We introduce a new consistency-based approach for defining and solving nonnegative/positive matrix and tensor completion problems. The novelty of the framework is that instead of artificially making the problem well-posed in the form of an application-arbitrary optimization problem, e.g., minimizing a bulk structural measure such as rank or norm, we show that a single property/constraint: preserving unit-scale consistency, guarantees the existence of both a solution and, under relatively weak support assumptions, uniqueness. The framework and solution algorithms also generalize directly to tensors of arbitrary dimensions while maintaining computational complexity that is linear in problem size for fixed dimension d. In the context of recommender system (RS) applications, we prove that two reasonable properties that should be expected to hold for any solution to the RS problem are sufficient to permit uniqueness guarantees to be established within our framework. Key theoretical contributions include a general unit-consistent tensor-completion framework with proofs of its properties, e.g., consensus-order and fairness, and algorithms with optimal runtime and space complexities, e.g., O(1) term-completion with preprocessing complexity that is linear in the number of known terms of the matrix/tensor. From a practical perspective, the seamless ability of the framework to generalize to exploit high-dimensional structural relationships among key state variables, e.g., user and product attributes, offers a means for extracting significantly more information than is possible for alternative methods that cannot generalize beyond direct user-product relationships. Finally, we propose our consensus ordering property as an admissibility criterion for any proposed RS method.

翻译：我们提出了一种基于一致性的新框架，用于定义和求解非负/正矩阵与张量补全问题。该框架的创新之处在于：无需通过形式化应用场景中任意设定的优化问题（如最小化秩或范数等全局结构度量）来人为保证问题适定性，而是证明单一性质/约束——保持单位尺度一致性（unit-scale consistency）——即可确保解的存在性，并在相对较弱的支撑假设下保证解的唯一性。该框架及求解算法可直接推广至任意维度的张量，同时保持计算复杂度与问题规模呈线性关系（固定维度d时）。在推荐系统（RS）应用场景中，我们证明了任何RS问题解应满足的两个合理性质，足以在框架内建立唯一性保证。关键理论贡献包括：提出通用单位一致性张量补全框架并证明其性质（如共识序与公平性），以及设计具有最优运行时与空间复杂度的算法（例如，预处理复杂度与矩阵/张量已知项数量呈线性关系的O(1)项补全）。从实践角度看，该框架能够无缝泛化以利用关键状态变量（如用户与商品属性）的高维结构关系，从而提取远超传统仅依赖直接用户-商品关系方法所能获取的信息量。最后，我们提出将共识序性质作为任意推荐系统方法应满足的可接受性准则。