# Identifiability of Nonnegative Tensor Decompositions via Positive Scattering

arXiv: https://arxiv.org/abs/2609.11606 · HTML: https://mlsanity.com/paper/2609.11606/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 21 references that are in the map.
- First posted on arXiv 10 Sep 2026.

## Builds on (most cited first, 1 of 1)

- [When are Overcomplete Topic Models Identifiable? Uniqueness of Tensor Tucker Decompositions with Structured Sparsity](https://mlsanity.com/paper/1308.2853/) (2013)

## Abstract (arXiv)

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 +∞, 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.