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. This is remarkable because it obviates the need for heuristic-based statistical or AI methods despite what appear to be distinctly human/subjective variables at the heart of the problem. 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.

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（例如最小化秩或范数等整体结构度量）来形成任意应用场景的优化问题，而是证明单一性质/约束——保持单位尺度一致性——即可保证解的存在性，且在相对宽松的支持假设条件下保证解的唯一性。该框架及求解算法可直接推广至任意维度的张量，同时对于固定维度d，计算复杂度与问题规模呈线性关系。在推荐系统（RS）应用场景中，我们证明：任何推荐系统解都理应满足的两个合理性质，足以在该框架内建立唯一性保证。这一结论意义重大，因为尽管问题核心涉及明显的人类/主观变量，但该框架无需依赖基于启发式的统计或人工智能方法。关键理论贡献包括：提出通用的单位一致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O(1)项补全，且预处理复杂度与矩阵/张量已知项数量线性相关）。从实践角度看，该框架能够无缝推广以利用关键状态变量（如用户与产品属性）间的高维结构关系，从而提取显著多于传统方法的信息——传统方法无法超越直接用户-产品关系进行泛化。