High-rank subtensors of high-rank tensors

2022/07/16 by Thomas Karam, Karam, Thomas · 1 citation
Engineering · Mathematics · #11B30 #15A69 #68R05 #Combinatorics (math.CO) #Elasticity and Material Modeling #FOS: Mathematics #Sparse and Compressive Sensing Techniques #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2207.08030

openalex publication_date 2022/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let d ≥ 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 X₁ × … × X_d such that the restriction has R-rank at least l and the sets X₁, …, 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 X₁, …, X_d to be pairwise disjoint.

Cited by

• Algebraic aspects of the polynomial Littlewood-Offord problem

Related

• On subtensors of high partition rank
• The interplay between bounded ranks of tensors arising from partitions
• Tensor rank and dimension expanders
• A Tensor Rank Theory and Maximum Full Rank Subtensors
• An Improvement on Ranks of Explicit Tensors