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Life sciences · Preprint
arXiv · September 10, 2026
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This is a preprint mathematics paper introducing positive scattering as a sufficient condition for certifying identifiability in nonnegative tensor decompositions. The work is theoretical, presenting novel algebraic conditions and proofs but containing no empirical validation, clinical data, or experimental results.
Preprint.
A threshold of 2|S|-2 guarantees minimality and nonnegative rank A threshold of 2|S|-1 guarantees uniqueness among nonnegative decompositions of the same length Mode costs are exactly 0, 1, or +∞, yielding exact activation characterization via graph connectivity
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This is a theoretical mathematics paper presenting novel sufficient conditions for tensor decomposition identifiability; it establishes mathematical framework and proof concepts rather than empirical validation or clinical evidence.
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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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