Subrank and optimal reduction of scalar multiplications to generic tensors

Abstract

The subrank of a tensor measures how much a tensor can be diagonalized. We determine this parameter precisely for essentially all (i.e., generic) tensors. Namely, we prove for generic tensors in $V\otimes V\otimes V$ with $dim\left(V\right)=n$ that the subrank is $\Theta \left(\sqrt{n}\right)$. Our result significantly improves on the previous upper bound from the work of Strassen (1991) and Bürgisser (1990) which was $n^{2/3+o\left(1\right)}$. Our result is tight up to an additive constant. Our full result covers not only 3-tensors but also $k$-tensors, for which we find that the generic subrank is $\Theta \left(n^{1/(k-1)}\right)$. Moreover, as an application, we prove that the subrank is not additive under the direct sum. As a consequence of our result, we obtain several large separations between the subrank and tensor methods that have received much interest recently, notably the slice rank (Tao, 2016), analytic rank (Gowers–Wolf, 2011; Lovett, 2018; Bhrushundi–Harsha–Hatami–Kopparty–Kumar, 2020), geometric rank (Kopparty–Moshkovitz–Zuiddam, 2020), and G-stable rank (Derksen, 2020). Our proofs of the lower bounds rely on a new technical result about an optimal decomposition of tensor space into structured subspaces, which we think may be of independent interest.