Advances in Combinatorics· 2026Q1
High-rank subtensors of high-rank tensors
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- Q1SCImago
- 2026year
Short summary
Researchers proved that any high-rank tensor can be restricted to a smaller sub-tensor (defined by product of sets $X_1 \times \dots \times X_d$) that retains high rank, with the size of these sets bounded by functions of the target rank.
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Key points
- Introduces functions $F_{d,R}$ and $G_{d,R}$ for bounding the size of subtensors preserving rank.
- Demonstrates that an order-$d$ tensor with $R$-rank $\ge G_{d,R}(l)$ can be restricted to a subtensor with $R$-rank $\ge l$.
- The sets defining the restriction ($X_1, \dots, X_d$) have sizes bounded by $F_{d,R}(l)$.
- The proof methods allow for pairwise disjoint sets $X_i$ under a natural condition.
AI-generated from the title and abstract; the full text is not read.
Abstract
Let $d \ge 2$ be a positive integer. We show that for a class of notions $R$ of rank for order-$d$ tensors, which includes in particular the tensor rank, the slice rank and the partition rank, there exist functions $F_{d,R}$ and $G_{d,R}$ such that if an order-$d$ tensor has $R$-rank at least $G_{d,R}(l)$ then we can restrict its entries to a product of sets $X_1 \times \dots \times X_d$ such that the restriction has $R$-rank at least $l$ and the sets $X_1, \dots, X_d$ each have size at most $F_{d,R}(l)$. Furthermore, our proof methods allow us to show that under a very natural condition we can require the sets $X_1, \dots, X_d$ to be pairwise disjoint.
The authors' abstract, as published at the source. Advances in Combinatorics, 2026 · DOI ↗
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Field: Computational Mathematics
Computational MathematicsMathematics