations; and (3) offers robustness by exploiting provable uniqueness of missing-value imputation. We provide a rigorous mathematical description of the method, including its generalization from matrix to tensor form to permit representation and exploitation of complex structural relationships among sets of user and product attributes. We argue that our analysis suggests a structured means for defining latent-space projections that can permit provable performance properties to be established for machine learning methods.

翻译：在本文中，我们提出了一种名为“移位一致性”的新约束，用于解决推荐系统背景下的矩阵/张量补全问题。我们的方法可证明地保证了若干关键数学性质：(1) 满足近期建立的推荐系统可容许性准则；(2) 满足一种公平性定义，消除了一类用户恶意影响系统推荐的特定潜在机会；(3) 通过利用缺失值插补的可证明唯一性提供鲁棒性。我们给出了该方法的严格数学描述，包括其从矩阵形式到张量形式的推广，以允许表示并利用用户和产品属性集合之间的复杂结构关系。我们论证，我们的分析提出了一种定义潜在空间投影的结构化手段，从而能够为机器学习方法建立可证明的性能性质。