Identifiability of Nonnegative Tensor Decompositions via Positive Scattering

• URL: http://arxiv.org/abs/2609.11606v1
• Date: Thu, 10 Sep 2026 14:26:45 GMT
• Title: Identifiability of Nonnegative Tensor Decompositions via Positive Scattering
• Abstract summary: We show that positivity provides additional information that is not captured by dimension and independence alone.<n>We combine this additional source of identifiability with the dimension budget underlying the Lovitz--Petrov generalization of Kruskal's theorem.<n>The resulting criterion can strictly certify sparse nonnegative tensor decompositions beyond the reach of Kruskal and Lovitz--Petrov conditions.
• Score: 4.894024268234817
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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