Classification of perfect tensor formats with generic unique decomposition
Let $d \ge 3$ and $n_1, \dots, n_d \ge 2$. The format $(n_1, \dots, n_d)$ is perfect if $$r_* = \frac{n_1 \cdots n_d}{1 + \sum_{i=1}^{d} (n_i - 1)}$$ is an integer. Generic means on a nonempty Zariski-open subset of $\mathbb{C}^{n_1} \otimes \cdots \otimes \mathbb{C}^{n_d}$, and two decompositions into rank-one tensors $v_1 \otimes \cdots \otimes v_d$ are the same if their summands agree up to order. A generic tensor of format $(2, k, k)$ has a unique decomposition of minimal length, by the Weierstrass–Kronecker canonical form of matrix pencils, and Hauenstein, Oeding, Ottaviani and Sommese proved the same for $(3, 4, 5)$ and $(2, 2, 2, 3)$, with $6$ and $4$ summands. When $n_1 \le \cdots \le n_d$ and $n_d \ge \prod_{i < d} n_i - \sum_{i < d} (n_i - 1)$, known results, which they collect, show that among perfect formats only $(2, k, k)$ has this property.
Problem. Is it true that a generic tensor of a perfect format $(n_1, \dots, n_d)$ has a unique decomposition of minimal length as a sum of rank-one tensors if and only if, up to permuting the factors, the format is $(2, k, k)$ with $k \ge 2$, $(3, 4, 5)$ or $(2, 2, 2, 3)$?
Source: Hauenstein, Oeding, Ottaviani and Sommese (2019), Conjecture 1.3.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- AG — Algebraic Geometry
- Secondary
- NA
- Keywords
- generic ranks, identifiability, OpenProblemsInNLA, tensor decompositions
- Literature checked
- Oct. 6, 2026 by clemens-admin
References
@article{hauenstein2019,
author = {Hauenstein, Jonathan D. and Oeding, Luke and Ottaviani, Giorgio and Sommese, Andrew J.},
title = {Homotopy techniques for tensor decomposition and perfect identifiability},
journal = {J. Reine Angew. Math.},
volume = {753},
year = {2019},
pages = {1--22},
doi = {10.1515/crelle-2016-0067},
eprint = {1501.00090},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-tr12,
author = {Townsend, Alex},
title = {{TR-12}: Classification of perfect tensor formats with generic unique decomposition},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-12/README.md},
note = {Read at commit e375a6f, 2026-09-30},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00129, "Classification of perfect tensor formats with generic unique decomposition", clopen.solutions.