Generic identifiability of tensors at strictly subcritical ranks
Let $d \ge 3$ and $n_1 \ge \cdots \ge n_d \ge 2$, and put $$r_* = \frac{n_1 \cdots n_d}{1 + \sum_{i=1}^{d} (n_i - 1)}, \qquad b = \prod_{i=2}^{d} n_i - \sum_{i=2}^{d} (n_i - 1).$$ A tensor of rank $r$ in $\mathbb{C}^{n_1} \otimes \cdots \otimes \mathbb{C}^{n_d}$ is identifiable if its decomposition as a sum of $r$ rank-one tensors $v_1 \otimes \cdots \otimes v_d$ is unique up to the order of the summands. A generic tensor of rank $r$ is one outside some proper Zariski-closed subset of the Zariski closure of the set of tensors of rank at most $r$. Chiantini, Ottaviani and Vannieuwenhoven (2014) proved the statement below when $n_1 \cdots n_d \le 15000$.
Problem. For every integer $1 \le r < r_*$, is a generic tensor of rank $r$ in $\mathbb{C}^{n_1} \otimes \cdots \otimes \mathbb{C}^{n_d}$ identifiable, except when $n_1 > b$ and $r \ge b$, or when the format $(n_1, \dots, n_d)$ and rank $r$ are $(4, 4, 3)$ and $5$, $(n, n, 2, 2)$ and $2n - 1$ with $n \ge 2$, $(4, 4, 4)$ and $6$, $(6, 6, 3)$ and $8$, or $(2, 2, 2, 2, 2)$ and $5$?
Source: Bocci, Chiantini and Ottaviani (2014) and Chiantini, Ottaviani and Vannieuwenhoven (2014), over ℂ; stated as a conjecture by Chiantini, Ottaviani and Vannieuwenhoven (2017), Conjecture 3.4.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- AG — Algebraic Geometry
- Secondary
- NA
- Keywords
- identifiability, OpenProblemsInNLA, secant varieties, tensor decompositions
- Literature checked
- Oct. 6, 2026 by clemens-admin
References
@article{chiantini2017,
author = {Chiantini, Luca and Ottaviani, Giorgio and Vannieuwenhoven, Nick},
title = {Effective Criteria for Specific Identifiability of Tensors and Forms},
journal = {SIAM J. Matrix Anal. Appl.},
volume = {38},
number = {2},
year = {2017},
pages = {656--681},
doi = {10.1137/16M1090132},
eprint = {1609.00123},
archivePrefix = {arXiv},
}
@article{bocci2014,
author = {Bocci, Cristiano and Chiantini, Luca and Ottaviani, Giorgio},
title = {Refined methods for the identifiability of tensors},
journal = {Ann. Mat. Pura Appl. (4)},
volume = {193},
number = {6},
year = {2014},
pages = {1691--1702},
doi = {10.1007/s10231-013-0352-8},
eprint = {1303.6915},
archivePrefix = {arXiv},
}
@article{chiantini2014,
author = {Chiantini, Luca and Ottaviani, Giorgio and Vannieuwenhoven, Nick},
title = {An Algorithm for Generic and Low-Rank Specific Identifiability of Complex Tensors},
journal = {SIAM J. Matrix Anal. Appl.},
volume = {35},
number = {4},
year = {2014},
pages = {1265--1287},
doi = {10.1137/140961389},
eprint = {1403.4157},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-tr11,
author = {Townsend, Alex},
title = {{TR-11}: Generic identifiability of tensors at strictly subcritical ranks},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-11/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-00128, "Generic identifiability of tensors at strictly subcritical ranks", clopen.solutions.