Subrank and optimal reduction of scalar multiplications to generic tensors

The subrank of a tensor measures how much a ten-sor can be diagonalized. We determine this parame-ter precisely for essentially all (i.e., generic) tensors.Namely, we prove for generic tensors in 𝑉 ⊗ 𝑉 ⊗ 𝑉 withdim(𝑉) = 𝑛 that the subrank is Θ(√𝑛). Our result sig-nificantly improves on the previous upper bound fromthe work of Strassen (1991) and Bürgisser (1990) whichwas 𝑛 2∕3+𝑜(1). Our result is tight up to an additive con-stant. Our full result covers not only 3-tensors but also𝑘-tensors, for which we find that the generic subrankis Θ(𝑛 1∕(𝑘−1) ). Moreover, as an application, we provethat the subrank is not additive under the direct sum.As a consequence of our result, we obtain several largeseparations between the subrank and tensor methodsthat have received much interest recently, notably theslice 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). Ourproofs of the lower bounds rely on a new technicalresult about an optimal decomposition of tensor spaceinto structured subspaces, which we think may be ofindependent interest