Frobenius inner products minimize the algebraic complexity of rank-one approximation
Fix integers $k \ge 1$, $n_1, \dots, n_k \ge 2$ and $d_1, \dots, d_k \ge 1$ with $d_1 + \cdots + d_k \ge 3$. Let $$W_{\mathbb{R}} = \bigotimes_{j=1}^{k} \operatorname{Sym}^{d_j}(\mathbb{R}^{n_j}),$$ let $W_{\mathbb{C}}$ be its complexification, and let $$X = \{c\, v_1^{\otimes d_1} \otimes \cdots \otimes v_k^{\otimes d_k} : c \in \mathbb{C},\ v_j \in \mathbb{C}^{n_j}\} \subset W_{\mathbb{C}}$$ be the affine cone over the Segre–Veronese variety. A positive definite symmetric bilinear form $Q$ on $W_{\mathbb{R}}$ is extended complex-bilinearly (without complex conjugation) to $W_{\mathbb{C}}$, and its Euclidean distance degree $\operatorname{ED}_Q(X)$ is the number of critical points of $Z \mapsto Q(A - Z, A - Z)$ on the smooth nonzero points of $X$, for Zariski-generic $A \in W_{\mathbb{C}}$, counted with algebraic multiplicity. Let $Q_F$ be the Frobenius inner product: the restriction to $W_{\mathbb{R}}$ of the entrywise inner product on $\bigotimes_j (\mathbb{R}^{n_j})^{\otimes d_j}$, so that the entries of symmetric tensors are weighted by their repetitions. Kozhasov, Muniz, Qi and Sodomaco proved $\operatorname{ED}_Q(X) \ge \operatorname{ED}_{Q_F}(X)$ for every $Q$ close enough to $Q_F$, and for every $Q$ when $k = 1$ and $n_1 = 2$ (binary forms) and when $k = 1$ and $n_1 = d_1 = 3$ (ternary cubics). They also proved it for matrices and for symmetric matrices, the formats with $d_1 + \cdots + d_k = 2$.
Problem. Is $\operatorname{ED}_Q(X) \ge \operatorname{ED}_{Q_F}(X)$ for every such $k$, $n_1, \dots, n_k$ and $d_1, \dots, d_k$ and every positive definite symmetric bilinear form $Q$ on $W_{\mathbb{R}}$?
Source: Kozhasov, Muniz, Qi and Sodomaco (2027), Conjecture 3.9 (first posted as arXiv:2309.15105, 2023); OpenProblemsInNLA states the formats with d₁ + ⋯ + dₖ ≥ 3, the order-two formats being their Theorems 4.1 and 4.2.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- AG — Algebraic Geometry
- Secondary
- NA, OC
- Keywords
- Euclidean distance degrees, OpenProblemsInNLA, rank-one approximations, Segre–Veronese varieties
- Literature checked
- Oct. 8, 2026 by clemens-admin
References
@article{kozhasov2027,
author = {Kozhasov, Khazhgali and Muniz, Alan and Qi, Yang and Sodomaco, Luca},
title = {On the minimal algebraic complexity of the rank-one approximation problem for general inner products},
journal = {Math. Comp.},
volume = {96},
number = {363},
year = {2027},
pages = {335--373},
doi = {10.1090/mcom/4176},
eprint = {2309.15105},
archivePrefix = {arXiv},
}
@article{draisma2016,
author = {Draisma, Jan and Horobe{\c{t}}, Emil and Ottaviani, Giorgio and Sturmfels, Bernd and Thomas, Rekha R.},
title = {The {E}uclidean Distance Degree of an Algebraic Variety},
journal = {Found. Comput. Math.},
volume = {16},
number = {1},
year = {2016},
pages = {99--149},
doi = {10.1007/s10208-014-9240-x},
eprint = {1309.0049},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-tr17,
author = {Townsend, Alex},
title = {{TR-17}: Frobenius inner products minimize the algebraic complexity of rank-one approximation},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-17/README.md},
note = {Read at commit eb7c9d6, 2026-10-07},
}
Resolution claims
For every Segre–Veronese cone and every positive definite real inner product on its ambient partially symmetric tensor space, its Euclidean distance degree is at least the degree for the Frobenius inner product. The argument uses a basis of Chern–Schwartz–MacPherson classes indexed by product-linear sections. Every corresponding signed Euler characteristic is at least one: a positive multihomogeneous form has a coercive …
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-00212, "Frobenius inner products minimize the algebraic complexity of rank-one approximation", clopen.solutions.