Tensor rank and dimension expanders

TL;DR

This paper establishes a lower bound on the rank of tensors derived from dimension expanders, providing explicit constructions with high tensor rank and extending results to border rank over real and complex fields.

Contribution

It introduces a lower bound on tensor rank from dimension expanders and constructs explicit high-rank tensors based on these expanders.

Findings

Constructed explicit tensors with rank at least (2 - epsilon) times n.

Extended tensor rank results to border rank over real and complex numbers.

Provided explicit tensor constructions using dimension expanders.

Abstract

We prove a lower bound on the rank of tensors constructed from families of linear maps that `expand' the dimension of every subspace. Such families, called {\em dimension expanders} have been studied for many years with several known explicit constructions. Using these constructions we show that one can construct an explicit $[D]\times [n] \times [n]$-tensor with rank at least $(2 - \epsilon)n$, with $D$ a constant depending on $\epsilon$. Our results extend to border rank over the real or complex numbers.