非负张量分解通过正散射的可辨识性
Identifiability of Nonnegative Tensor Decompositions via Positive Scattering
- Columbia University（哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文提出正散射项量化非负张量分解的额外可辨识性，结合维度预算给出两个充分条件，并通过正分裂不等式证明，能严格证明超出Kruskal和Lovitz--Petrov条件的稀疏分解。
中文摘要 AI 辅助
张量分解的可辨识性通常通过因子族上的线性代数条件来建立。然而，对于非负分解，正性提供了维度和独立性单独无法捕捉的额外信息：非负项不能相互抵消，且其支撑集约束了竞争分解。我们引入一个正散射项来量化这一额外的可辨识性来源，并将其与Lovitz--Petrov对Kruskal定理推广所依据的维度预算相结合。对于每个分量子集，我们获得两个充分条件：阈值$2|S|-2$保证最小性和非负秩，而更强的阈值$2|S|-1$保证在相同长度的非负分解中的唯一性。关键结果是一个关于非负秩一张量不可约交换的正分裂不等式，它将维度约束与支撑集诱导的几何刚性相结合。尽管散射项是通过对中间因子空间的优化来定义的，我们证明其模式代价恰好为$0$、$1$或$+\infty$，从而在图连通性方面产生精确的激活刻画。由此产生的准则可以严格证明稀疏非负张量分解的可辨识性，这超出了Kruskal和Lovitz--Petrov条件的适用范围，包括那些即使在重塑后这些条件仍然失败的例子。在矩阵情形中，这两个准则分别简化为满秩分解和双侧可分性。
英文摘要
Identifiability of tensor decompositions is often established through linear-algebraic conditions on the factor families. For nonnegative decompositions, however, positivity provides additional information that is not captured by dimension and independence alone: nonnegative terms cannot cancel, and their supports constrain competing decompositions. We introduce a positive scattering term that quantifies this additional source of identifiability and combine it with the dimension budget underlying the Lovitz--Petrov generalization of Kruskal's theorem. For every subset of components, we obtain two sufficient conditions: a threshold of $2|S|-2$ guarantees minimality and nonnegative rank, while the stronger threshold $2|S|-1$ guarantees uniqueness among nonnegative decompositions of the same length. The key result is a positive splitting inequality for irreducible exchanges of nonnegative rank-one tensors, which combines the dimension constraint with support-induced geometric rigidity. Although the scattering term is defined through an optimization over intermediate factor spaces, we show that its mode costs are exactly $0$, $1$, or $+\infty$, yielding an exact activation characterization in terms of graph connectivity. The resulting criterion can strictly certify sparse nonnegative tensor decompositions beyond the reach of Kruskal and Lovitz--Petrov conditions, including examples for which those conditions fail even after reshaping. In the matrix case, the two criteria reduce respectively to full-rank factorization and two-sided separability.
