Rank of tensors with size 2 x ... x 2

TL;DR

This paper investigates the ranks of small hypercubes of tensors, providing characterizations, proofs of known maximal ranks, and evidence for typical ranks, with implications for larger tensor sizes.

Contribution

It characterizes the rank 3 tensors in 2x2x2 case using hyperdeterminants and offers new proofs and evidence regarding maximal and typical ranks of 2x2x2x2 tensors.

Findings

Rank 3 tensors in 2x2x2 are characterized by hyperdeterminants.

Maximal rank of 2x2x2x2 complex tensors is 4.

Evidence suggests 5 is a typical rank for 2x2x2x2 real tensors.

Abstract

We study an upper bound of ranks of $n$-tensors with size $2\times\cdots\times2$ over the complex and real number field. We characterize a $2\times 2\times 2$ tensor with rank 3 by using the Cayley's hyperdeterminant and some function. Then we see another proof of Brylinski's result that the maximal rank of $2\times2\times2\times2$ complex tensors is 4. We state supporting evidence of the claim that 5 is a typical rank of $2\times2\times2\times2$ real tensors. Recall that Kong and Jiang show that the maximal rank of $2\times2\times2\times2$ real tensors is less than or equal to 5. The maximal rank of $2\times2\times2\times2$ complex (resp. real) tensors gives an upper bound of the maximal rank of $2\times\cdots\times 2$ complex (resp. real) tensors.