Real subrank of order-three tensors
Benjamin Biaggi, Jan Draisma, Sarah Eggleston
TL;DR
This work analyzes the real subrank $Q(T)$ of order-three tensors by relating it to the complex subrank $Q_{\,\mathbb C}(T)$, establishing a general lower bound $Q(T)\ge\lfloor\sqrt{Q_{\,\mathbb C}(T)}\rfloor$ and, in the maximal complex subrank case, a stronger $Q(T)\ge n/2$ for $T\in{\mathbb R^n}^{\otimes3}$. It proves that typical subranks are consecutive and develops a geometric method to bound subrank from above via intersections with the Segre variety, enabling explicit results for formats like $3\times3\times5$ and $4\times4\times4$. The paper catalogs typical subranks for several small order-three formats, showing, for example, that $2\times2\times2$ has typical subranks $1,2$, while $3\times3\times5$ has typical subranks $2,3$, and demonstrates that the quaternion multiplication tensor in $\mathbb{R}^{4\otimes3}$ has subrank $2$, with $2$ being typical in a neighborhood. Finally, it analyzes subranks of direct sums of real division algebras, proving additive behavior $Q(f_n)=nQ(f_1)$ and providing tight bounds $Q(f_n)\le nd$ with $d=\dim_\mathbb{R}D/2$, exact for $D=\mathbb C$ or $\mathbb H$ and conjectural for the octonions.
Abstract
We study the subrank of real order-three tensors and give an upper bound to the subrank of a real tensor given its complex subrank. Using similar arguments to those used by Bernardi-Blekherman-Ottaviani, we show that all subranks between the minimal typical subrank and the maximal typical subrank, which equals the generic subrank, are also typical. We then study small tensor formats with more than one typical subrank. In particular, we construct a $3 \times 3 \times 5$-tensor with subrank $2$ and show that the subrank of the $4 \times 4 \times 4$-quaternion multiplication tensor is $2$. Finally, we consider the tensor associated to componentwise complex multiplication in $\mathbb{C}^n$ and show that this tensor has real subrank $n$ - informally, no more than $n$ real scalar multiplications can be carried out using a device that does $n$ complex scalar multiplications. We also prove a version of this result for other real division algebras.