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：并非通过人为构造适定性问题（如最小化秩或范数等全局结构度量）来引入应用任意的优化目标，而是证明单一性质/约束——保持单位尺度一致性——既能保证解的存在性，也能在相对较弱的支持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，且在固定维度d下保持与问题规模呈线性关系的计算复杂度。在推荐系统（RS）应用场景中，我们证明了推荐系统问题任何合理解都应具备的两项性质，足以在此框架内建立唯一性保证。关键理论贡献包括：通用单位一致性张量补全框架及其性质证明（如共识序与公平性），以及具有最优运行时与空间复杂度的算法（例如，基于预处理复杂度与已知矩阵/张量项数呈线性关系的O(1)项补全）。从实践角度看，该框架能无缝利用关键状态变量（如用户与产品属性）间的高维结构关系，相比仅能处理直接用户-产品关系的替代方法，可提取显著更多的信息。最后，我们提出将共识序性质作为任何推荐系统方法的可接受性准则。