Infinite mean condition number in every identifiable tensor format
Let $d \ge 3$, $n_1, \dots, n_d \ge 2$ and $r \ge 2$ be such that complex tensors of rank $r$ in $\mathbb{C}^{n_1} \otimes \cdots \otimes \mathbb{C}^{n_d}$ are generically identifiable. Let $A$ be a random real tensor of rank $r$ in $\mathbb{R}^{n_1} \otimes \cdots \otimes \mathbb{R}^{n_d}$ with density proportional to $e^{-\|A\|_F^2/2}$ with respect to the Euclidean volume on the smooth locus of identifiable real tensors of rank $r$ (tensors are sampled by volume, not as sums of independent Gaussian rank-one terms). Let $\kappa(A)$ be the condition number of its tensor rank decomposition: the spectral norm of the derivative at $A$ of a local inverse of the addition map $(A_1, \dots, A_r) \mapsto A_1 + \cdots + A_r$ on $r$-tuples of rank-one tensors, with the Frobenius norm and its product norm. Beltrán, Breiding and Vannieuwenhoven proved $\mathbb{E}\,\kappa(A) = \infty$ for $r = 2$, and for $r \ge 3$ when $n_1, \dots, n_d \ge 3$ and complex tensors of rank $r - 2$ in $\mathbb{C}^{n_1 - 2} \otimes \cdots \otimes \mathbb{C}^{n_d - 2}$ are also generically identifiable.
Problem. Is $\mathbb{E}\,\kappa(A) = \infty$ for every such $d$, $n_1, \dots, n_d$ and $r$?
Source: Beltrán, Breiding and Vannieuwenhoven (2023), Conjecture 1.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- NA — Numerical Analysis
- Secondary
- AG, PR
- Keywords
- condition numbers, OpenProblemsInNLA, random tensors, tensor decompositions
- Literature checked
- Oct. 6, 2026 by clemens-admin
References
@article{beltran2023,
author = {Beltr{\'a}n, Carlos and Breiding, Paul and Vannieuwenhoven, Nick},
title = {The Average Condition Number of Most Tensor Rank Decomposition Problems is Infinite},
journal = {Found. Comput. Math.},
volume = {23},
number = {2},
year = {2023},
pages = {433--491},
doi = {10.1007/s10208-022-09551-1},
eprint = {1903.05527},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-tr05,
author = {Townsend, Alex},
title = {{TR-05}: Infinite mean condition number in every identifiable tensor format},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-05/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 It will not change if the problem is solved, reclassified or withdrawn.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-00125, "Infinite mean condition number in every identifiable tensor format", clopen.solutions.