Identifiability of Nonnegative Tensor Decompositions via Positive Scattering
Updated 4 h ago · first seen 11 Sept 2026
paper_01M294FR47RNXTZZ6ASKXJWNE4
- Published
- 11 Sept 2026
- T1 · 4 h ago
- arXiv
- 2609.11606
- T1 · 4 h ago
- Category
- stat.ML
- T1 · 4 h ago
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.
Authors 2
Haoming Wang, Ming Yuan
Specification
- Official page
Source:arXiv (Atom API + RSS)T1observed 4 h agohigh
- Arxiv announce type
- cross
Source:arXiv (Atom API + RSS)T1observed 4 h agohigh
- arXiv id
- 2609.11606
Source:arXiv (Atom API + RSS)T1observed 4 h agohigh
- Categories
- stat.ML, cs.LG, math.CO, math.ST, stat.TH
Source:arXiv (Atom API + RSS)T1observed 4 h agohigh
Source:arXiv (Atom API + RSS)T1observed 4 h agohigh
- Primary category
- stat.ML
Source:arXiv (Atom API + RSS)T1observed 4 h agohigh
- Published
- 11 Sept 2026
Source:arXiv (Atom API + RSS)T1observed 4 h agohigh
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4 h ago
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- Authors
- Haoming Wang, Ming Yuan
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Claim history
Official pageofficial_url1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| https://arxiv.org/abs/2609.11606 | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Abstractabstract1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| 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. | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Arxiv announce typearxiv_announce_type1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| cross | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
arXiv idarxiv_id1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| 2609.11606 | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Categoriescategories1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| stat.ML, cs.LG, math.CO, math.ST, stat.TH | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
PDFpdf_url1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| https://arxiv.org/pdf/2609.11606 | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Primary categoryprimary_category1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| stat.ML | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Publishedpublished_at1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| 11 Sept 2026 | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
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New paper: Identifiability of Nonnegative Tensor Decompositions via Positive Scattering
arxiv
| Source | Document | Type | Tier | Last observed | Snapshots |
|---|---|---|---|---|---|
| arXiv (Atom API + RSS) | rss.arxiv.org/rss/cs.LG | feed | T1· Official | 4 h ago | 1 |
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