Tensor rank and dimension expanders
Zeev Dvir
TL;DR
This work introduces a dimension-spreading framework that converts small, structured families of linear maps into explicit high-rank tensors. It proves a lower bound $R(T^{\mathcal{A}})\ge n+t-s$ for $(s,t)$-dimension-spreading sets of $n\times n$ matrices and extends the result to border rank over $\mathbb{R}$ or $\mathbb{C}$. By leveraging dimension expanders, the authors show how to construct explicit $(D,n,n)$-tensors with rank near $2n$ (specifically $(2-\epsilon)n$ for suitable $\epsilon$) using constant-size, field-independent expansions. They provide explicit paths from dimension expanders (notably the BY13 monotone-expander construction) to dimension-spreading maps, yielding poly$(1/\epsilon)$-sized spreading families and corresponding high-rank tensors, with broad implications for explicit tensor rank lower bounds and related complexity questions.
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 $(2 - ε)n$, with $D$ a constant depending on $ε$. Our results extend to border rank over the real or complex numbers.