Monotonicity of the average number of critical rank-one approximations
For $p \ge 3$ and $2 \le n_1 \le \cdots \le n_p$, let $G \in \mathbb{R}^{n_1} \otimes \cdots \otimes \mathbb{R}^{n_p}$ have independent standard normal entries, and let $a(n_1, \dots, n_p)$ be the expected number of critical points of $Z \mapsto \|G - Z\|_F^2$ on the manifold of nonzero real rank-one tensors $Z$, each critical tensor counted once (the number is finite almost surely). Draisma and Horobeţ express $a(n_1, \dots, n_p)$ as a Gaussian integral over a space of dimension $1 + \sum_{i < j} (n_i - 1)(n_j - 1)$. They note that the analogous inequality holds with equality for matrices ($p = 2$) and for the generic number of complex critical points, which Friedland and Ottaviani computed.
Problem. Is $$a(n_1, \dots, n_{p-1}, n_p) \le a(n_1, \dots, n_{p-1}, n_p - 1)$$ for every $p \ge 3$ and all $2 \le n_1 \le \cdots \le n_p$ with $n_p - 1 > \sum_{i=1}^{p-1} (n_i - 1)$?
Source: Draisma and Horobeţ (2016), Conjecture 1.3.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- NA — Numerical Analysis
- Secondary
- AG, PR
- Keywords
- critical points, OpenProblemsInNLA, random tensors, rank-one approximations
- Literature checked
- Oct. 6, 2026 by clemens-admin
References
@article{draisma2016,
author = {Draisma, Jan and Horobe{\c{t}}, Emil},
title = {The average number of critical rank-one approximations to a tensor},
journal = {Linear Multilinear Algebra},
volume = {64},
number = {12},
year = {2016},
pages = {2498--2518},
doi = {10.1080/03081087.2016.1164660},
eprint = {1408.3507},
archivePrefix = {arXiv},
}
@article{friedland2014,
author = {Friedland, Shmuel and Ottaviani, Giorgio},
title = {The Number of Singular Vector Tuples and Uniqueness of Best Rank-One Approximation of Tensors},
journal = {Found. Comput. Math.},
volume = {14},
number = {6},
year = {2014},
pages = {1209--1242},
doi = {10.1007/s10208-014-9194-z},
eprint = {1210.8316},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-tr16,
author = {Townsend, Alex},
title = {{TR-16}: Monotonicity of the average number of critical rank-one approximations},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-16/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-00130, "Monotonicity of the average number of critical rank-one approximations", clopen.solutions.