Finite mean angular condition number for identifiable tensor decomposition
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 $\Psi$ be a local inverse at $A$ of the addition map $(A_1, \dots, A_r) \mapsto A_1 + \cdots + A_r$ on $r$-tuples of rank-one tensors, and let $p(B) = B/\|B\|_F$. The angular condition number of $A$ is $$\kappa_{\mathrm{ang}}(A) = \bigl\|D(p^{\times r} \circ \Psi)(A)\bigr\|_2,$$ the spectral norm of the derivative, along the locus, of the map to the individually normalised summands, with the Frobenius norm and its product norm. Beltrán, Breiding and Vannieuwenhoven proved $\mathbb{E}\,\kappa_{\mathrm{ang}}(A) < \infty$ for $r = 2$, and that the condition number without the normalisation, $\|D\Psi(A)\|_2$, has infinite mean for $r = 2$.
Problem. Is $\mathbb{E}\,\kappa_{\mathrm{ang}}(A) < \infty$ for every such $d$, $n_1, \dots, n_d$ and every $r \ge 3$?
Source: Beltrán, Breiding and Vannieuwenhoven (2023), Conjecture 2, stated there for every rank; the case r = 2 is their Theorem 3.
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. 8, 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-tr06,
author = {Townsend, Alex},
title = {{TR-06}: Finite mean angular condition number for identifiable tensor decomposition},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-06/README.md},
note = {Read at commit eb7c9d6, 2026-10-07},
}
Resolution claims
The manuscript proves that the angular condition number of the tensor rank decomposition has finite mean for random real tensors of order $d \ge 3$, mode sizes $n_j \ge 2$ and rank $r \ge 3$, in every format where complex tensors of rank $r$ are generically identifiable. The input has density proportional to $e^{-\|A\|^2/2}$ with respect to volume on the …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-08 | Needs verification → Solution claimed | clemens-admin | Listed with the record of its solution, unreviewed. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00208, "Finite mean angular condition number for identifiable tensor decomposition", clopen.solutions.