Geometric rank of tensors and subrank of matrix multiplication

Geometric rank of tensors and subrank of matrix multiplication, Discrete Analysis 2023:1, 25 pp. The rank of a matrix is a parameter of obvious importance, so it is natural to wonder what the right definition is for the rank of a higher-dimensional matrix -- that is, of a tensor (which can be thought of as a $d$-dimensional array of elements of a field $\mathbb F$). This question turns out not to have a straightforward answer: there are many different natural definitions for the rank of a tensor that specialize to matrix rank when the tensor has dimension 2. However, it also turns out that several of these definitions are useful and have important applications. One approach to defining a notion of rank is first to decide what the rank-1 tensors should be and then to define the rank of a tensor $T$ to be the smallest number of rank-1 tensors needed to generate $T$ as a linear combination. If $d=3$ and we define a rank-1 tensor to be one of the form $T(x,y,z)=a(x)b(y)c(z)$, then we obtain the notion of _tensor rank_. If instead we define a rank-1 tensor to be one of the form $a(x)b(y,z)$, $a(y)b(x,z)$ or $a(z)b(x,y)$, then we obtain the notion of _slice rank_, which was introduced by Tao and played a key role in his reformulation of the famous solution to the cap-set problem. A different approach, which works when $\mathbb F$ is the finite field $\mathbb F_p$ is to observe (following a straightforward calculation) that if $M$ is a matrix of rank $r$, then $s(M)=\mathbb E_{x,y}\exp(2\pi iM(x,y)/p)=p^{-r}$. If we wanted, we could therefore define the rank of $M$ to be $\log_p(1/s(M))$. This has an obvious generalization to higher-order tensors. For instance, if $d=3$ then we define $s(T)$ to be $\mathbb E_{x,y,z}\exp(2\pi iT(x,y,z)/p)$ and we can consider the quantity $\log_p(1/s(T))$ as a kind of rank of $T$. This notion was introduced by Gowers and Wolf, who called it the _analytic rank_ of the tensor. However, in contrast with tensor rank and slice rank, the definition of the analytic rank depends on the field being finite, and it is not obvious what an analogous notion should be for fields of characteristic zero. This question was explicitly asked by Lovett, and is answered in this paper, where the authors introduce yet another notion of rank, which they call _geometric rank_. For a 3-tensor $T$, the geometric rank is defined as follows. Let $T$ have coefficients $T(i,j,k)$ for $(i,j,k)\in [n_1]\times [n_2]\times [n_3]$, where $[n]$ denotes the set $\{1,2,