New conditions improve uniqueness of nonnegative tensor decompositions
Identifiability of Nonnegative Tensor Decompositions via Positive Scattering
Machine Learning
Summary
Figuring out the parts that make up complex data arranged in multiple dimensions, called tensors, is hard because different parts can fit the data equally well. The authors show that knowing all parts must be positive adds useful clues, making it easier to uniquely identify these parts. They introduce a mathematical way to measure this extra information and prove conditions that guarantee the parts are both minimal and unique under positivity. Their work can identify cases where older methods fail, especially when the data parts are sparse or structured.
What this means in practice
- •For signal processing engineers: Confirm uniqueness of positive component decompositions in multi-dimensional signals for improved source separation.
- •For machine learning developers: Verify minimal and unique nonnegative decompositions in model features to improve interpretability of sparse data representations.
A theory result. No direct application yet.
Authors
Haoming Wang, Ming Yuan
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.